Geodesics, singularities, and blow-ups · Content 1 Singular Spaces 2 Problems in the geometry and analysis of singular spaces 3 Resolution of singularities, blow-up 4 Geodesics:

Geodesics, singularities, and blow-ups

Daniel Grieser

(Carl von Ossietzky Universitat Oldenburg)

7.9.2012

(joint work with Vincent Grandjean)

WE Heraeus-Seminar ’Algebro-Geometric Methods in FundamentalPhysics’

Physikzentrum Bad HonnefSeptember 3-7, 2012

Daniel Grieser (Oldenburg) Geodesics, singularities, and blow-ups 7.9.2012 1 / 16

Summer school

’Singular Analysis’

(for Ph.D. students andpostdocs)

19.-21. September 2012Uni Oldenburg, W01 1-117

http://www.math-conf.

uni-hannover.de/ana12/


http://www.math-conf.uni-hannover.de/ana12/

http://www.math-conf.uni-hannover.de/ana12/

Content

1 Singular Spaces

2 Problems in the geometry and analysis of singular spaces

3 Resolution of singularities, blow-up

4 Geodesics: Results

5 Method

Please ask!


Singular Spaces

Singular space

Union of smooth manifolds (strata) of different dimensions, e.g.(semi)-algebraic set

X = Xreg ∪ Xsing

Xreg = highest-dimensional stratum


Problems

Metric on Xreg:X ⊂ (Rn, geucl) (or other smooth Riemannian manifold)– induced metric g on Xreg: restriction of geucl– also ∃ other interesting classes of metrics

Problems

Geometry: geodesics, curvature etc. of (Xreg, g) near Xsing

Analysis (PDE): e.g. ∆g= Laplace-Beltrami operator on Xreg

Behavior of solutions of ∆gu = f near Xsing

wave propagation on Xindex theory (Atiyah-Singer) etc.

Generalize 20th century ’smooth’ mathematics to singular spaces!

(Kondratiev, Cheeger, Melrose, Mazzeo, Seeley, Schulze,. . . )


Problems


Problems








Problems


Problems








Relation to this conference

Geodesics near space-time singularity

Blow-up technique from algebraic geometry

But: No symmetry no explicit formulas!


Resolution of singularities

One can get a singular space as image of a smooth space under a ’simple’map (’blow-down’):

X β−→ X

(x , y , z) 7→ (xz , yz , z)

Resolution problem

Given X , find β so that the preimage X is smooth!

(X , β) is called a resolution of X (≈ ’singular sets of coordinates’ on X )

Hironaka 1964: Every algebraic X can be resolved by a sequence of’blow-ups’, i.e. by a composition of maps of the form above


Resolution of singularities

One can get a singular space as image of a smooth space under a ’simple’map (’blow-down’):

X β−→ X

(x , y , z) 7→ (xz , yz , z)

Resolution problem

Given X , find β so that the preimage X is smooth!

(X , β) is called a resolution of X (≈ ’singular sets of coordinates’ on X )

Hironaka 1964: Every algebraic X can be resolved by a sequence of’blow-ups’, i.e. by a composition of maps of the form above


Blow-ups and metrics

Let g = induced metric on X and g = corresponding metric on X .Note: g is only positive semi-definite (degenerate) at ∂X .

Example: Cone r

f

g = dr2 + r2 dφ2 g = Euclidean metric

singular vs. degenerate

regular space X singular space Xwith ↔ withdegenerate metric Riemannian metric

Open problem: Understand degenerations of g for general resolutionsDaniel Grieser (Oldenburg) Geodesics, singularities, and blow-ups 7.9.2012 8 / 16

Degenerate metrics

Consequences of metric degeneration:(in the example of g = dr2 + r2dφ2):

∆g = ∂2r + 1r ∂r + 1

r2∂2φ has singular coefficients

Geodesic flow is singular at r = 0

Aside: Conjectured general normal form for resolved (degenerate)metrics:

g =∑j

ajd(rαj )2 +∑k

bk(rβk dφk)2

on Rl+ × Rn−l , where r = (r1, . . . , rl), αj , βk ∈ Nl

0 and aj , bk smooth


Degenerate metrics

Consequences of metric degeneration:(in the example of g = dr2 + r2dφ2):

∆g = ∂2r + 1r ∂r + 1

r2∂2φ has singular coefficients

Geodesic flow is singular at r = 0

Aside: Conjectured general normal form for resolved (degenerate)metrics:

g =∑j

ajd(rαj )2 +∑k

bk(rβk dφk)2

on Rl+ × Rn−l , where r = (r1, . . . , rl), αj , βk ∈ Nl

0 and aj , bk smooth


Geodesics

(M, g) Riemannian manifoldMetric: g = gij(x)dx idx j : TM → R, x ∈ M

Dual metric: g∗ = g ij(x)ξiξj : T ∗M → R, (x , ξ) ∈ T ∗M

Definition of geodesic

x(t) part of a solution of the Hamiltonian system on T ∗M with energyE = 1

2g∗:

x = Eξ ξ = −Ex

Geodesics = locally shortest curves on MFact: Given x0, v0, there is a unique geodesic with x(0) = x0, x(0) = v0This defines the exponential map, i.e. normal coordinates.

Problem

Is this true if x0 is a singularity of a singular space?How do the geodesics starting at x0 behave? exp map?


Geodesics





2g∗:

x = Eξ ξ = −Ex

Geodesics = locally shortest curves on MFact: Given x0, v0, there is a unique geodesic with x(0) = x0, x(0) = v0

This defines the exponential map, i.e. normal coordinates.

Problem



Geodesics





2g∗:

x = Eξ ξ = −Ex


Problem



Geodesics





2g∗:

x = Eξ ξ = −Ex


Problem



Results: Perturbed cone

X = ←

X

r

f

General conic metric:g = dr2 + r2h(r , φ, dr , dφ)

φ = (φ1, . . . , φn−1)

Theorem (Melrose, Wunsch 2001)

For every φ ∈ ∂X there is a unique geodesic starting at φ. These geodesicsfoliate smoothly a neighborhood of ∂X .

Corollary

On X there is a unique geodesic leaving the singular point in any giventangential direction.exp map exists, is smooth, is local diffeomorphism. Have normalcoordinates.


Results: Perturbed cusp

X = ←

X

r

f

General cusp metric: g =

(1− 2r2S(φ))dr2 + r4h(r , φ, dr , dφ)

S : ∂X → R describesshape of cross section

Theorem

a) If S = const then as for cone: Smooth exp map, normal coordinates.

b) S not constant: exp map exists, is not smooth.

c) If S varies strongly then exp map not injective locally.

Examples: a) centered circle, b) off-center circle / far off-center circle

a) b): or



X = ←

X

r

f


(1− 2r2S(φ))dr2 + r4h(r , φ, dr , dφ)


Theorem





a) b): or



X = ←

X

r

f


(1− 2r2S(φ))dr2 + r4h(r , φ, dr , dφ)


Theorem





a) b): or


Results: Mix of cone and cusp (dimX = 2)

X = ←

X

r

f

Type of metric near φ = 0: g =

dr2 + r2(r2 + φ2)dφ2

(transition from cone to cusp)

Theorem

For φ 6= 0 like the cone, but exp map is non-smooth at φ = 0.

exp map is log− smooth after blow-up.

Corollary

Hardt’s conjecture (distance function on algebraic set is subanalytic) isFALSE.


Results: Mix of cone and cusp (dimX = 2)

X = ←

X

r

f

Type of metric near φ = 0: g =

dr2 + r2(r2 + φ2)dφ2

(transition from cone to cusp)

Theorem

For φ 6= 0 like the cone, but exp map is non-smooth at φ = 0.

exp map is log− smooth after blow-up.

Corollary

Hardt’s conjecture (distance function on algebraic set is subanalytic) isFALSE.


Method

Coordinates (r , φ, ξ, η) on T ∗X , then geodesics = solutions of

r = Eξ, φ = Eη, ξ = −Er , η = −Eφ (∗)

Problem: Energy is singular, e.g. cone: E = 12

(ξ2 + η2

r2

)Rescale angular momentum θ := η

r to make E smooth

Rescale time: Multiply (∗) by r , obtain vector field V in (r , φ, ξ, θ)

V is smooth on cT ∗X , tangential to boundary r = 0

Starting points of geodesics = critical points of V

Analyze linearizations of V at critical points, use invariant manifoldtheorem; important: V is normally hyperbolic, i.e. not too degenerate


Method

Main issue

Rescaling θ = ηr makes E smooth, but the symplectic form

dx ∧ dξ + dφ ∧ dη degenerate (For cusp: θ = ηr3

, not ηr2

)

Flow of V:

Cusp with mildly varying S

Cone-Cusp: V is not smooth on T ∗X ,need to blow up its singularities incT ∗X


Remarks, outlook

For general singularities, need to understand degeneracies of resolvedmetrics to higher order

Expected picture for isolated surface singularity:(if tangent cone is not a single ray)

Related problem: Geodesics almost hitting the singularity

Apply methods and results to wave propagation

Reference: arXiv:1205.4554


Remarks, outlook

For general singularities, need to understand degeneracies of resolvedmetrics to higher order

Expected picture for isolated surface singularity:(if tangent cone is not a single ray)

Related problem: Geodesics almost hitting the singularity

Apply methods and results to wave propagation

Reference: arXiv:1205.4554


Documents

Geodesics, singularities, and blow-ups · Content 1 Singular Spaces 2 Problems in the geometry and analysis of singular spaces 3 Resolution of singularities, blow-up 4 Geodesics: