Periodic Reporting for period 5 - CURVATURE (Optimal transport techniques in the geometric analysis of spaces with curvature bounds)
Reporting period: 2024-04-01 to 2025-04-30
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 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), and in sub-Riemannian settings.
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 provide a new non-smooth geometric and analytical setting for Einstein's theory of gravity.
This theme has been investigated by the PI and by the PDRA Clemens Saemann (with collaborators).
Ricci) curvature bounds.
The program involves analysis and geometry, with strong connections to probability and mathematical physics. The problems have been 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), and in sub-Riemannian settings.
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 provide a new non-smooth geometric and analytical setting for Einstein's theory of gravity.
This theme has been investigated by the PI and by the PDRA Clemens Saemann (with collaborators).
THEME I:
Brué, Naber, Semola (Annals of Math, 2025): Milnor conjectured that the fundamental group of a complete Riemannian manifold with Ric ≥ 0 is finitely generated. This paper disproves it with a 7D example having π₁ ≈ ℚ/ℤ.
Kapovitch, Mondino (Geometry & Topology): Studied topological regularity of RCD(K,N) spaces. Defined boundary in this setting and proved stability under Gromov-Hausdorff convergence.
Brué, Naber, Semola (Inventiones Math): Proved that boundaries in noncollapsed RCD(K,N) spaces are (N-1)-rectifiable and homeomorphic to manifolds away from codim-2 singular sets.
Brué, Mondino, Semola (GAFA): Solved a conjecture on metric measure boundaries, proving vanishing measure on RCD(K,N) spaces without boundary. Clarifies structure of infinite geodesics in Alexandrov spaces.
Mondino, Semola (Memoirs AMS): Developed pointwise and viscosity Laplacian bounds on RCD(K,N) spaces. Proved PDE principles for perimeter-minimizing sets and regularity of their boundaries.
Mondino, Semola (AJM): Proved Lipschitz continuity of harmonic maps from RCD(K,N) to CAT(0) spaces. Established synthetic Bochner-Eells-Sampson inequality, generalizing classical smooth results.
THEME II:
Cavalletti, Mondino (Analysis & PDE): Provided precise formulas for the Laplacian of distance functions on RCD(K,N) spaces, capturing both smooth and singular parts.
Ciosmak (JMPA): Extended localization to vector-valued 1-Lipschitz maps. Constructed partitions of ℝⁿ into convex sets and showed conditional log-concavity for log-concave measures. Partial confirmation of Klartag's conjecture.
Ciosmak (Calc. Var. PDE): Developed optimal transport theory for vector measures. Disproved a conjecture by Klartag with a counterexample, but proved it under AC marginals. Generalized Kantorovich duality.
Barilari, Mondino, Rizzi (Memoirs AMS): Proposed a unified synthetic curvature theory covering both Riemannian and sub-Riemannian geometries using gauge metric measure spaces.
Kristaly, Mondino (Proc. LMS): Proved sharp isoperimetric inequality for clamped plate eigenvalues in RCD(0,N) spaces (close to N=2,3). Analyzed rigidity and stability using volume ratios and Bessel functions.
THEME III:
Mondino, Suhr (JEMS): Formulated Einstein equations via optimal transport. Linked Ricci curvature, energy-momentum tensor, and cosmological constant through entropy convexity.
Cavalletti, Mondino (Cambridge J. Math): Developed optimal transport theory in Lorentzian synthetic spaces. Defined timelike Ricci lower bounds and extended Hawking singularity theorem to Lorentzian length spaces.
Cavalletti, Manini, Mondino (Comm. Math. Phys): Studied optimal transport on null hypersurfaces. Gave a transport-based characterization of the Null Energy Condition. Proved sharp and rigid versions of Hawking area theorem.
Cavalletti, Manini, Mondino (preprint): Introduced synthetic null hypersurfaces. Defined entropy-based null energy condition stable under convergence. Extended Penrose singularity theorem to continuous spacetimes.
Mondino, Saemann (preprint): Defined Lorentzian Gromov-Hausdorff convergence via causal diamonds. Proved a compactness theorem and connected it to causal set theory.
These studies collectively advance the synthetic treatment of curvature, topology, and boundary regularity in both Riemannian and Lorentzian, smooth and non-smooth settings. They merge deep insights from geometric analysis, PDEs, and relativity through the lens of optimal transport and metric measure spaces.
Brué, Naber, Semola (Annals of Math, 2025): Milnor conjectured that the fundamental group of a complete Riemannian manifold with Ric ≥ 0 is finitely generated. This paper disproves it with a 7D example having π₁ ≈ ℚ/ℤ.
Kapovitch, Mondino (Geometry & Topology): Studied topological regularity of RCD(K,N) spaces. Defined boundary in this setting and proved stability under Gromov-Hausdorff convergence.
Brué, Naber, Semola (Inventiones Math): Proved that boundaries in noncollapsed RCD(K,N) spaces are (N-1)-rectifiable and homeomorphic to manifolds away from codim-2 singular sets.
Brué, Mondino, Semola (GAFA): Solved a conjecture on metric measure boundaries, proving vanishing measure on RCD(K,N) spaces without boundary. Clarifies structure of infinite geodesics in Alexandrov spaces.
Mondino, Semola (Memoirs AMS): Developed pointwise and viscosity Laplacian bounds on RCD(K,N) spaces. Proved PDE principles for perimeter-minimizing sets and regularity of their boundaries.
Mondino, Semola (AJM): Proved Lipschitz continuity of harmonic maps from RCD(K,N) to CAT(0) spaces. Established synthetic Bochner-Eells-Sampson inequality, generalizing classical smooth results.
THEME II:
Cavalletti, Mondino (Analysis & PDE): Provided precise formulas for the Laplacian of distance functions on RCD(K,N) spaces, capturing both smooth and singular parts.
Ciosmak (JMPA): Extended localization to vector-valued 1-Lipschitz maps. Constructed partitions of ℝⁿ into convex sets and showed conditional log-concavity for log-concave measures. Partial confirmation of Klartag's conjecture.
Ciosmak (Calc. Var. PDE): Developed optimal transport theory for vector measures. Disproved a conjecture by Klartag with a counterexample, but proved it under AC marginals. Generalized Kantorovich duality.
Barilari, Mondino, Rizzi (Memoirs AMS): Proposed a unified synthetic curvature theory covering both Riemannian and sub-Riemannian geometries using gauge metric measure spaces.
Kristaly, Mondino (Proc. LMS): Proved sharp isoperimetric inequality for clamped plate eigenvalues in RCD(0,N) spaces (close to N=2,3). Analyzed rigidity and stability using volume ratios and Bessel functions.
THEME III:
Mondino, Suhr (JEMS): Formulated Einstein equations via optimal transport. Linked Ricci curvature, energy-momentum tensor, and cosmological constant through entropy convexity.
Cavalletti, Mondino (Cambridge J. Math): Developed optimal transport theory in Lorentzian synthetic spaces. Defined timelike Ricci lower bounds and extended Hawking singularity theorem to Lorentzian length spaces.
Cavalletti, Manini, Mondino (Comm. Math. Phys): Studied optimal transport on null hypersurfaces. Gave a transport-based characterization of the Null Energy Condition. Proved sharp and rigid versions of Hawking area theorem.
Cavalletti, Manini, Mondino (preprint): Introduced synthetic null hypersurfaces. Defined entropy-based null energy condition stable under convergence. Extended Penrose singularity theorem to continuous spacetimes.
Mondino, Saemann (preprint): Defined Lorentzian Gromov-Hausdorff convergence via causal diamonds. Proved a compactness theorem and connected it to causal set theory.
These studies collectively advance the synthetic treatment of curvature, topology, and boundary regularity in both Riemannian and Lorentzian, smooth and non-smooth settings. They merge deep insights from geometric analysis, PDEs, and relativity through the lens of optimal transport and metric measure spaces.