Periodic Reporting for period 4 - CURVATURE (Optimal transport techniques in the geometric analysis of spaces with curvature bounds)
Reporting period: 2022-10-01 to 2024-03-31
The unifying goal of the CURVATURE project is to develop new strategies and tools in order to attack fundamental questions in the theory of smooth and non-smooth spaces satisfying (mainly
Ricci) curvature bounds.
The program involves analysis and geometry, with strong connections to probability and mathematical physics. The problems have been (and will be) attacked by an innovative merging
of geometric analysis and optimal transport techniques.
The project is composed of three inter-connected themes which, altogether, give a unique unifying insight of smooth and non-smooth structures satisfying Ricci Curvature bounds.
Below, the goal of each theme is briefly recalled.
Theme I investigates the structure of non smooth spaces with Ricci curvature bounded below in a synthetic sense. More precisely the focus is on the class of RCD(K,N) spaces. The class of RCD(K,N) metric measure spaces includes as remarkable sub-classes: measured Gromov-Hausdorff limits of smooth manifolds with lower Ricci curvature bounds and finite dimensional Alexandrov spaces with lower sectional curvature bounds. This theme has been investigated both by the PI and by the PDRA Daniele Semola (with collaborators),
Theme II is devoted to geometric and functional inequalities for both smooth and non-smooth, finite and infinite dimensional spaces satisfying Ricci curvature lower bounds. Theme II also includes technical developements of Optimal Transport for vector valued measures, a topic which is strictly connected to the tools used in the theme (i.e. the localisation technique).
This theme has been investigated both by the PI and by the PDRAs Daniele Semola and Krzysztof Ciosmak (with collaborators).
Theme III investigates optimal transport in the Lorentzian setting, where the Ricci curvature plays a key role in Einstein's equations of general relativity. The ultimate goal is to get a weak formulation of Einstein's
equations in a singular setting. This theme has been investigated by the PI with collaborators.
Ricci) curvature bounds.
The program involves analysis and geometry, with strong connections to probability and mathematical physics. The problems have been (and will be) attacked by an innovative merging
of geometric analysis and optimal transport techniques.
The project is composed of three inter-connected themes which, altogether, give a unique unifying insight of smooth and non-smooth structures satisfying Ricci Curvature bounds.
Below, the goal of each theme is briefly recalled.
Theme I investigates the structure of non smooth spaces with Ricci curvature bounded below in a synthetic sense. More precisely the focus is on the class of RCD(K,N) spaces. The class of RCD(K,N) metric measure spaces includes as remarkable sub-classes: measured Gromov-Hausdorff limits of smooth manifolds with lower Ricci curvature bounds and finite dimensional Alexandrov spaces with lower sectional curvature bounds. This theme has been investigated both by the PI and by the PDRA Daniele Semola (with collaborators),
Theme II is devoted to geometric and functional inequalities for both smooth and non-smooth, finite and infinite dimensional spaces satisfying Ricci curvature lower bounds. Theme II also includes technical developements of Optimal Transport for vector valued measures, a topic which is strictly connected to the tools used in the theme (i.e. the localisation technique).
This theme has been investigated both by the PI and by the PDRAs Daniele Semola and Krzysztof Ciosmak (with collaborators).
Theme III investigates optimal transport in the Lorentzian setting, where the Ricci curvature plays a key role in Einstein's equations of general relativity. The ultimate goal is to get a weak formulation of Einstein's
equations in a singular setting. This theme has been investigated by the PI with collaborators.
We quickly describe the content of the most relevant papers produced in the context of the corresponding theme.
THEME I.
1) On the topology and the boundary of N-dimensional RCD(K,N) spaces, by Vitali Kapovitch, Andrea Mondino, published in "Geometry & Topology".
Brief summary: We establish topological regularity and stability of N-dimensional RCD(K,N) spaces (up to a small singular set), also called non-collapsed RCD(K,N) in the literature. We also introduce the notion of a boundary of such spaces and study its properties, including its behavior under Gromov-Hausdorff convergence.
2) Boundary regularity and stability for spaces with Ricci bounded below, by Elia Bruè, Aaron Naber, Daniele Semola, preprint arXiv:2011.08383.
Brief summary: This paper studies the structure and stability of boundaries in noncollapsed RCD(K,N) spaces, that is, metric-measure spaces with Ricci curvature bounded below. The main structural result is that the boundary ∂X is homeomorphic to a manifold away from a set of codimension 2, and is N−1 rectifiable. These results are new even for Gromov-Hausdorff limits of smooth manifolds with boundary, and require new techniques.
THEME II.
1) New formulas for the Laplacian of distance functions and applications, by Fabio Cavalletti, Andrea Mondino, published by "Analysis and PDE".
Brief summary: We prove an exact representation formula for the Laplacian of the distance (and more generally for an arbitrary 1-Lipschitz function) in the framework of metric measure spaces satisfying Ricci curvature lower bounds. Such a representation formula makes apparent the classical upper bounds and also some new lower bounds, together with a precise description of the singular part.
2) Leaves decompositions in Euclidean spaces, by Krzysztof Ciosmak, published by "Journal de Mathématiques Pures et Appliquées ".
Brief summary: We partly extend the localisation technique to the multiple constraints setting. For a given 1-Lipschitz map u:R^n→R^m, m≤n, we define and prove the existence of a partition of R^n, up to a set of Lebesgue measure zero, into maximal closed convex sets such that the restriction of u is an isometry on these sets. We consider a disintegration, with respect to this partition, of a log-concave measure. We prove that for almost every set of the partition of dimension m, the associated conditional measure is log-concave. This partially confirms a conjecture of Klartag.
THEME III.
1) An optimal transport formulation of the Einstein equations of general relativity, by Andrea Mondino, Stefan Suhr, accepted for publication by "Journal of the European Math. Society".
Brief summary: The goal of the paper is to give an optimal transport formulation of the full Einstein equations of general relativity, linking the (Ricci) curvature of a space-time with the cosmological constant and the energy-momentum tensor. The result gives a new connection between general relativity and optimal transport; moreover it gives a mathematical reinforcement of the strong link between general relativity and thermodynamics/information theory that emerged in the physics literature of the last years.
2) Optimal transport in Lorentzian synthetic spaces, synthetic timelike Ricci curvature lower bounds and applications, by Fabio Cavalletti, Andrea Mondino, preprint arXiv:2004.08934.
Brief summary:The goal of the present work is three-fold. The first goal is to set foundational results on optimal transport in Lorentzian synthetic spaces (several results are new even for smooth Lorentzian manifolds). The second one is to give a synthetic notion of "timelike Ricci curvature bounded below and dimension bounded above" for a Lorentzian space using optimal transport. This notion is proved to be stable under a suitable weak convergence of Lorentzian synthetic spaces. The third goal is to draw applications, most notably extending volume comparisons and Hawking singularity Theorem (in sharp form) to the synthetic setting.
THEME I.
1) On the topology and the boundary of N-dimensional RCD(K,N) spaces, by Vitali Kapovitch, Andrea Mondino, published in "Geometry & Topology".
Brief summary: We establish topological regularity and stability of N-dimensional RCD(K,N) spaces (up to a small singular set), also called non-collapsed RCD(K,N) in the literature. We also introduce the notion of a boundary of such spaces and study its properties, including its behavior under Gromov-Hausdorff convergence.
2) Boundary regularity and stability for spaces with Ricci bounded below, by Elia Bruè, Aaron Naber, Daniele Semola, preprint arXiv:2011.08383.
Brief summary: This paper studies the structure and stability of boundaries in noncollapsed RCD(K,N) spaces, that is, metric-measure spaces with Ricci curvature bounded below. The main structural result is that the boundary ∂X is homeomorphic to a manifold away from a set of codimension 2, and is N−1 rectifiable. These results are new even for Gromov-Hausdorff limits of smooth manifolds with boundary, and require new techniques.
THEME II.
1) New formulas for the Laplacian of distance functions and applications, by Fabio Cavalletti, Andrea Mondino, published by "Analysis and PDE".
Brief summary: We prove an exact representation formula for the Laplacian of the distance (and more generally for an arbitrary 1-Lipschitz function) in the framework of metric measure spaces satisfying Ricci curvature lower bounds. Such a representation formula makes apparent the classical upper bounds and also some new lower bounds, together with a precise description of the singular part.
2) Leaves decompositions in Euclidean spaces, by Krzysztof Ciosmak, published by "Journal de Mathématiques Pures et Appliquées ".
Brief summary: We partly extend the localisation technique to the multiple constraints setting. For a given 1-Lipschitz map u:R^n→R^m, m≤n, we define and prove the existence of a partition of R^n, up to a set of Lebesgue measure zero, into maximal closed convex sets such that the restriction of u is an isometry on these sets. We consider a disintegration, with respect to this partition, of a log-concave measure. We prove that for almost every set of the partition of dimension m, the associated conditional measure is log-concave. This partially confirms a conjecture of Klartag.
THEME III.
1) An optimal transport formulation of the Einstein equations of general relativity, by Andrea Mondino, Stefan Suhr, accepted for publication by "Journal of the European Math. Society".
Brief summary: The goal of the paper is to give an optimal transport formulation of the full Einstein equations of general relativity, linking the (Ricci) curvature of a space-time with the cosmological constant and the energy-momentum tensor. The result gives a new connection between general relativity and optimal transport; moreover it gives a mathematical reinforcement of the strong link between general relativity and thermodynamics/information theory that emerged in the physics literature of the last years.
2) Optimal transport in Lorentzian synthetic spaces, synthetic timelike Ricci curvature lower bounds and applications, by Fabio Cavalletti, Andrea Mondino, preprint arXiv:2004.08934.
Brief summary:The goal of the present work is three-fold. The first goal is to set foundational results on optimal transport in Lorentzian synthetic spaces (several results are new even for smooth Lorentzian manifolds). The second one is to give a synthetic notion of "timelike Ricci curvature bounded below and dimension bounded above" for a Lorentzian space using optimal transport. This notion is proved to be stable under a suitable weak convergence of Lorentzian synthetic spaces. The third goal is to draw applications, most notably extending volume comparisons and Hawking singularity Theorem (in sharp form) to the synthetic setting.
Below, we give an overview of the progress beyond the state of the art and on the expected results until the end of the project for each theme.
Theme I. Study the properties and the sharp regularity of minimal boundaries in the framework of RCD(K,N) spaces.
Theme II. Develop a unifying approach to Ricci Curvature lower bounds via optimal transport for Riemannian and sub-Riemannian spaces, including both smooth and non-smooth structures. This can be seen as an answer to a long-standing "great unification problem" in the field. This is a challenging project with high risk/high gain, and we plan to make good progress by the end of the granting period.
Theme III. Establish geometric applications of paper 2) of Theme III. A particularly interesting application would be an isoperimetric inequality for (possibly non-smooth) space-times with timelike Ricci Curvature bounded below.
Theme I. Study the properties and the sharp regularity of minimal boundaries in the framework of RCD(K,N) spaces.
Theme II. Develop a unifying approach to Ricci Curvature lower bounds via optimal transport for Riemannian and sub-Riemannian spaces, including both smooth and non-smooth structures. This can be seen as an answer to a long-standing "great unification problem" in the field. This is a challenging project with high risk/high gain, and we plan to make good progress by the end of the granting period.
Theme III. Establish geometric applications of paper 2) of Theme III. A particularly interesting application would be an isoperimetric inequality for (possibly non-smooth) space-times with timelike Ricci Curvature bounded below.