Anti-Self-Dual 4-Manifolds, Quasi-Fuchsian Groups, & Almost-K …claude/auckland.pdf · Anti-Self-Dual 4-Manifolds, Quasi-Fuchsian Groups, & Almost-K ahler Geometry Claude LeBrun

Anti-Self-Dual 4-Manifolds,

Quasi-Fuchsian Groups, &

Almost-Kahler Geometry

Claude LeBrunStony Brook University

Conformal and Symplectic GeometryUniversity of Auckland, 7 February, 2018

1

Most recent results joint with

2


Christopher J. BishopStony Brook University

3



e-print: arXiv:1708.03824 [math.DG]

4



e-print: arXiv:1708.03824 [math.DG]

To appear in Comm. An. Geom.

5

Themes of this conference:

Conformal Geometry


Symplectic GeometryJJJJJJJJJJJJJJJJJJJJ]

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-�

6

Theme of this talk:

Conformal Geometry


Symplectic Geometry

7

Theme of this talk:

Conformal Geometry


Symplectic GeometryJJJJJJJJJJ JJJJJJJJJJ]

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-�

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Theme of this talk:

Conformal Geometry



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Talk concerns phenomena specific to dimension 4.

9

Theme of this talk:

Conformal Geometry



�

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-�

Talk concerns phenomena specific to dimension 4.

Higher dimensions are demonstrably different.

10

Let (M2m, ω) be a compact symplectic manifold.

11


12


Thus, ω is a 2-form with dω = 0 and ω∧m 6= 0.

13



14



By convention, orient M so that ω∧m > 0.

15




⇒ ∃ compatible almost-complex structures J :

16





J : TM → TM, J2 = −1,

17





J : TM → TM, J2 = −1,

J∗ω = ω, ω(v, Jv) > 0 ∀v 6= 0.

18





J : TM → TM, J2 = −1,

J∗ω = ω, ω(v, Jv) > 0 ∀v 6= 0.

19





J : TM → TM, J2 = −1,

J∗ω = ω, ω(v, Jv) > 0 ∀v 6= 0.

Leads to theory of J -holomorphic curves,

20





J : TM → TM, J2 = −1,

J∗ω = ω, ω(v, Jv) > 0 ∀v 6= 0.

Leads to theory of J -holomorphic curves,Gromov-Witten invariants, etc.

21





J : TM → TM, J2 = −1,

J∗ω = ω, ω(v, Jv) > 0 ∀v 6= 0.

Leads to theory of J -holomorphic curves,Gromov-Witten invariants, etc.

Imitates Kahler geometry in a non-Kahler setting.

22





J : TM → TM, J2 = −1,

J∗ω = ω, ω(v, Jv) > 0 ∀v 6= 0.

23





J : TM → TM, J2 = −1,

J∗ω = ω, ω(v, Jv) > 0 ∀v 6= 0.

=⇒ g := ω(·, J ·) is a Riemannian metric.

24





J : TM → TM, J2 = −1,

J∗ω = ω, ω(v, Jv) > 0 ∀v 6= 0.


Such g are called almost-Kahler metrics, because

25





J : TM → TM, J2 = −1,

J∗ω = ω, ω(v, Jv) > 0 ∀v 6= 0.


Such g are called almost-Kahler metrics, because

Kahler ⇐⇒ J integrable.

26

Discussion involves three intertwined structures:

27


J

g

ωJJJJJJJJJJJJJJJJJJJJ]

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-�

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J

g


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-�

Any two algebraically determine the third.

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J

g


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-�

Any two algebraically determine the third.

For example, can avoid explicitly mentioning J .

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J

g


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-�

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g

ω

��

32


g

ω

��

Lemma.An oriented Riemannian manifold (M2m, g)is almost-Kahler w/ respect to the 2-form ω ⇐⇒

33


g

ω

��


34


g

ω

��


35


g

ω

��


36


g

ω

��

Lemma.An oriented Riemannian manifold (M2m, g)is almost-Kahler w/ respect to the 2-form ω ⇐⇒• |ω|g ≡

√m,

37


g

ω

��


√m,

• dω = 0, and

38


g

ω

��


√m,

• dω = 0, and

• ∗ω =ω∧(m−1)

(m− 1)!.

39


g

ω

��


√m,

• ω is a harmonic 2-form, and

• ∗ω =ω∧(m−1)

(m− 1)!.

40

Simplifies dramatically when m = 2:

g

ω

��


√m,


• ∗ω =ω∧(m−1)

(m− 1)!.

41


g

ω

��

Lemma.An oriented Riemannian 4-manifold (M, g)is almost-Kahler w/ respect to the 2-form ω ⇐⇒• |ω|g ≡

√2,


• ∗ω = ω.

42

On oriented (M4, g),

Λ2 = Λ+ ⊕ Λ−

43


Λ2 = Λ+ ⊕ Λ−

Λ+ self-dual 2-forms:(+1)-eigenspace of ∗ : Λ2→ Λ2.

44


Λ2 = Λ+ ⊕ Λ−


Λ− anti-self-dual 2-forms:(−1)-eigenspace of ∗ : Λ2→ Λ2.

45


g

ω

��


√2,


• ∗ω = ω.

46


g

ω

��


√2,

• ω is a self-dual harmonic 2-form.

47


Λ2 = Λ+ ⊕ Λ−



48


Λ2 = Λ+ ⊕ Λ−



But Hodge star

∗ : Λ2→ Λ2

is conformally invariant on middle-dimensional forms:

49


Λ2 = Λ+ ⊕ Λ−



But Hodge star

∗ : Λ2→ Λ2

is conformally invariant on middle-dimensional forms:

Only depends on the conformal class

[g] := {u2g | u : M → R+}.

50

On oriented (M4, [g]),

Λ2 = Λ+ ⊕ Λ−



Only depends on the conformal class

[g] := {u2g | u : M → R+}.

51


g

ω

��


√2,

• ω is a self-dual harmonic 2-form.

52


[g]

ω

��

Proposition.A conformal class [g] on a smoothcompact oriented 4-manifold M is represented byan almost-Kahler metric g iff it carries a self-dual harmonic 2-form ω that is 6= 0 everywhere.

53


[g]

ω

��


Moreover, the set of conformal classes [g] on Mthat carry such a harmonic form ω is open inthe C2 topology.

54

Hodge theory:

H2(M,R) = {ϕ ∈ Γ(Λ2) | dϕ = 0, d ∗ ϕ = 0}.

55

Hodge theory:

H2(M,R) = {ϕ ∈ Γ(Λ2) | dϕ = 0, d ∗ ϕ = 0}.Since ∗ is involution of RHS, =⇒

H2(M,R) = H+g ⊕H−g ,

56

Hodge theory:


H2(M,R) = H+g ⊕H−g ,

where

H±g = {ϕ ∈ Γ(Λ±) | dϕ = 0}self-dual & anti-self-dual harmonic forms.

57

Hodge theory:


H2(M,R) = H+g ⊕H−g ,

where


One can choose a basis for H±g that depends

continuously on g in the C1,α topology.

58


[g]

ω

��



59

Simplifies dramatically when m = 2:“Conformal classes of symplectic type”



60

Hodge theory:


H2(M,R) = H+g ⊕H−g ,

where




61

Hodge theory:


H2(M,R) = H+g ⊕H−g ,

where




In particular, the numbers

b±(M) = dimH±gare independent of g, and so are invariants of M .

62

b±(M)?

63

Best understood in terms of intersection pairing

64


H2(M,R)×H2(M,R) −→ R

( [ϕ] , [ψ] ) 7−→∫Mϕ ∧ ψ

65


H2(M,R)×H2(M,R) −→ R

( [ϕ] , [ψ] ) 7−→∫Mϕ ∧ ψ

Diagonalize:

+1. . .

+1︸︷︷︸b+(M)

b−(M)

−1. . .

−1

.

66


H2(M,R)×H2(M,R) −→ R

( [ϕ] , [ψ] ) 7−→∫Mϕ ∧ ψ

Diagonalize:

+1. . .

+1︸︷︷︸b+(M)

b−(M)

−1. . .

−1

.

67


H2(M,R)×H2(M,R) −→ R

( [ϕ] , [ψ] ) 7−→∫Mϕ ∧ ψ

Diagonalize:

+1. . .

+1︸︷︷︸b+(M)

b−(M)

−1. . .

−1

.

b2(M) = b+(M) + b−(M)

68


H2(M,R)×H2(M,R) −→ R

( [ϕ] , [ψ] ) 7−→∫Mϕ ∧ ψ

Diagonalize:

+1. . .

+1︸︷︷︸b+(M)

b−(M)

−1. . .

−1

.

τ (M) = b+(M)− b−(M)

“Signature” of M .

69

Signature defined in terms of intersection pairing

H2(M,R)×H2(M,R) −→ R

( [ϕ] , [ψ] ) 7−→∫Mϕ ∧ ψ

Diagonalize:

+1. . .

+1︸︷︷︸b+(M)

b−(M)

−1. . .

−1

.

τ (M) = b+(M)− b−(M)

Signature of M .

70

Signature defined in terms of intersection pairing

H2(M,R)×H2(M,R) −→ R

( [ϕ] , [ψ] ) 7−→∫Mϕ ∧ ψ

τ (M) = b+(M)− b−(M)

71

Signature defined in terms of intersection pairing,

but also expressible as a curvature integral:

τ (M) =1

12π2

∫M

(|W+|2 − |W−|2

)dµ

= 〈p1(M), [M ]〉

72



τ (M) =1

12π2

∫M

(|W+|2 − |W−|2

)dµ

= 〈13p1(M), [M ]〉

(Thom-Hirzebruch Signature Formula)

73



τ (M) =1

12π2

∫M

(|W+|2 − |W−|2

)dµ

= 〈p1(M), [M ]〉

Has major consequences in conformal geometry.

74


Λ2 = Λ+ ⊕ Λ−

75

Riemann curvature of g

R : Λ2→ Λ2

76


R : Λ2→ Λ2

splits into 4 irreducible pieces:

77


R : Λ2→ Λ2


R =

W+ + s

12 r

r W− + s12

78


R : Λ2→ Λ2


Λ+∗ Λ−∗

Λ+ W+ + s12 r

Λ− r W− + s12

79


R : Λ2→ Λ2


Λ+∗ Λ−∗

Λ+ W+ + s12 r

Λ− r W− + s12

where

s = scalar curvature

r = trace-free Ricci curvature

W+ = self-dual Weyl curvature (conformally invariant)

W− = anti-self-dual Weyl curvature

80


R : Λ2→ Λ2


Λ+∗ Λ−∗

Λ+ W+ + s12 r

Λ− r W− + s12

where

s = scalar curvature

r = trace-free Ricci curvature

W+ = self-dual Weyl curvature (conformally invariant)

W− = anti-self-dual Weyl curvature ′′

81

For M4 compact, the Weyl functional

82


W ([g]) =

∫M

(|W+|2 + |W−|2

)dµg

83


W ([g]) =

∫M

(|W+|2 + |W−|2

)dµg

measures the deviation from conformal flatness,

84


W ([g]) =

∫M

(|W+|2 + |W−|2

)dµg

measures the deviation from conformal flatness,because (M4, g) is locally conformally flat ⇐⇒its Weyl curvature W = W+ + W− vanishes.

85


W ([g]) =

∫M

(|W+|2 + |W−|2

)dµg


86


W ([g]) =

∫M

(|W+|2 + |W−|2

)dµg


87


W ([g]) =

∫M

(|W+|2 + |W−|2

)dµg


Basic problems: For given smooth compact M4,

88


W ([g]) =

∫M

(|W+|2 + |W−|2

)dµg



•What is inf W ?

89


W ([g]) =

∫M

(|W+|2 + |W−|2

)dµg



•What is inf W ?

•Do there exist minimizers?

90


W ([g]) =

∫M

(|W+|2 + |W−|2

)dµg


But we’ve already noted that

12π2τ (M) =

∫M

(|W+|2 − |W−|2

)dµg

is a topological invariant.

91


W ([g]) =

∫M

(|W+|2 + |W−|2

)dµg



12π2τ (M) =

∫M

(|W+|2 − |W−|2

)dµg


So Weyl functional is essentially equivalent to

[g] 7−→∫M|W+|2dµg

92


W ([g]) =

∫M

(|W+|2 + |W−|2

)dµg



12π2τ (M) =

∫M

(|W+|2 − |W−|2

)dµg


In particular, metrics with W+ ≡ 0 minimize W .

93


W ([g]) =

∫M

(|W+|2 + |W−|2

)dµg



12π2τ (M) =

∫M

(|W+|2 − |W−|2

)dµg



If g has W+ ≡ 0, it is said to be anti-self-dual.

94


W ([g]) =

∫M

(|W+|2 + |W−|2

)dµg



12π2τ (M) =

∫M

(|W+|2 − |W−|2

)dµg



If g has W+ ≡ 0, it is said to be anti-self-dual.(ASD)

95

Twistor picture of anti-self-duality condition:

96


Oriented (M4, g)! (Z, J).

97



Z = S(Λ+), J : TZ → TZ, J2 = −1:

98



Z = S(Λ+), J : TZ → TZ, J2 = −1:

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Z = S(Λ+), J : TZ → TZ, J2 = −1:

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Theorem (Atiyah-Hitchin-Singer). (Z, J) is a com-plex 3-manifold iff W+ = 0.

100



Z = S(Λ+), J : TZ → TZ, J2 = −1:

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rr S(Λ+)

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Reconceptualizes earlier work by Penrose.

101



Z = S(Λ+), J : TZ → TZ, J2 = −1:

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rr S(Λ+)

M 4

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Z = S(Λ+), J : TZ → TZ, J2 = −1:

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rr S(Λ+)

M 4

↓

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Motivates study of ASD metrics,and yields methods for constructing them.

103

So ASD metrics are linked to complex geometry. . .

104

A different link with complex geometry:

105


If (M4, g, J) is a Kahler surface, then [g] is ASD

106


If (M4, g, J) is a Kahler surface, then [g] is ASD⇐⇒ the scalar curvature s of g is identically zero.

107



Scalar-flat Kahler surfaces:

108



Scalar-flat Kahler surfaces:special case of cscK manifolds,

109



Scalar-flat Kahler surfaces:special case of cscK manifolds,and so of extremal Kahler manifolds.

110



Scalar-flat Kahler surfaces:special case of cscK manifolds,and so of extremal Kahler manifolds.

Results proved about SFK in ’90s foreshadowedmany more recent results about general case.

111




112




Classification up to diffeomorphism: compact case

113




Classification up to diffeomorphism: (compact case)

114




Classification up to diffeomorphism: (compact case)

115




Classification up to diffeomorphism:

•Ricci-flat case (ignore from now on)

116






•Non-Ricci-flat case

117






–

–

–


118






–K3

–

–


119






–K3

– T 4

–


120






–K3

– T 4

– eight specific finite quotients of these


121






–K3

– T 4

– eight specific finite quotients of these


122







123







–

–

–

–

124







–CP2#kCP2, k ≥ 10

–

–

–

125

Convention:

CP2 = reverse oriented CP2.

126

Convention:


Connected sum #:

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127

Convention:


Connected sum #:

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128

Convention:


Connected sum #:

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Convention:


Connected sum #:

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Convention:


Connected sum #:

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Convention:


Connected sum #:

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Convention:


Connected sum #:

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Convention:


Connected sum #:

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Convention:


Connected sum #:

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Blowing Up: M M#CP2

135

Convention:


Connected sum #:

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Convention:


Connected sum #:

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Blowing Up: M M#CP2

137

Convention:


Connected sum #:

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138







–CP2#kCP2, k ≥ 10

–

–

–

139







–CP2#kCP2, k ≥ 10

– (T 2 × S2)#kCP2, k ≥ 1

–

–

140







–CP2#kCP2, k ≥ 10

– (T 2 × S2)#kCP2, k ≥ 1

– Σ× S2

–

141







–CP2#kCP2, k ≥ 10

– (T 2 × S2)#kCP2, k ≥ 1

– Σ× S2 and Σ×S2,

–

142







–CP2#kCP2, k ≥ 10

– (T 2 × S2)#kCP2, k ≥ 1

– Σ× S2 and Σ×S2, genus Σ ≥ 2

–

143







–CP2#kCP2, k ≥ 10

– (T 2 × S2)#kCP2, k ≥ 1

– Σ× S2 and Σ×S2, genus Σ ≥ 2

– (Σ× S2)#kCP2, k ≥ 1

144

Notice that the 4-manifolds CP2#kCP2

145

Notice that the 4-manifolds CP2#kCP2do not admit scalar-flat Kahler metrics when k ≤ 9.

146


Plausible conjecture:

147


Plausible conjecture:these manifolds don’t admit any ASD metrics.

148



Stronger conjecture:

149



Stronger conjecture:any metric on one of these manifolds M satisfies∫

M|W+|2dµ ≥

4π2

3(9− k)

150




M|W+|2dµ ≥

4π2

3(2χ + 3τ )(M)

151




M|W+|2dµ ≥

4π2

3(2χ + 3τ )(M)

Theorem (Gursky ’98).True for conformal classesof positive Yamabe constant.

152




M|W+|2dµ ≥

4π2

3(2χ + 3τ )(M)

Theorem (Gursky ’98).True for conformal classesof positive Yamabe constant.

Theorem (L ’15). True for conformal classes ofsymplectic type.

153

Last result indicates that almost-Kahler conditiongives extra control on ASD conformal geometry.

154


Inyoung Kim ’16: classification of almost-KahlerASD roughly the same as in scalar-flat Kahler case.

155



Does this say anything about general ASD metrics?

156




Almost-Kahler ASD metrics sweep out an open setin the ASD moduli space.

157

Example.

Σ

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158

Example.

Σ

S2

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..........................................................

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...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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159

Example.

M = Σ× S2

Σ

S2

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...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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160

Example.

M = Σ× S2

K = −1Σ

S2

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...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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161

Example.

M = Σ× S2K = +1

K = −1Σ

S2

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162

Example.

M = Σ× S2K = +1

K = −1Σ

S2

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..........................................................

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...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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Product is scalar-flat Kahler.

163

Example.

M = Σ× S2K = +1

K = −1Σ

S2

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..........................................................

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...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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Product is scalar-flat Kahler.

164

Example.

M = Σ× S2K = +1

K = −1Σ

S2

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...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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Product is scalar-flat Kahler.For both orientations!

165

Example.

M = Σ× S2K = +1

K = −1Σ

S2

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...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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Product is scalar-flat Kahler.For both orientations!W+ = 0.

166

Example.

M = Σ× S2K = +1

K = −1Σ

S2

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...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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Product is scalar-flat Kahler.For both orientations!W± = 0.

167

Example.

M = Σ× S2K = +1

K = −1Σ

S2

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...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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Product is scalar-flat Kahler.For both orientations!W± = 0.Locally conformally flat!

168

Example.

M = Σ× S2K = +1

K = −1Σ

S2

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M = H2×S2= S4 − S1

169

Example.

M = Σ× S2K = +1

K = −1Σ

S2

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M = H2×S2 = S4 − S1

170

Example.

M = Σ× S2K = +1

K = −1Σ

S2

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M = H2×S2 = S4 − S1

π1(Σ) ↪→ SO+(1, 2)×SO(3) ↪→ SO+(1, 5)

171

Example.

M = Σ× S2K = +1

K = −1Σ

S2

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................................................................................................................................

M = H2×S2 = S4 − S1

π1(Σ) ↪→ SO+(1, 2)× SO(3)↪→ SO+(1, 5)

172

Example.

M = Σ× S2K = +1

K = −1Σ

S2

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.............................

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................................................................................................................................

M = H2×S2 = S4 − S1

π1(Σ) ↪→ SO+(1, 2)× SO(3) ↪→ SO+(1, 5)

173

Example.

M = Σ× S2K = +1

K = −1Σ

S2

......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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Scalar-flat Kahler deformations: 12(g−1) moduli

174

Example.

M = Σ× S2K = +1

K = −1Σ

S2

......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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Scalar-flat Kahler deformations: 12(g−1) moduliLocally conformally flat def’ms: 30(g − 1) moduli

175

Example.

M = Σ× S2K = +1

K = −1Σ

S2

......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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..........................................................

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Scalar-flat Kahler deformations: 12(g−1) modulialmost-Kahler ASD deformat’ns: 30(g−1) moduli

176





177





Is this subset also closed?

178






Does one get entire connected components this way?

179






Does one get entire connected components this way?

Alas, No!

180

Theorem A.Consider 4-manifold M = Σ× S2,

181

Theorem A.Consider 4-manifold M = Σ× S2,

182

Theorem A.Consider 4-manifold M = Σ× S2,where Σ compact Riemann surface of genus g .

183

Theorem A.Consider 4-manifolds M = Σ× S2,where Σ compact Riemann surface of genus g .

184


Then ∀ even g � 0, ∃ family [gt], t ∈ [0, 1], oflocally-conformally-flat classes on M , such that

185



186



187



188



189



190



• ∃ scalar-flat Kahler metric g0 ∈ [g0]; but

191




• @ almost-Kahler metric g ∈ [g1].

192





Same method simultaneously proves. . .

193

Theorem B. Fix an integer k ≥ 2, and thenconsider the 4-manifolds M = (Σ× S2)#kCP2,

194

Theorem B. Fix an integer k ≥ 2, and thenconsider the 4-manifolds M = (Σ× S2)#kCP2,

195

Theorem B. Fix an integer k ≥ 2, and thenconsider the 4-manifolds M = (Σ× S2)#kCP2,where Σ compact Riemann surface of genus g .

196


197


Then ∀ even g � 0, ∃ family [gt], t ∈ [0, 1], ofanti-self-dual conformal classes on M , such that

198



199




200





201





Proof hinges on a construction of hyperbolic 3-manifolds.

202





Proof hinges on a construction of hyperbolic 3-manifolds.

We begin by revisiting hyperbolic metrics on Σ.

203

H2.........................................................

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204

H2.........................................................

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205

H2.........................................................

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206

H2.........................................................

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π1(Σ) ↪→ SO+(1, 2) = PSL(2,R)∩ ∩

SO+(1, 3) = PSL(2,C)

207

H2.........................................................

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π1(Σ) ↪→ SO+(1, 2) = PSL(2,R)∩ ∩

SO+(1, 3) = PSL(2,C)

208

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209

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Limit Set...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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210

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........

H3

Limit Set...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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...........................................................

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.......................................

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...........

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.......................................

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.......

.............................................

..........................

.......................... ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. .............

..........................

π1(Σ)∼=−→ Γ ⊂ PSL(2,R) Fuchsian group

211

........................................................

........................................

.................................

.............................

...........................

........................

.......................

......................

....................

....................

...................

.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.................................

........................................

.........................................................

..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

........

H3

Limit Set...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

....................................................................................

...........................................................

.............................................

....................................

..............................................................................................

......................................................................................................................................................

..........................................................................................................................................

................................................................................................................

..................................................

......................................................................................................................................................

.........................

..............................

............................

...................................................................................

...................................................................................

.......................................................

............................

.......................................................................

................................................................................

...............................................................

...............................................................................................................................................

.......................................

.....................

...........

................................................................................

......................................................................................................................................

.......................................

................................

...............................................................................................................................................

.......

.............................................

..........................

.......................... ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. .............

..........................

π1(Σ)∼=−→ Γ ⊂ PSL(2,C) Fuchsian group

212

.......

.......

.......

.......

.......

.......

...................................................................................................................................................................................................................................................

.....................................

.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..........................................

..............................

.........................

.....................

....................

................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.......................................................................

.........................................

......................................................

................... .............

............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. .........................

213

.......

.......

.......

.......

.......

.......

...................................................................................................................................................................................................................................................

.....................................

.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..........................................

..............................

.........................

.....................

....................

................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.......................................................................

.........................................

......................................................

................... .............

............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. .........................

Fuchsian quasi-Fuchsian

214

.......

.......

.......

.......

.......

.......

...................................................................................................................................................................................................................................................

.....................................

.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..........................................

..............................

.........................

.....................

....................

...................................................................................................................................................................................................... .......

.......

.......

.......

.......

.......

...................................................................................................................................................................................................................................................

.....................................

.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..........................................

..............................

.........................

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................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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..........................................................................................................................................................

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...............................

..........................................................................................................................................................................................................................................................

.............................................

..............................................................................

......................................................................................................

........................................

....................................................................

..........................................................................................

...............................

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....................................

.......................................................................................................................

................... .............

............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ................... ......

....................

....................................................................... ................. ...

.....

...................................

... ................. ........................ ............... ...........

............... .......................... ................... .....

......... .............

............... ................ ....................................... ..................... ................ ...

...............

....................


215

.......

.......

.......

.......

.......

.......

...................................................................................................................................................................................................................................................

.....................................

.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..........................................

..............................

.........................

.....................

....................

...................................................................................................................................................................................................... .......

.......

.......

.......

.......

.......

...................................................................................................................................................................................................................................................

.....................................

.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..........................................

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.........................

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....................

................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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...........................

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...............................

..........................................................................................................................................................................................................................................................

.............................................

..............................................................................

......................................................................................................

........................................

....................................................................

..........................................................................................

...............................

...................................................

....................................

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................... .............

............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ................... ......

....................

....................................................................... ................. ...

.....

...................................

... ................. ........................ ............... ...........

............... .......................... ................... .....

......... .............

............... ................ ....................................... ..................... ................ ...

...............

....................


216

.......

.......

.......

.......

.......

.......

...................................................................................................................................................................................................................................................

.....................................

.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..........................................

..............................

.........................

.....................

....................

...................................................................................................................................................................................................... .......

.......

.......

.......

.......

.......

...................................................................................................................................................................................................................................................

.....................................

.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..........................................

..............................

.........................

.....................

....................

................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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..........................................................................................

...............................

..........................................................................................................................................................................................................................................................

.............................................

..............................................................................

......................................................................................................

........................................

....................................................................

..........................................................................................

...............................

...................................................

....................................

.......................................................................................................................

................... .............

............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ................... ......

....................

....................................................................... ................. ...

.....

...................................

... ................. ........................ ............... ...........

............... .......................... ................... .....

......... .............

............... ................ ....................................... ..................... ................ ...

...............

....................


π1(Σ)∼=−→ Γ ⊂ PSL(2,C) quasi-Fuchsian group

217

.......

.......

.......

.......

.......

.......

...................................................................................................................................................................................................................................................

.....................................

.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..........................................

..............................

.........................

.....................

....................

...................................................................................................................................................................................................... .......

.......

.......

.......

.......

.......

...................................................................................................................................................................................................................................................

.....................................

.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..........................................

..............................

.........................

.....................

....................

................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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...........................

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........................................

....................................................................

..........................................................................................

...............................

..........................................................................................................................................................................................................................................................

.............................................

..............................................................................

......................................................................................................

........................................

....................................................................

..........................................................................................

...............................

...................................................

....................................

.......................................................................................................................

................... .............

............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ................... ......

....................

....................................................................... ................. ...

.....

...................................

... ................. ........................ ............... ...........

............... .......................... ................... .....

......... .............

............... ................ ....................................... ..................... ................ ...

...............

....................



of Bers type

218

.......

.......

.......

.......

.......

.......

...................................................................................................................................................................................................................................................

.....................................

.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..........................................

..............................

.........................

.....................

....................

...................................................................................................................................................................................................... .......

.......

.......

.......

.......

.......

...................................................................................................................................................................................................................................................

.....................................

.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..........................................

..............................

.........................

.....................

....................

................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.......................................................................

.........................................

............................................... ...................................

...........................

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............................................................................................................

........................................

....................................................................

..........................................................................................

...............................

..........................................................................................................................................................................................................................................................

.............................................

..............................................................................

......................................................................................................

........................................

....................................................................

..........................................................................................

...............................

...................................................

....................................

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................... .............

............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ................... ......

....................

....................................................................... ................. ...

.....

...................................

... ................. ........................ ............... ...........

............... .......................... ................... .....

......... .............

............... ................ ....................................... ..................... ................ ...

...............

....................



of Bers type

Quasi-conformally conjugate to Fuchsian.

219

H3

........................................................

........................................

.................................

.............................

...........................

........................

.......................

......................

....................

....................

...................

.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.................................

........................................

.........................................................

..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

........

...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

....................................................................................

...........................................................

.............................................

....................................

..............................................................................................

......................................................................................................................................................

..........................................................................................................................................

................................................................................................................

..................................................

......................................................................................................................................................

.........................

..............................

............................

...................................................................................

...................................................................................

.......................................................

............................

.......................................................................

................................................................................

...............................................................

...............................................................................................................................................

.......................................

.....................

...........

................................................................................

......................................................................................................................................

.......................................

................................

...............................................................................................................................................

.......

.............................................

..........................

.......................... ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. .............

..........................

220

.....................................................................................................................................................................................................................................................................................................................................................................................................

.....................................................................................................................................................................................................................................................................................................................................................................................

................

................................................................................. .....................................................

......................................................................................................................................

.....................................................................................................................................................................................................................................................................................................................................................................................................

.....................................................................................................................................................................................................................................................................................................................................................................................

................

................................................................................. .....................................................

......................................................................................................................................

.................................................................

.................................................................

.................................................................

.................................................................

............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

221

X = H3/Γ

.....................................................................................................................................................................................................................................................................................................................................................................................................

.....................................................................................................................................................................................................................................................................................................................................................................................

................

................................................................................. .....................................................

......................................................................................................................................

.....................................................................................................................................................................................................................................................................................................................................................................................................

.....................................................................................................................................................................................................................................................................................................................................................................................

................

................................................................................. .....................................................

......................................................................................................................................

.................................................................

.................................................................

.................................................................

.................................................................

............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

222

X = H3/Γ

.....................................................................................................................................................................................................................................................................................................................................................................................................

.....................................................................................................................................................................................................................................................................................................................................................................................

................

................................................................................. .....................................................

......................................................................................................................................

.....................................................................................................................................................................................................................................................................................................................................................................................................

.....................................................................................................................................................................................................................................................................................................................................................................................

................

................................................................................. .....................................................

......................................................................................................................................

.................................................................

.................................................................

.................................................................

.................................................................

............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

Γ Fuchsian

223

X = H3/Γ

.....................................................................................................................................................................................................................................................................................................................................................................................................

.....................................................................................................................................................................................................................................................................................................................................................................................

................

................................................................................. .....................................................

......................................................................................................................................

.....................................................................................................................................................................................................................................................................................................................................................................................................

.....................................................................................................................................................................................................................................................................................................................................................................................

................

................................................................................. .....................................................

......................................................................................................................................

.................................................................

.................................................................

.................................................................

.................................................................

............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

Γ Fuchsian

X ≈ Σ× R

224

X = H3/Γ

....................................

....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.................................

.....................................................................................................................................................................................................................................................................................................................................................................................

................

................................................................................. .....................................................

......................................................................................................................................

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.................................................

.......................................................................................................................................................................................................................................................................................................................................................................................................................................

................

................................................................................. .....................................................

......................................................................................................................................

.................................................................

.................................................................

.................................................................

.................................................................

................

................

.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..................

..................

..................

..................

..................

............

Γ quasi-Fuchsian

X ≈ Σ× R

225

X = H3/Γ

....................................

....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.................................

.....................................................................................................................................................................................................................................................................................................................................................................................

................

................................................................................. .....................................................

......................................................................................................................................

.....................................................................................................................................................................................................................................................................................................................................................................................................

.................................................

.......................................................................................................................................................................................................................................................................................................................................................................................................................................

................

................................................................................. .....................................................

......................................................................................................................................

.................................................................

.................................................................

.................................................................

.................................................................

................

................

.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..................

..................

..................

..................

..................

............

Γ quasi-Fuchsian

X ≈ Σ× R

Freedom: two points in Teichmuller space.

226

X = H3/Γ

....................................

....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.................................

.....................................................................................................................................................................................................................................................................................................................................................................................

................

................................................................................. .....................................................

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.......................................................................................................................................................................................................................................................................................................................................................................................................................................

................

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.................................................................

.................................................................

.................................................................

................

................

.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..................

..................

..................

..................

..................

............

Γ quasi-Fuchsian

X ≈ Σ× R

227

X = H3/Γ

....................................

....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.................................

.....................................................................................................................................................................................................................................................................................................................................................................................

................

................................................................................. .....................................................

......................................................................................................................................

.....................................................................................................................................................................................................................................................................................................................................................................................................

.................................................

.......................................................................................................................................................................................................................................................................................................................................................................................................................................

................

................................................................................. .....................................................

......................................................................................................................................

.................................................................

.................................................................

.................................................................

.................................................................

................

................

.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..................

..................

..................

..................

..................

............

Γ quasi-Fuchsian

X ≈ Σ× [0, 1]

228

s

s

1

0

f.......................................................................................................................................

...........................

X = H3/Γ

....................................

....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.................................

.....................................................................................................................................................................................................................................................................................................................................................................................

................

................................................................................. .....................................................

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.....................................................................................................................................................................................................................................................................................................................................................................................................

.................................................

.......................................................................................................................................................................................................................................................................................................................................................................................................................................

................

................................................................................. .....................................................

......................................................................................................................................

.................................................................

.................................................................

.................................................................

.................................................................

................

................

.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..................

..................

..................

..................

..................

............

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

....

Γ quasi-Fuchsian

X ≈ Σ× [0, 1]

Tunnel-Vision function:

f : X → [0, 1]

229

s

s

1

0

f.......................................................................................................................................

...........................

X = H3/Γ

....................................

....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.................................

.....................................................................................................................................................................................................................................................................................................................................................................................

................

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.................................................................

.................................................................

.................................................................

................

................

.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..................

..................

..................

..................

..................

............

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

....

Γ quasi-Fuchsian

X ≈ Σ× [0, 1]


f : X → [0, 1]

∆f = 0

230

.......

.......

.......

.......

.......

.......

...................................................................................................................................................................................................................................................

.....................................

.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..........................................

..............................

.........................

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....................

...................................................................................................................................................................................................... .......

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...................................................................................................................................................................................................................................................

.....................................

.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..........................................

..............................

.........................

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................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.......................................................................

.........................................

............................................... ...................................

...........................

...................................................................................................................................................

..........................................................................................................................................................

.............................................................................................................................................

..............................................................................

............................................................................................................

........................................

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..........................................................................................

...............................

..........................................................................................................................................................................................................................................................

.............................................

..............................................................................

......................................................................................................

........................................

....................................................................

..........................................................................................

...............................

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................... .............

............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ................... ......

....................

....................................................................... ................. ...

.....

...................................

... ................. ........................ ............... ...........

............... .......................... ................... .....

......... .............

............... ................ ....................................... ..................... ................ ...

...............

....................


231

s

s

1

0

f.......................................................................................................................................

...........................

X = H3/Γ

....................................

....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.................................

.....................................................................................................................................................................................................................................................................................................................................................................................

................

................................................................................. .....................................................

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.................................................

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................

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......................................................................................................................................

.................................................................

.................................................................

.................................................................

.................................................................

................

................

.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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....

Γ quasi-Fuchsian

X ≈ Σ× [0, 1]


f : X → [0, 1]

∆f = 0

232

X = H3/Γ

....................................

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233

X = H3/Γ

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Construction of conformally flat 4-manifolds:

234

X = H3/Γ

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M = [X × S1]/ ∼

235

X = H3/Γ

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M = [X × S1]/ ∼

∼: crush ∂X × S1 to ∂X .

236

X = H3/Γ

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M = [X × S1]/ ∼

237

X = H3/Γ

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M = [X × S1]/ ∼

g = f (1− f )[h + dt2]

238

X = H3/Γ

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M = [X × S1]/ ∼

g = f (1− f )[h + dt2]

239

X = H3/Γ

.....................................................................................................................................................................................................................................................................................................................................................................................................

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............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................


M = [X × S1]/ ∼

g = f (1− f )[h + dt2]

Fuchsian case: Σ× S2 scalar-flat Kahler.

240

X = H3/Γ

....................................

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.................................................................

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241

X = H3/Γ

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Construction of ASD 4-manifolds:

242

s ss

ssX = H3/Γ

....................................

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Choose k points p1, . . . , pk ∈ X

243

s

s

1

0

f.......................................................................................................................................

...........................

s ss

ssX = H3/Γ

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....



satisfying∑kj=1 f (pj) ∈ Z.

244

s

s

1

0

f.......................................................................................................................................

...........................

s ss

ssX = H3/Γ

....................................

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....



satisfying∑kj=1 f (pj) ∈ Z.

Can do if k 6= 1.

245

s ss

ssX = H3/Γ

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Let Gj be the Green’s function of pj:

246

s ss

ssX = H3/Γ

....................................

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Let Gj be the Green’s function of pj:

∆Gj = 2πδpj, Gj → 0 at ∂X

247

s ss

ssX = H3/Γ

....................................

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Let Gj be the Green’s function of pj, and set

V = 1 +

k∑j=1

Gj.

248

s ss

ssX = H3/Γ

....................................

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V = 1 +

k∑j=1

Gj.

249

s ss

ssX = H3/Γ

....................................

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............


V = 1 +

k∑j=1

Gj.

Choose P → (X−{p1, . . . , pk}) circle bundle withconnection form θ such that

dθ = ?dV .

250

s ss

ssX = H3/Γ

....................................

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g = f (1− f )[V h + V −1θ2]

V = 1 +

k∑j=1

Gj

dθ = ?dV

251

s ss

ssX = H3/Γ

....................................

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............


g = f (1− f )[V h + V −1θ2]

V = 1 +

k∑j=1

Gj

dθ = ?dV

252

s ss

ssX = H3/Γ

....................................

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............


g = f (1− f )[V h + V −1θ2]

M = P ∪ {p1, . . . , pk} ∪ ∂X

253

s ss

ssX = H3/Γ

....................................

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............


g = f (1− f )[V h + V −1θ2]

M = P ∪ {p1, . . . , pk} ∪ ∂X↓ ↓ ↓ ↓X = X − {p1, . . . , pk} ∪ {p1, . . . , pk} ∪ ∂X

254

s ss

ssX = H3/Γ

....................................

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............


g = f (1− f )[V h + V −1θ2]

M = P ∪ {p1, . . . , pk} ∪ ∂X

255

s ss

ssX = H3/Γ

....................................

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............


g = f (1− f )[V h + V −1θ2]

M = P ∪ {p1, . . . , pk} ∪ ∂X

≈(Σ× S2)#kCP2 scalar-flat Kahler

256

.......................................................................................................................

..........................

s s

Σ

X = Σ× [0, 1]

.......

.......

.......

.......

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.......................................

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..............................................

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..........................................................................................................................................................................................

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.................................................................................................................................................................................................................................................................

.......................................................................................................................................................

.................

..........................................................................................................................................................................................

.............................

.................................................................................................................................................................................................................................................................

.......................................................................................................................................................

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...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ....

............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ....

...........................................................................................................................................................................................................

.......................................................................................................................................................................................................

..............................................................................................................................................................

.........................................................................................................................................................................

257

s ss

ssX = H3/Γ

....................................

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g = f (1− f )[V h + V −1θ2]

M = P ∪ {p1, . . . , pk} ∪ ∂X

≈(Σ× S2)#kCP2 scalar-flat Kahler

258

s ss

ssX = H3/Γ

.....................................................................................................................................................................................................................................................................................................................................................................................................

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g = f (1− f )[V h + V −1θ2]

M = P ∪ {p1, . . . , pk} ∪ ∂X

Fuchsian case: (Σ× S2)#kCP2 scalar-flat Kahler

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Γ quasi-Fuchsian

X ≈ Σ× [0, 1]


f : X → [0, 1]

∆f = 0

260

Theorem.Let (M, [g]) be ASD manifold arisingfrom a quasi-Fuchsian 3-manifold (X, h) and aconfiguration of points in X.

Then ∃ almost-Kahler g ∈ [g] ⇐⇒ tunnel-visionfunction f : X → (0, 1) has no critical points.

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Proof.

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Proof.

b+[(Σ× S2)#kCP2] = 1.

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Proof.

b+[(Σ× S2)#kCP2] = 1.

ω = df ∧ θ + V ? df .

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Proof.

b+[(Σ× S2)#kCP2] = 1.

ω = df ∧ θ + V ? df .

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Lemma.For any piecewise smooth Jordan curveγ ⊂ C and any ε > 0, there is a positive integerN such that, for every compact oriented surfaceΣ of genus g ≥ N , there is quasi-Fuchsian groupΓ ∼= π1(Σ) whose limit set Λ(Γ) ⊂ C ⊂ CP1 iswithin Hausdorff distance ε of γ.

If γ is invariant under ζ 7→ −ζ, and if g is even,we can also arrange for Λ(Γ) to also be invariantunder reflection through the origin.

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Kia Ora!

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Documents

Anti-Self-Dual 4-Manifolds, Quasi-Fuchsian Groups, & Almost-K …claude/auckland.pdf · Anti-Self-Dual 4-Manifolds, Quasi-Fuchsian Groups, & Almost-K ahler Geometry Claude LeBrun