arXiv research

In 2D, synthetic curvature notions match.

Proves equivalence of two types of Ricci curvature bounds.

Study compares synthetic and distributional Ricci curvature bounds.

This survey introduces synthetic timelike Ricci curvature bounds in Lorentzian spaces.

Unified study of Riemannian and sub-Riemannian geometries with synthetic Ricci curvature bounds.

The study proposes conjectures on limit spaces of Riemannian manifolds with Ricci curvature.

Paper investigates rigidity of synthetic Ricci curvature bounds on manifolds.

Survey on curvature bounds and isoperimetric inequalities.

Rigidity of cones with bounded Ricci curvature proven.

Study inradius collapsed manifolds with lower Ricci curvature bounds, proving properties of their limits.

Stability of timelike Ricci bounds in low-regularity spacetimes.

Study solves Gel'fand's inverse problem in non-smooth spaces with Ricci curvature bounds.

Study new Ricci bounds for metric measure spaces, preserving properties under time changes.

The paper extends Lorentzian splitting theorems under curvature-dimension bounds.

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 examines how curvature-dimension conditions transform under time change for diffusions.

Entropy study on synthetic spaces with curvature bounds.

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 compares nonsmooth spaces with integrable Ricci bounds.

The report explores conditions for positive or non-negative scalar curvature in 3-manifolds and weak Ricci curvature bounds.

Uniform heat kernel bounds lead to synthetic Ricci curvature conditions for Lipschitz manifolds.

Counterexample disproves gluing theorem for MCP metric spaces.

We simplify evaluation of Ollivier-Ricci curvature bounds in hypergraphs.

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…

Develops optimal transport in Lorentzian spaces with synthetic curvature bounds.

Sharp inequality for Lorentzian spaces with timelike Ricci bounds.

Survey on gluing constructions under lower curvature bounds.

The paper characterizes synthetic upper Ricci bounds for metric measure spaces using heat flow, entropy, and optimal transport.

Study of spectral gaps in non-smooth spaces with bounded Ricci curvature.

Study on cones over metric spaces with curvature bounds.

The paper explores sharp isoperimetric properties on non-compact spaces with Ricci bounds.

Uniformly Euclidean metrics with isolated singularities on certain manifolds are Ricci flat and have nonnegative synthetic Ricci curvature.

Sharp bounds on Cheeger's isoperimetric constant and Laplacian eigenvalue in metric measure spaces.

Let $(M,g)$ be a smooth Riemannian manifold and $\mathsf{G}$ a compact Lie group acting on $M$ effectively and by isometries. It is well known that a lower bound of the sectional curvature of $(M,g)$ is again a bound for the curvature of the quotient space, which is an Alexandrov space of curvature bounded below. Moreo…

Sharp inequality in spaces with non-negative Ricci curvature.

We prove that if $(X,\mathsf{d},\mathfrak{m})$ is a metric measure space with $\mathfrak{m}(X)=1$ having (in a synthetic sense) Ricci curvature bounded from below by $K>0$ and dimension bounded above by $N\in [1,\infty)$, then the classic Lévy-Gromov isoperimetric inequality (together with the recent sharpening counter…

The study establishes conditions for stratified spaces to satisfy RCD(K, N) curvature-dimension condition.

Upper bounds on revised first Betti number and torus stability for RCD spaces.

Synthetic scalar curvature defined via Gaussian integrals, applied to manifolds and flows.

Sharp eigenvalue bounds on metric measure spaces extend Cheng's theorem.

Unified approach to various energy conditions in spacetime geometry.

The study constructs a Lorentzian length space and explores its properties and relationships with metric and causal geometry.

The study finds bounds on Ricci curvature for possibly collapsed spaces.

The study proves timelike Ricci bounds for low regularity spacetimes using optimal transport.

Sharp isoperimetric inequality proven for specific metric measure spaces.

Ricci flow controls curvature on manifolds with bounds.

Survey on warped products and their curvature properties.

Study compares isoperimetric profiles on manifolds with integral Ricci curvature bounds.