AnalyticalSynthetic Geometry

AimsMethods

Further plansReferences

Curvature bounds in low-regularity geometry

Nathalie Tassotti

Presentation of Doctoral Thesis ProjectAdvisor: Privatdoz. Dr. James D.E. Grant

14 October, 2011

Nathalie Tassotti Curvature bounds in low-regularity geometry

AnalyticalSynthetic Geometry

AimsMethods

Further plansReferences

Outline

Curvature bounds in Riemannian geometry

Possible approaches

Aims

Methods

Further plans

Nathalie Tassotti Curvature bounds in low-regularity geometry

AnalyticalSynthetic Geometry

AimsMethods

Further plansReferences

Outline

Curvature bounds in Riemannian geometry

Possible approaches

Aims

Methods

Further plans

Nathalie Tassotti Curvature bounds in low-regularity geometry

AnalyticalSynthetic Geometry

AimsMethods

Further plansReferences

Outline

Curvature bounds in Riemannian geometry

Possible approaches

Aims

Methods

Further plans

Nathalie Tassotti Curvature bounds in low-regularity geometry

AnalyticalSynthetic Geometry

AimsMethods

Further plansReferences

Outline

Curvature bounds in Riemannian geometry

Possible approaches

Aims

Methods

Further plans

Nathalie Tassotti Curvature bounds in low-regularity geometry

AnalyticalSynthetic Geometry

AimsMethods

Further plansReferences

Outline

Curvature bounds in Riemannian geometry

Possible approaches

Aims

Methods

Further plans

Nathalie Tassotti Curvature bounds in low-regularity geometry

AnalyticalSynthetic Geometry

AimsMethods

Further plansReferences

Curvature bounds in Riemannian geometry

Classical Riemannian geometry:

study Riemannian metrics with curvature bounds.

theorems: Myers’s theorem, Cartan–Hadamard theorem,Bishop–Gromov relative volume comparison theorem, spheretheorem, Toponogov comparison theorem

’Problem’: proofs use the exponential map, requires themetric to be at least C 2.

In C 1,1 still possible to find unique solutions to geodesicequations, not possible for lower regularity

Nathalie Tassotti Curvature bounds in low-regularity geometry

AnalyticalSynthetic Geometry

AimsMethods

Further plansReferences

Curvature bounds in Riemannian geometry

Classical Riemannian geometry:

study Riemannian metrics with curvature bounds.

theorems: Myers’s theorem, Cartan–Hadamard theorem,Bishop–Gromov relative volume comparison theorem, spheretheorem, Toponogov comparison theorem

’Problem’: proofs use the exponential map, requires themetric to be at least C 2.

In C 1,1 still possible to find unique solutions to geodesicequations, not possible for lower regularity

Nathalie Tassotti Curvature bounds in low-regularity geometry

AnalyticalSynthetic Geometry

AimsMethods

Further plansReferences

Curvature bounds in Riemannian geometry

Classical Riemannian geometry:

study Riemannian metrics with curvature bounds.

theorems: Myers’s theorem, Cartan–Hadamard theorem,Bishop–Gromov relative volume comparison theorem, spheretheorem, Toponogov comparison theorem

’Problem’: proofs use the exponential map, requires themetric to be at least C 2.

In C 1,1 still possible to find unique solutions to geodesicequations, not possible for lower regularity

Nathalie Tassotti Curvature bounds in low-regularity geometry

AnalyticalSynthetic Geometry

AimsMethods

Further plansReferences

Curvature bounds in Riemannian geometry

Classical Riemannian geometry:

study Riemannian metrics with curvature bounds.

theorems: Myers’s theorem, Cartan–Hadamard theorem,Bishop–Gromov relative volume comparison theorem, spheretheorem, Toponogov comparison theorem

’Problem’: proofs use the exponential map, requires themetric to be at least C 2.

In C 1,1 still possible to find unique solutions to geodesicequations, not possible for lower regularity

Nathalie Tassotti Curvature bounds in low-regularity geometry

AnalyticalSynthetic Geometry

AimsMethods

Further plansReferences

Approaches

Analytical

Synthetic geometry

Nathalie Tassotti Curvature bounds in low-regularity geometry

AnalyticalSynthetic Geometry

AimsMethods

Further plansReferences

Approaches

Analytical

Synthetic geometry

Nathalie Tassotti Curvature bounds in low-regularity geometry

AnalyticalSynthetic Geometry

AimsMethods

Further plansReferences

Analytical approach

Distributional tensor fields

Extensions of standard elements of differential geometry in acoordinate-invariant way (LeFloch–Mardare)

Algebras of generalized functions (Steinbauer–Vickers)

Obtain description of the curvature for metrics which havecurvature well-defined as a distribution/lie in a particularSobolev space.

Question: minimal regularity conditions?

Nathalie Tassotti Curvature bounds in low-regularity geometry

AnalyticalSynthetic Geometry

AimsMethods

Further plansReferences

Analytical approach

Distributional tensor fields

Extensions of standard elements of differential geometry in acoordinate-invariant way (LeFloch–Mardare)

Algebras of generalized functions (Steinbauer–Vickers)

Obtain description of the curvature for metrics which havecurvature well-defined as a distribution/lie in a particularSobolev space.

Question: minimal regularity conditions?

Nathalie Tassotti Curvature bounds in low-regularity geometry

AnalyticalSynthetic Geometry

AimsMethods

Further plansReferences

Analytical approach

Distributional tensor fields

Extensions of standard elements of differential geometry in acoordinate-invariant way (LeFloch–Mardare)

Algebras of generalized functions (Steinbauer–Vickers)

Obtain description of the curvature for metrics which havecurvature well-defined as a distribution/lie in a particularSobolev space.

Question: minimal regularity conditions?

Nathalie Tassotti Curvature bounds in low-regularity geometry

AnalyticalSynthetic Geometry

AimsMethods

Further plansReferences

Analytical approach

Distributional tensor fields

Extensions of standard elements of differential geometry in acoordinate-invariant way (LeFloch–Mardare)

Algebras of generalized functions (Steinbauer–Vickers)

Obtain description of the curvature for metrics which havecurvature well-defined as a distribution/lie in a particularSobolev space.

Question: minimal regularity conditions?

Nathalie Tassotti Curvature bounds in low-regularity geometry

AnalyticalSynthetic Geometry

AimsMethods

Further plansReferences

Analytical approach

Distributional tensor fields

Extensions of standard elements of differential geometry in acoordinate-invariant way (LeFloch–Mardare)

Algebras of generalized functions (Steinbauer–Vickers)

Obtain description of the curvature for metrics which havecurvature well-defined as a distribution/lie in a particularSobolev space.

Question: minimal regularity conditions?

Nathalie Tassotti Curvature bounds in low-regularity geometry

AnalyticalSynthetic Geometry

AimsMethods

Further plansReferences

IdeaAlexandrov spaces

Idea

Take a classical theorem characterizing a manifold thatsatisfies a curvature bound and use the consequences as adefinition of a curvature bound in a more general context.

Example: Toponogov comparison theorem

Obtain description of the curvature-bounds for spaces whichare not necessarily manifolds

Nathalie Tassotti Curvature bounds in low-regularity geometry

AnalyticalSynthetic Geometry

AimsMethods

Further plansReferences

IdeaAlexandrov spaces

Idea

Take a classical theorem characterizing a manifold thatsatisfies a curvature bound and use the consequences as adefinition of a curvature bound in a more general context.

Example: Toponogov comparison theorem

Obtain description of the curvature-bounds for spaces whichare not necessarily manifolds

Nathalie Tassotti Curvature bounds in low-regularity geometry

AnalyticalSynthetic Geometry

AimsMethods

Further plansReferences

IdeaAlexandrov spaces

Idea

Take a classical theorem characterizing a manifold thatsatisfies a curvature bound and use the consequences as adefinition of a curvature bound in a more general context.

Example: Toponogov comparison theorem

Obtain description of the curvature-bounds for spaces whichare not necessarily manifolds

Nathalie Tassotti Curvature bounds in low-regularity geometry

AnalyticalSynthetic Geometry

AimsMethods

Further plansReferences

IdeaAlexandrov spaces

Alexandrov spaces with curvature bounded below

Definition

A metric space (X , d) whose metric is intrinsic, i.e.

d(x , y) := inf{L(γ)|γ is an admissible curve connecting x and y},

is called a length space.

Examples: Metric spaces (X , d), Riemannian manifolds(M, g), etc.

Definition

An Alexandrov space with curvature bounded below is a lengthspace which satisfies the Toponogov triangle comparison test.

Nathalie Tassotti Curvature bounds in low-regularity geometry

AnalyticalSynthetic Geometry

AimsMethods

Further plansReferences

IdeaAlexandrov spaces

Alexandrov spaces with curvature bounded below

Definition

A metric space (X , d) whose metric is intrinsic, i.e.

d(x , y) := inf{L(γ)|γ is an admissible curve connecting x and y},

is called a length space.

Examples: Metric spaces (X , d), Riemannian manifolds(M, g), etc.

Definition

An Alexandrov space with curvature bounded below is a lengthspace which satisfies the Toponogov triangle comparison test.

Nathalie Tassotti Curvature bounds in low-regularity geometry

AnalyticalSynthetic Geometry

AimsMethods

Further plansReferences

Aims

Aims

analyse connections between approaches

work out minimal analytical conditions for well-defined notionof curvature/geodesics in appropriate function spaces

possible enhancement in regularity by imposing curvaturebounds

do these metrics define synthetic-geometric structures thatsatisfy the corresponding curvature bound?

Nathalie Tassotti Curvature bounds in low-regularity geometry

AnalyticalSynthetic Geometry

AimsMethods

Further plansReferences

Aims

Aims

analyse connections between approaches

work out minimal analytical conditions for well-defined notionof curvature/geodesics in appropriate function spaces

possible enhancement in regularity by imposing curvaturebounds

do these metrics define synthetic-geometric structures thatsatisfy the corresponding curvature bound?

Nathalie Tassotti Curvature bounds in low-regularity geometry

AnalyticalSynthetic Geometry

AimsMethods

Further plansReferences

Aims

Aims

analyse connections between approaches

work out minimal analytical conditions for well-defined notionof curvature/geodesics in appropriate function spaces

possible enhancement in regularity by imposing curvaturebounds

do these metrics define synthetic-geometric structures thatsatisfy the corresponding curvature bound?

Nathalie Tassotti Curvature bounds in low-regularity geometry

AnalyticalSynthetic Geometry

AimsMethods

Further plansReferences

Aims

Aims

analyse connections between approaches

work out minimal analytical conditions for well-defined notionof curvature/geodesics in appropriate function spaces

possible enhancement in regularity by imposing curvaturebounds

do these metrics define synthetic-geometric structures thatsatisfy the corresponding curvature bound?

Nathalie Tassotti Curvature bounds in low-regularity geometry

AnalyticalSynthetic Geometry

AimsMethods

Further plansReferences

AnalyticalSynthetic geometrySmoothing the metricPossible conclusions

Method

Start with (M, g), M a smooth manifold, g sufficiently regularfor the curvature tensor to be well-defined as a distribution

Assumption: ∃ K such that

R(X ,Y ,X ,Y )− K(g(X ,X )g(Y ,Y )− g(X ,Y )2

)≥ 0

in the sense of distributions, for all X ,Y ∈ X(M).

Approximate g by smooth metrics, {gn}n∈N which satisfy, inthe classical sense, a curvature bound of the form

Rgn(X ,Y ,X ,Y )− Kn

(gn(X ,X )gn(Y ,Y )− gn(X ,Y )2

)≥ 0

Kn are constants and Kn → K as n→∞.

Nathalie Tassotti Curvature bounds in low-regularity geometry

AnalyticalSynthetic Geometry

AimsMethods

Further plansReferences

AnalyticalSynthetic geometrySmoothing the metricPossible conclusions

Method

Start with (M, g), M a smooth manifold, g sufficiently regularfor the curvature tensor to be well-defined as a distribution

Assumption: ∃ K such that

R(X ,Y ,X ,Y )− K(g(X ,X )g(Y ,Y )− g(X ,Y )2

)≥ 0

in the sense of distributions, for all X ,Y ∈ X(M).

Approximate g by smooth metrics, {gn}n∈N which satisfy, inthe classical sense, a curvature bound of the form

Rgn(X ,Y ,X ,Y )− Kn

(gn(X ,X )gn(Y ,Y )− gn(X ,Y )2

)≥ 0

Kn are constants and Kn → K as n→∞.

Nathalie Tassotti Curvature bounds in low-regularity geometry

AnalyticalSynthetic Geometry

AimsMethods

Further plansReferences

AnalyticalSynthetic geometrySmoothing the metricPossible conclusions

Method

Start with (M, g), M a smooth manifold, g sufficiently regularfor the curvature tensor to be well-defined as a distribution

Assumption: ∃ K such that

R(X ,Y ,X ,Y )− K(g(X ,X )g(Y ,Y )− g(X ,Y )2

)≥ 0

in the sense of distributions, for all X ,Y ∈ X(M).

Approximate g by smooth metrics, {gn}n∈N which satisfy, inthe classical sense, a curvature bound of the form

Rgn(X ,Y ,X ,Y )− Kn

(gn(X ,X )gn(Y ,Y )− gn(X ,Y )2

)≥ 0

Kn are constants and Kn → K as n→∞.

Nathalie Tassotti Curvature bounds in low-regularity geometry

AnalyticalSynthetic Geometry

AimsMethods

Further plansReferences

AnalyticalSynthetic geometrySmoothing the metricPossible conclusions

Toponogov comparison theorem: (M, dgn) define A. spaceswith curvature ≥ Kn.

(M, dgn) converge to (M, dg) in the Gromov–Hausdorfftopology.

(M, dg) is an A. space with curvature bounded below by K ′

for all K ′ < K .

(M, g) defines, via the distance function dg, an A. space withcurvature bounded below by K ′, for all K ′ < K .

Nathalie Tassotti Curvature bounds in low-regularity geometry

AnalyticalSynthetic Geometry

AimsMethods

Further plansReferences

AnalyticalSynthetic geometrySmoothing the metricPossible conclusions

Toponogov comparison theorem: (M, dgn) define A. spaceswith curvature ≥ Kn.

(M, dgn) converge to (M, dg) in the Gromov–Hausdorfftopology.

(M, dg) is an A. space with curvature bounded below by K ′

for all K ′ < K .

(M, g) defines, via the distance function dg, an A. space withcurvature bounded below by K ′, for all K ′ < K .

Nathalie Tassotti Curvature bounds in low-regularity geometry

AnalyticalSynthetic Geometry

AimsMethods

Further plansReferences

AnalyticalSynthetic geometrySmoothing the metricPossible conclusions

Toponogov comparison theorem: (M, dgn) define A. spaceswith curvature ≥ Kn.

(M, dgn) converge to (M, dg) in the Gromov–Hausdorfftopology.

(M, dg) is an A. space with curvature bounded below by K ′

for all K ′ < K .

(M, g) defines, via the distance function dg, an A. space withcurvature bounded below by K ′, for all K ′ < K .

Nathalie Tassotti Curvature bounds in low-regularity geometry

AnalyticalSynthetic Geometry

AimsMethods

Further plansReferences

AnalyticalSynthetic geometrySmoothing the metricPossible conclusions

Toponogov comparison theorem: (M, dgn) define A. spaceswith curvature ≥ Kn.

(M, dgn) converge to (M, dg) in the Gromov–Hausdorfftopology.

(M, dg) is an A. space with curvature bounded below by K ′

for all K ′ < K .

(M, g) defines, via the distance function dg, an A. space withcurvature bounded below by K ′, for all K ′ < K .

Nathalie Tassotti Curvature bounds in low-regularity geometry

AnalyticalSynthetic Geometry

AimsMethods

Further plansReferences

AnalyticalSynthetic geometrySmoothing the metricPossible conclusions

Smoothing the metric

The sense in which the sequence of smooth approximations gn

approach g is determined by the regularity of g.

What are the minimal regularity conditions on g, s.t. thereexist approximating gn?

Regularity improvement by curvature bounds?

Nathalie Tassotti Curvature bounds in low-regularity geometry

AnalyticalSynthetic Geometry

AimsMethods

Further plansReferences

AnalyticalSynthetic geometrySmoothing the metricPossible conclusions

Smoothing the metric

The sense in which the sequence of smooth approximations gn

approach g is determined by the regularity of g.

What are the minimal regularity conditions on g, s.t. thereexist approximating gn?

Regularity improvement by curvature bounds?

Nathalie Tassotti Curvature bounds in low-regularity geometry

AnalyticalSynthetic Geometry

AimsMethods

Further plansReferences

AnalyticalSynthetic geometrySmoothing the metricPossible conclusions

Smoothing the metric

The sense in which the sequence of smooth approximations gn

approach g is determined by the regularity of g.

What are the minimal regularity conditions on g, s.t. thereexist approximating gn?

Regularity improvement by curvature bounds?

Nathalie Tassotti Curvature bounds in low-regularity geometry

AnalyticalSynthetic Geometry

AimsMethods

Further plansReferences

AnalyticalSynthetic geometrySmoothing the metricPossible conclusions

A class of low-regularity metrics with curvature bounded below inan appropriate analytical sense have properties of A. spaces withcurvature bounded below:

non-branching of geodesics

splitting theorems

volume monotonicity theorems

bounds on Hausdorff-dimension of the singular set

Nathalie Tassotti Curvature bounds in low-regularity geometry

AnalyticalSynthetic Geometry

AimsMethods

Further plansReferences

AnalyticalSynthetic geometrySmoothing the metricPossible conclusions

A class of low-regularity metrics with curvature bounded below inan appropriate analytical sense have properties of A. spaces withcurvature bounded below:

non-branching of geodesics

splitting theorems

volume monotonicity theorems

bounds on Hausdorff-dimension of the singular set

Nathalie Tassotti Curvature bounds in low-regularity geometry

AnalyticalSynthetic Geometry

AimsMethods

Further plansReferences

AnalyticalSynthetic geometrySmoothing the metricPossible conclusions

A class of low-regularity metrics with curvature bounded below inan appropriate analytical sense have properties of A. spaces withcurvature bounded below:

non-branching of geodesics

splitting theorems

volume monotonicity theorems

bounds on Hausdorff-dimension of the singular set

Nathalie Tassotti Curvature bounds in low-regularity geometry

AnalyticalSynthetic Geometry

AimsMethods

Further plansReferences

AnalyticalSynthetic geometrySmoothing the metricPossible conclusions

A class of low-regularity metrics with curvature bounded below inan appropriate analytical sense have properties of A. spaces withcurvature bounded below:

non-branching of geodesics

splitting theorems

volume monotonicity theorems

bounds on Hausdorff-dimension of the singular set

Nathalie Tassotti Curvature bounds in low-regularity geometry

AnalyticalSynthetic Geometry

AimsMethods

Further plansReferences

AnalyticalSynthetic geometrySmoothing the metricPossible conclusions

A class of low-regularity metrics with curvature bounded below inan appropriate analytical sense have properties of A. spaces withcurvature bounded below:

non-branching of geodesics

splitting theorems

volume monotonicity theorems

bounds on Hausdorff-dimension of the singular set

Nathalie Tassotti Curvature bounds in low-regularity geometry

AnalyticalSynthetic Geometry

AimsMethods

Further plansReferences

Further plans

Analysis of low-regularity metrics with lower bounds on theRicci-tensor

Conditions to define metric measure spaces with Riccicurvature bounded below in the sense of Lott-Villani, Sturm

Long term goal: study Lorentzian metrics of low regularityand develop a synthetic geometric approach to singularitytheorems in General Relativity

Nathalie Tassotti Curvature bounds in low-regularity geometry

AnalyticalSynthetic Geometry

AimsMethods

Further plansReferences

Further plans

Analysis of low-regularity metrics with lower bounds on theRicci-tensor

Conditions to define metric measure spaces with Riccicurvature bounded below in the sense of Lott-Villani, Sturm

Long term goal: study Lorentzian metrics of low regularityand develop a synthetic geometric approach to singularitytheorems in General Relativity

Nathalie Tassotti Curvature bounds in low-regularity geometry

AnalyticalSynthetic Geometry

AimsMethods

Further plansReferences

Further plans

Analysis of low-regularity metrics with lower bounds on theRicci-tensor

Conditions to define metric measure spaces with Riccicurvature bounded below in the sense of Lott-Villani, Sturm

Long term goal: study Lorentzian metrics of low regularityand develop a synthetic geometric approach to singularitytheorems in General Relativity

Nathalie Tassotti Curvature bounds in low-regularity geometry

AnalyticalSynthetic Geometry

AimsMethods

Further plansReferences

Y. Burago, M. Gromov, and G. Perel′man, A. D.Alexandrov spaces with curvatures bounded below, Russian Math.Surveys, 47 (1992), pp. 1–58.

P. G. LeFloch and C. Mardare, Definition and stability ofLorentzian manifolds with distributional curvature, Port. Math.(N.S.), 64 (2007), pp. 535–573.

J. Lott and C. Villani, Ricci curvature for metric-measurespaces via optimal transport, Ann. of Math. (2), 169 (2009),pp. 903–991.

R. Steinbauer and J. A. Vickers, On the Geroch-Traschenclass of metrics, Classical Quantum Gravity, 26 (2009), pp. 065001.

K.-T. Sturm, On the geometry of metric measure spaces. I & II,Acta Math., 196 (2006), pp. 65–177.

Nathalie Tassotti Curvature bounds in low-regularity geometry