CR Geometry, Mappings into Spheres, and Sums-Of-Squares ...pebenfel/CR Rigidity - III_handout.pdf · Rigidity in CR Geometry. A basic problem. Let M = Mn ‘ ˆC n+1 be a Levi nondegenerate

CR Geometry, Mappings into Spheres, andSums-Of-Squares

Lecture III

Peter Ebenfelt

University of California, San Diego

Available at http://www.math.ucsd.edu/∼pebenfel/.

Peter Ebenfelt (UCSD) CR Geometry, Mappings into Spheres, and Sums-Of-Squares Lecture IIIAvailable at http://www.math.ucsd.edu/∼pebenfel/. 1

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Outline - Lecture III

1 Transversal CR immersions into hyperquadrics

2 Basic Rigidity Results

3 The CR Second Fundamental Form and Gauss Equation

4 Analysis of the Gauss Equation; an example

5 References


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Rigidity in CR Geometry. A basic problem.

Let M = Mn` ⊂ Cn+1 be a Levi nondegenerate hypersurface of CR

dimension n ≥ 1 and Levi signature 0 ≤ ` ≤ n/2.

Consider F(M,QN`′ ): the space of transversal CR immersions into the

hyperquadric QN`′ ⊂ CN+1,

f : M → QN`′ , T 1,0

f (p)QN`′ + f∗T

1,0p Cn+1 = T 1,0

f (p)CN+1.

Equivalence relation in F(M,QN`′ ): f1 ∼ f2 ⇐⇒ ∃T ∈ Aut(QN

`′ )such that f2 = T f1.

Basic Problem:

Describe the deformation space F(M,QN`′ )/ ∼.

”Rigidity” in this context means F(M,QN`′ )/ ∼ consists of a single

point.


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A necessary condition on QN`′ .

Proposition 1

If there exists a transversal f : Mn` → QN

`′ , i.e. F(Mn` ,Q

N`′ ) 6= ∅, then

N ≥ n and `′ ≥ `.

Proof. A direct consequence of Levi form invariance. Pick contact formsθ, θ′, and local (1, 0)-vector frames L1, . . . , Ln, L′1, . . . , L

′N for M = Mn

`

and QN`′ , respectively. The Levi forms of M and QN

`′ are represented byn× n and N ×N matrices E and E ′; WLOG with signatures (`, n− `) and(`′,N − `′). We have f ∗θ′ = aθ for some real-valued function a on M.

Exercise: f is transversal at p ∈ M ⇐⇒ a(p) 6= 0.

In the chosen (1, 0)-vector frames, (f∗)|T 1,0M is represented by an n × Nmatrix B, and we have by Levi form invariance: aE = BE ′B∗. Theconclusion now easily follows from standard linear algebra.


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Existence of transversal f : M → QN`′ . Basic results.

There are real-analytic Levi nondegenerate hypersurfaces M = Mn` of

any signature ` and CR dimension n for which F(M,QN`′ ) = ∅ for all

N and `′. ”Counting” argument (local) [7]; see also [1].

If M is real-algebraic, then ∃N ≥ n, `′ ≥ ` such that F(M,QN`′ ) 6= ∅;

in general `′ > `. ”Diagonalization” argument [11].

There are compact, real-algebraic, strictly pseudoconvex M such thatF(M, S2N+1) = ∅ for every N. [8]; recall S2N+1 = QN

0 .

A Basic Example

Let X = X n+1 ⊂ CN+1 be complex analytic variety of dimension n with anisolated singularity at 0 ∈ X .

The Milnor links Mε := X ∩ S2N+1ε , for sufficiently small radii ε > 0,

are compact strictly pseudoconvex hypersurfaces in X that admit (byconstruction) transversal CR embeddings fε : Mε → S2N+1.


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CR complexity of M = Mn` .

Definition 1

(i) The CR complexity of M = Mn` is given by

µ(M) := minN − n : F(M,QN` ) 6= ∅.

(ii) More generally, for k ≥ 0, the CR k-complexity of M = Mn` is given by

µk(M) := minN − n : F(M,QN`+k) 6= ∅, N ≥ 2(`+ k).

We note that 0 ≤ µ(M) = µ0(M) ≤ ∞.

Assume `+ k < N/2. Then,

µk+1(M) ≤ µk(M) + 1.


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Basic rigidity results f : M = Mn` → QN

` .

Theorem 1 (see [4, 5])

Assume µ(M) <∞, and let f0 ∈ F(M,QN0` ) with N0 − n = µ(M). If

f ∈ F(M,QN` ) and

(N − n) + µ(M) < n,

then f = T L f0 for some T ∈ Aut(QN` ); L denotes the standard linear

embedding z = (z1, . . . , zn+1) 7→ (z , 0).

Applies to M = Qn0 = S2n+1, µ(M) = 0, f0 = Id . Faran, Webster

[6, 12].

Theorem 2 (see [2])

Assume µ(M) < `, and let f0 ∈ F(M,QN0` ) with N0 − n = µ(M). If

f ∈ F(M,QN` ), then f = T L f0 for some T ∈ Aut(QN

` ).

Super-rigidity. Applies to M = Qn` , ` > 0. Baouendi–Huang [3].


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Adapted pseudohermitian frames.

Recall: A choice of contact form θ on M = Mn` defines a pseudohermitian

structure. A (local) CR coframe (θ, θα) is admissible if dθ = ihαβθα ∧ θβ.

Consider a transversal CR map f : M → M = MN`′ , and fix an admissible

CR coframe (θ, θα) on M.

An adapted (local) CR coframe on M

There exists a CR coframe (θ, θA) (locally) on M near f (M) such that

f ∗(θ, θA) = (θ, θα, 0).

Moreover, if the Levi form hαβ is diagonal with ±1 on diagonal, then we

can choose (θ, θA) such that hAB is diagonal with ±1 on diagonal.

Side preserving (hAB) = I ′`′,N−`′ vs. side reversing (hAB) = I ′N−`′,`′ .


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The CR Second Fundamental Form (SFF) of

f : Mn` → MN

`′ .

Fix (locally) a CR coframe (θ, θα) on M; (hαβ) = I`,n−`. Let (θ, θA) be an

adapted CR coframe on MN`′ near f (M); (hAB) = I ′`′,N−`′ . By pulling back

to Md θA = ωB

A ∧ θB + τA ∧ θ, ωAB + ωBA = 0

we deduceωα

β = ωαβ, τα = τα,

and0 = ωβ

a ∧ θβ + τ a ∧ θ; a = N − n + 1, . . . ,N.

By using τB = ABC θ

C , we conclude that, pulled back to M,

τ a = 0, ωβa := ωβ

a = ωβaαθ

α, ωβaα = ωα

aβ.


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The SFF of f : M → M , cont’d.

Definition 2

For p ∈ M, the SFF of f : M → M at p,

SFFp : T 1,0p M × T 1,0

p M → T 1,0f (p)M/f∗(T

1,0p M),

is defined bySFFp(Lα, Lβ) := ωα

aβ[La].

We shall view SFF as a tensor ωαaβ on Cn × CN−n × Cn. The

pseudohermitian connections yield covariant differentals

∇ω aα β = dω a

α β − ω aµ βω

µα + ω b

α βωa

b − ω aα µω

µβ .

We write ω aα β;γ to denote the component in the direction θγ and define

inductively:

∇ω aγ1 γ2;γ3...γj

= dω aγ1 γ2;γ3...γj

+ ω bγ1 γ2;γ3...γj

ω ab −

j∑l=1

ω aγ1 γ2;γ3...µ...γj

ω µγl.


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Main steps in proof of Thm 1.

Let f0 : M = Mn` → QN0

` , f : M → QN` be transversal immersions,

N0 = n + µ(M) ≤ N. Fix CR coframe (θ, θα) on M. In adapted CRcoframes for f0 and f , the SFFs are (ωα

aβ)N0−n

a=1 and (ωαaβ)N−na=1 .

1. Show: After suitable choice of adapted CR normal frames:

ωγ1aγ2;...γk+2

= ωγ1aγ2;...γk+2

, k = 0, 1, 2, . . . . (1)

where extra zeros has been added to ωγ1aγ2;...γk+2

for a > N0 − n.

2. Show: The identities (1) imply, under additional technicalassumption (always OK in context of Thm 1), ∃T ∈ Aut(QN

` ) suchthat f = T L f0.

Here, we will focus on Step 1. For Step 2, see [4].


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The (CR) Gauss Equation.

Gauss Equation for f : M = Mn` → QN

`′

Sαβµν =− habωa

α µωb

β ν

+ω aγ αω

γ

aβhµν + ω a

γ µωγ

aβhαν + ω a

γ αωγaνhµβ + ω a

γ µωγaνhαβ

n + 2

−ω aγ δω

γ δa

(n + 1)(n + 2)(hαβhµν + hανhµβ).

The ”essential” features of this equation can be captured in asimpler, point-wise polynomial identity:

S(ζ, ζ) = −〈ω(ζ), ω(ζ)〉′ + A(ζ, ζ)) 〈ζ, ζ〉 .

This will be explained in more detail in a few slides.


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Sketch of proof of Gauss Equation.

Inspecting the structure equation for ωαβ on QN

`′ ,

dωαβ = ωα

A ∧ ωAβ + Rα

βABθ

A ∧ θB + . . . ,

and pulling back to M, we conclude:

Rαβνµ = Rα

βνµ − ωαaνωβaµ. (2)

A symmetric tensor Tαβνµ can be uniquely decomposed as

Tαβνµ = T 0αβνµ

+ (hαβ)⊗ (T 1νµ),

where T 0αβνµ

is the trace-free part of Tαβνµ, i.e., T 0αανµ = 0.

Taking the trace-free part of (2) and using SABCD = 0 on QN`′ yields

Gauss Equation.


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Gauss Equation for sectional curvature.

Define polynomials in ζ = (ζ1, . . . , ζn) and ζ:

S(ζ, ζ) := Sαβνµζαζνζβζµ, ωa(ζ) := ωα

aνζαζν ,

and Hermitian forms

〈ζ, ζ〉 := hαβζαζβ, 〈τ, τ〉′ := habτ

aτb.

〈·, ·〉 has rank n and signature `,

〈·, ·〉′ has rank N − n and signature either `′ − ` (side preserving) orN − `′ − ` (side reversing). Note: side reversing requires `′ + ` ≥ n.

Gauss Equation

S(ζ, ζ) = −〈ω(ζ), ω(ζ)〉′ + A(ζ, ζ)) 〈ζ, ζ〉 .


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Analysis of the Gauss Equation. An example (Thm 1).

Let f0 : M = Mn` → QN0

` , f : M → QN` be transversal immersions,

N0 = n + µ(M) ≤ N. Fix CR coframe (θ, θα) on M. In adapted CRcoframes for f0 and f , the SFFs are (ωα

aβ)N0−n

a=1 and (ωαaβ)N−na=1 .

Recall, `′ = ` =⇒ (hab) = I(N0−n)×(N0−n), (hab) = I(N−n)×(N−n).

Gauss Equations become:

S(ζ, ζ) = −||ω(ζ)||2 + A(ζ, ζ) 〈ζ, ζ〉 ,S(ζ, ζ) = −||ω(ζ)||2 + A(ζ, ζ) 〈ζ, ζ〉 ;

||ω(ζ)||2 has µ(M) = N0 − n terms and ||ω(ζ)||2 has N − n terms.

Subtracting the two Gauss equations, we obtain:

− ||ω(ζ)||2 + ||ω(ζ)||2 = B(ζ, ζ) 〈ζ, ζ〉 . (3)


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Huang’s Lemma.

Huang’s Lemma (see [9])

Let 1 ≤ k < n; g1, . . . , gk , f1, . . . , fk holomorphic functions near 0 ∈ Cn

such thatk∑

i=1

gi (ζ)fi (ζ) = B(ζ, ζ) 〈ζ, ζ〉 .

Then B(ζ, ζ) ≡ 0.

Applying Huang’s Lemma to subtracted Gauss Equations (3) =⇒

||ω(ζ)||2 = ||ω(ζ)||2.

Linear Algebra =⇒ ∃U ∈ U(CN−n):

ω(ζ) = (ω(ζ), 0)U. (4)


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Changing adapted normal frame.

We write U = (Uab), where a, b = 1, . . . ,N − n. (4) =⇒

ωαaβ = Ub

aωαbβ,

where we have added extra zeros to ωαbβ for b > N0 − n.

We change the adapted CR frame for f by Lb 7→ UbaLa =⇒

ωαaβ = ωα

aβ. (5)

We note freedom in choice of U due to zeros in ωαbβ. May make

further changes in La for a > N0 − n without altering (5).


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Conformally Flat Tensors (CFT).

A tensor Tα1...αk β1...βla1...am is conformally flat (a CFT) if it is of the

form (hαβ)⊗ (T ′α1...αk−1β1...βl−1

a1...am), i.e., a sum of terms each

having a factor hαi βj;

T a1...am(ζ, ζ) = Tα1...αk β1...βla1...amζα1 . . . ζαk ζβ1 . . . ζβ` = A(ζ, ζ) 〈ζ, ζ〉 .

The Gauss Equation can be written

Sαβνµ = −habωαaνωβ

bµ mod CFT . (6)

Since hαβ is parallel, i.e. ∇hαβ = 0, any covariant derivative of aconformally flat tensor is again conformally flat.

Lemma 1 (see [4], [2])

The covariant derivative tensor ωαaβ;γ is conformally flat (a CFT).


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Completing Step 1 in proof of Thm 1.

Taking repeated covariant derivatives of Gauss Equation (6), usingLemma 1, yields

Sγ1βγ2µ;γ3...γk= −habωγ1

aγ2;γ3...γk ωβ

bµ mod CFT ,

Sγ1βγ2µ;γ3...γk= −habωγ1

aγ2;γ3...γkωβ

bµ mod CFT .

(7)

Multiply both equations by ζγ1 . . . ζγk ζβζµ, introduce

Ωa(k)(ζ) := ωγ1

aγ2;γ3...γk ζ

γ1 . . . ζγk , Ωa(2) = ωa(ζ),

and subtract the two equations in (7) =⇒

N0−n∑a=1

(Ωa

(k)(ζ)− Ωa(k)(ζ)

)ωa(ζ) = B(ζ, ζ) 〈ζ, ζ〉 . (8)


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Completing Step 1 in proof of Thm 1; cont’d.

Applying Huang’s Lemma to (8), we conclude

N0−n∑a=1

(Ωa

(k)(ζ)− Ωa(k)(ζ)

)ωa(ζ) = 0,

which in tensor form becomes

N0−n∑a=1

(ωγ1aγ2;γ3...γk − ωγ1

aγ2;γ3...γk )ωβ

aµ = 0. (9)

Consider the normal vectors Vγ1...γk := (ωγ1aγ2;γ3...γk )N−na=1 ∈ CN−n,

and subspaces (essentially introduced by Lamel [10])

Ek := spanVγ1...γl : 2 ≤ l ≤ k ⊂ CN−n.


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Completion of Step 1 in special case.

We note that (9) means that the projection of

ωγ1aγ2;γ3...γk − ωγ1

aγ2;γ3...γk

on E2 = E2 vanishes.

If we assume that E2 = CN−n (which can only happen if N = N0),then we have completed Step 1:

ωγ1aγ2;...γk+2

= ωγ1aγ2;...γk+2

, k = 2, 3 . . . .

The proof of Thm 1 is now completed by completing Step 2.


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What to do if E2 6= CN−n?

One must then also differentiate the Gauss Equations in the θγ

directions, and use a commutation formula for covariant derivatives[2].

Doing so, and repeatedly using Huang’s Lemma as above, one maydeduce that El = El and the projection of

ωγ1aγ2;γ3...γk − ωγ1

aγ2;γ3...γk

on El , for any l , vanishes, after possibly additional changes of adaptednormal frame. This completes Step I.

If El = CN−n for some l , then Step 2 can be executed.

If the El = El stabilize at E∞ ( CN−n, then need to show f (M) iscontained in an affine plane section of QN

` of appropriate dimension.In the context of Thm 1, this is OK. More on this in next lecture...


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The End


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M. Salah Baouendi, Peter Ebenfelt, and Xiaojun Huang.Super-rigidity for CR embeddings of real hypersurfaces intohyperquadrics.Adv. Math., 219(5):1427–1445, 2008.

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Peter Ebenfelt, Xiaojun Huang, and Dmitry Zaitsev.The equivalence problem and rigidity for hypersurfaces embedded intohyperquadrics.


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Amer. J. Math., 127(1):169–191, 2005.

James J. Faran.The linearity of proper holomorphic maps between balls in the lowcodimension case.J. Differential Geom., 24(1):15–17, 1986.

Franc Forstneric.Embedding strictly pseudoconvex domains into balls.Trans. Amer. Math. Soc., 295(1):347–368, 1986.

X. Huang and M. Xiao.Chern-moser-weyl tensor and embeddings into hyperquadrics.preprint; https://arxiv.org/abs/1606.09145, 2012.

Xiaojun Huang.On a linearity problem for proper holomorphic maps between balls incomplex spaces of different dimensions.J. Differential Geom., 51:13–33, 1999.

Bernhard Lamel.


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Holomorphic maps of real submanifolds in complex spaces of differentdimensions.Pacific J. Math, 201(2):357–387, 2001.

S. M. Webster.Some birational invariants for algebraic real hypersurfaces.Duke Math. J., 45(1):39–46, 1978.

S. M. Webster.The rigidity of C-R hypersurfaces in a sphere.Indiana Univ. Math. J., 28(3):405–416, 1979.


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Documents

CR Geometry, Mappings into Spheres, and Sums-Of-Squares ...pebenfel/CR Rigidity - III_handout.pdf · Rigidity in CR Geometry. A basic problem. Let M = Mn ‘ ˆC n+1 be a Levi nondegenerate