Proves equivalence of two types of Ricci curvature bounds.
problem Equivalence of distributional and synthetic Ricci curvature bounds.
method Analyzes weighted Riemannian manifolds with specific smoothness conditions.
result Proves equivalence of Ricci curvature bounds under given conditions.
Study compares synthetic and distributional Ricci curvature bounds.
problem Comparing synthetic and distributional approaches to lower Ricci curvature bounds.
method Analyzes synthetic via weak displacement convexity and distributional via non-negativity of Ricci-tensor.
result Distributional bounds imply entropy bounds for C1 metrics and vice versa for C1,1 under convergence condition. Unified study of Riemannian and sub-Riemannian geometries with synthetic Ricci curvature bounds.
problem Unified framework for Riemannian and sub-Riemannian geometries.
method Study of gauge metric measure spaces.
result Unified synthetic Ricci curvature lower bounds for both Riemannian and sub-Riemannian structures.
Study inradius collapsed manifolds with lower Ricci curvature bounds, proving properties of their limits.
problem Characterizing limits of inradius collapsed manifolds with lower Ricci curvature bounds.
method Analyzing families of manifolds with specific curvature and boundary conditions, proving properties of the limits.
result Limits of inradius collapsed manifolds have at most two boundary components and a lower Ricci curvature bound.
Survey on curvature bounds and isoperimetric inequalities.
problem Interplay between curvature bounds and isoperimetric problems.
method Survey of classical and recent results.
result Recent developments on curvature bounds and isoperimetric problems.
Study new Ricci bounds for metric measure spaces, preserving properties under time changes.
problem Extend Ricci bounds to non-synthetic spaces and understand their behavior under time changes.
method Introduce distribution-valued lower Ricci bounds BE1(κ,∞), prove equivalence with gradient estimates, and show preservation under time changes. result Distribution-valued Ricci bounds BE1(κ,∞) are preserved under arbitrary time changes and imply sharp gradient estimates. The study proposes conjectures on limit spaces of Riemannian manifolds with Ricci curvature.
problem Understanding the regularity of limit spaces of Riemannian manifolds with Ricci curvature.
method Synthetic treatment of lower bounds on Ricci curvature for metric measure spaces.
result Several conjectures on the regularity of limit spaces.
We prove that in two dimensions the synthetic notions of lower bounds on sectional and on Ricci curvature coincide.
Stability of timelike Ricci bounds in low-regularity spacetimes.
problem Stability of synthetic timelike Ricci curvature bounds under C0-limits. method Constructing smooth approximations and analyzing limiting behavior via Lorentzian optimal transport.
result Impulsive gravitational waves satisfy synthetic timelike Ricci curvature lower bounds.
Study compares nonsmooth spaces with integrable Ricci bounds.
problem Comparing geometric and functional inequalities on nonsmooth spaces.
method Localization method and one-dimensional comparison estimates.
result Extension of comparison principles to nonsmooth settings.
In this paper we investigate Lott-Sturm-Villani's synthetic lower Ricci curvature bound on Riemannian manifolds with boundary. We prove several measure rigidity results for some important functional and geometric inequalities, which completely characterize CD(K,∞) condition and non-collapsed ${\rm CD}(K, …
Motivated by Perelman's Pseudo Locality Theorem for the Ricci flow, we prove that if a Riemannian manifold has Ricci curvature bounded below in a metric ball which moreover has almost maximal volume, then in a smaller ball (in a quantified sense) it holds an almost-euclidean isoperimetric inequality. The result is actu…
The paper explores sharp isoperimetric properties on non-compact spaces with Ricci bounds.
problem Sharp isoperimetric properties on non-compact spaces with Ricci bounds.
method Sharp isoperimetric comparison theorems and asymptotic isoperimetric properties.
result Almost regularity theorems and enhanced functional inequalities.
The paper examines how curvature-dimension conditions transform under time change for diffusions.
problem Transforming curvature-dimension conditions for diffusions under time change.
method Derives precise transformation formulas for synthetic lower Ricci bounds and curvature-dimension conditions.
result Precise formulas for curvature-dimension conditions under time change for diffusions and metric measure spaces.
Survey on gluing constructions under lower curvature bounds.
problem Understanding lower curvature bounds in various geometric contexts.
method Analyzes gluing constructions in smooth and non-smooth settings.
result Provides conjectures and theorems on synthetic lower Ricci curvature bounds.
Sharp inequality for Lorentzian spaces with timelike Ricci bounds.
problem Establishing bounds on achronal hypersurfaces in Lorentzian spaces.
method Optimal transport and synthetic TCDpe(K,N) spaces. result Sharp isoperimetric-type inequality for Lorentzian spaces.
Measure contraction property is a synthetic Ricci curvature lower bound for metric measure spaces. We consider Sasakian manifolds with non-negative Tanaka-Webster Ricci curvature equipped with the metric measure space structure defined by the sub-Riemannian metric and the Popp measure. We show that these spaces satisfy…
Study fine Pólya-Szegő inequalities in metric spaces with applications.
problem Fine Pólya-Szegő rearrangement inequalities in metric spaces.
method Theory of Sobolev and BV functions, synthetic Ricci bounds, isoperimetric inequality.
result New geometric and functional inequalities under Ricci lower bounds.
Measure contraction properties MCP(K,N) are synthetic Ricci curvature lower bounds for metric measure spaces which do not necessarily have smooth structures. It is known that if a Riemannian manifold has dimension N, then MCP(K,N) is equivalent to Ricci curvature bounded below by K. On the other hand, it was ob…
We simplify evaluation of Ollivier-Ricci curvature bounds in hypergraphs.
problem Computational challenges in evaluating Ollivier-Ricci curvature bounds in hypergraphs.
method Simplified approach with linear computational complexity.
result Significant improvements in evaluating Ollivier-Ricci curvature bounds.
We discuss various characterizations of synthetic upper Ricci bounds for metric measure spaces in terms of heat flow, entropy and optimal transport. In particular, we present a characterization in terms of semiconcavity of the entropy along certain Wasserstein geodesics which is stable under convergence of mm-spaces. A…
Uniform heat kernel bounds lead to synthetic Ricci curvature conditions for Lipschitz manifolds.
problem Establishing synthetic Ricci curvature conditions for Lipschitz manifolds.
method Uniform heat kernel bounds and synthetic Ricci curvature conditions.
result Uniform heat kernel bounds lead to synthetic Ricci curvature conditions for Lipschitz manifolds.
Uniform bounds for eigenvalues of Hodge Laplacian on manifolds with lower Ricci curvature.
problem Establishing bounds for eigenvalues of Hodge Laplacian under lower Ricci curvature.
method Using geometric assumptions including lower Ricci curvature, injectivity radius, and diameter bounds.
result Uniform eigenvalue bounds for the Hodge Laplacian and connection Laplacian.
This survey introduces synthetic timelike Ricci curvature bounds in Lorentzian spaces.
problem Synthetic timelike Ricci curvature bounds in non-smooth Lorentzian spaces.
method Optimal transport and entropy tools.
result Synthetic version of Hawking's singularity theorem and synthetic characterisation of Einstein's vacuum equations.
Abstract reviews known and open questions on spaces with lower Ricci bounds.
problem Understanding the structure and regularity of spaces with lower Ricci curvature bounds.
method Review of known results and presentation of open questions.
result Presentation of new open questions in the field.
Along the line of the Yang Conjecture, we give a new estimate on the lower bound of the first non-zero eigenvalue of a closed Riemannian manifold with negative lower bound of Ricci curvature in terms of the in-diameter and the lower bound of Ricci curvature.
Study on cones over metric spaces with curvature bounds.
problem Establishing curvature bounds for cones over metric spaces.
method Developed a localization technique to prove synthetic curvature bounds.
result Riemannian and Lorentzian cones over CD-spaces satisfy MCP and vice versa.
We prove that if (X,d,m) is a metric measure space with m(X)=1 having (in a synthetic sense) Ricci curvature bounded from below by K>0 and dimension bounded above by N∈[1,∞), then the classic Lévy-Gromov isoperimetric inequality (together with the recent sharpening counter…
We introduce the notions of `super-Ricci flows' and `Ricci flows' for time-dependent families of metric measure spaces (X,dt,mt)t∈I. The former property is proven to be stable under suitable space-time versions of mGH-convergence. Uniformly bounded families of super-Ricci flows are compact. In the spirit of t…
The paper proves inequalities under Bakry-Émery-Ricci curvature bounds.
problem Proving functional inequalities under lower Bakry-Émery-Ricci curvature bounds.
method Lower m-Bakry-Émery-Ricci curvature bounds with ε-range. result Proves Cheng type inequality and local Sobolev inequality.
Lower bounds for geodesic ball volume in 3D with Ricci curvature constraints.
problem Finding volume bounds in 3D manifolds with Ricci curvature limits.
method Providing lower bounds for geodesic ball volume with upper bounds on Ricci curvature.
result Established lower bounds for geodesic ball volume under Ricci curvature constraints.
Develops optimal transport in Lorentzian spaces with synthetic curvature bounds.
problem Synthetic curvature bounds for Lorentzian spaces.
method Optimal transport, convexity analysis of entropy functionals.
result Synthetic notion of timelike Ricci curvature lower bounds.
In this note, we provide some general discussion on the Ricci lower bound along Kähler-Ricci flow with singularity over closed manifold.
Survey on warped products and their curvature properties.
problem Understanding warped products and their curvature bounds.
method Construction and analysis of warped products between manifolds and metric spaces.
result Warp products have nice curvature properties, especially sectional and Ricci bounds.
Rigidity and almost rigidity of Sobolev inequalities on compact spaces with lower Ricci curvature bounds.
problem Characterizing and proving rigidity and almost rigidity of Sobolev inequalities on compact spaces with lower Ricci curvature bounds.
method Analysis of Riemannian manifolds and metric measure spaces with synthetic lower Ricci curvature bounds, using concentration compactness and Polya-Szego inequalities.
result Closed Riemannian manifolds with optimal Sobolev constant are isometric to the sphere, and almost equality implies close measure Gromov-Hausdorff convergence to a spherical suspension.
It is shown that the diameter of a compact shrinking Ricci soliton has a universal lower bound. This is proved by extending universal estimates for the first non-zero eigenvalue of Laplacian on compact Riemannian manifolds with lower Ricci curvature bound to a twisted Laplacian on compact shrinking Ricci solitons.
Surveying Ricci flow for weak lower scalar curvature bounds.
problem Creating local definitions for weak lower scalar curvature bounds for C0 metrics. method Using Ricci flow to define and analyze weak lower scalar curvature bounds.
result Properties and applications of Ricci flow in defining weak lower scalar curvature bounds.
The report explores conditions for positive or non-negative scalar curvature in 3-manifolds and weak Ricci curvature bounds.
problem Conditions for positive or non-negative scalar curvature in 3-manifolds and weak Ricci curvature bounds on non-smooth spaces.
method Description of results in dimension 3, exploration of weak forms of Ricci curvature, use of volume entropy and Bishop-Gromov inequality.
result Recent results on weak Ricci curvature bounds and conditions for positive or non-negative scalar curvature in 3-manifolds.
Extends Ricci flow theory to Kato-type curvature bounds, proving manifold properties.
problem Extending Ricci flow theory under Kato-type curvature bounds.
method Extends non-collapsed Ricci flow existence theory to Kato-type lower bounds.
result Compact three-dimensional non-collapsed strong Kato limit spaces are homeomorphic to smooth manifolds.
Study lower bounds on modified K-energy on Fano manifolds with Kähler-Ricci solitons.
problem Lower boundedness of modified K-energy on Fano manifolds.
method Extend Tosatti's method to study Fano manifolds with Kähler-Ricci solitons.
result Establish lower bounds on modified K-energy for Kähler-Ricci solitons.
Two-dimensional collapsed spaces with lower Ricci bounds are topological surfaces.
problem Topology of collapsed spaces with lower Ricci bounds
method Prove that collapsed spaces are topological surfaces
result Collapsed spaces are topological surfaces
We give an estimate on the lower bound of the first non-zero eigenvalue of the Laplacian for a closed Riemannian manifold with positive Ricci curvature in terms of the in-diameter and the lower bound of the Ricci curvature.
Study on Kähler manifolds shows rigidity of eigenvalues with positive Ricci bound.
problem Optimal rigidity results for eigenvalues on Kähler manifolds with positive Ricci lower bound.
method Established optimal rigidity results for eigenvalues on Kähler manifolds with positive Ricci lower bound.
result Complex projective space is the only Kähler manifold with the largest multiplicity of the first eigenvalue.
Lower Ricci curvature bound prevents first Betti number from dropping more than dimension in collapsing manifolds.
problem Understanding how the first Betti number behaves under manifold collapse with Ricci curvature bounds.
method Analyzing sequences of Riemannian manifolds with lower Ricci curvature bounds.
result The first Betti number cannot drop more than the dimension in collapsing manifolds.
We give new estimates on the lower bounds for the first closed or Neumann eigenvalue for a compact manifold with positive Ricci curvature in terms of the diameter and the lower bound of Ricci curvature. The results improve the previous estimates.
We give a new estimate on the lower bound for the first Dirichlet eigenvalue for a compact manifold with positive Ricci curvature in terms of the in-diameter and the lower bound of the Ricci curvature. The result improves the previous estimates.
The paper proves scalar curvature lower bounds along Ricci flow on compact manifolds.
problem Preserving scalar curvature lower bounds along Ricci flow.
method Analyzing Ricci flow on compact manifolds with initial metrics having scalar curvature lower bounds.
result If initial metric has scalar curvature lower bound, it is preserved along Ricci flow.
2-regular points found in spaces with lower Ricci curvature bound.
problem Characterizing points in spaces with lower Ricci curvature bound.
method Analyzing measured Gromov-Hausdorff limits of Riemannian manifolds.
result 2-regular points in interior geodesics of limit spaces are 2-rectifiable.