arXiv research

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.

Survey on curvature bounds and isoperimetric inequalities.

Stability of timelike Ricci bounds in low-regularity spacetimes.

We will study metric measure spaces $(X,d,m)$ beyond the scope of spaces with synthetic lower Ricci bounds. In particular, we introduce distribution-valued lower Ricci bounds BE$_1(κ,\infty)$ $\bullet$ for which we prove the equivalence with sharp gradient estimates, $\bullet$ the class of which will be preserved under…

We prove that in two dimensions the synthetic notions of lower bounds on sectional and on Ricci curvature coincide.

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

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

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 ${\rm CD}(K, \infty)$ condition and non-collapsed ${\rm CD}(K, …

Entropy study on synthetic spaces with curvature bounds.

Study compares nonsmooth spaces with integrable Ricci bounds.

We analyze Lorentzian spacetimes subject to curvature-dimension bounds using the Bakry-Émery-Ricci tensor. We extend the Hawking-Penrose type singularity theorem and the Lorentzian timelike splitting theorem to synthetic dimensions $N\le 1$, including all negative synthetic dimensions. The rigidity of the timelike spli…

Sharp inequality for Lorentzian spaces with timelike Ricci 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.

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

We show that the only metric measure space with the structure of an $N$-cone and with two-sided synthetic Ricci bounds is the Euclidean space ${\mathbb R}^{N+1}$ for $N$ integer. This is based on a novel notion of Ricci curvature upper bounds for metric measure spaces given in terms of the short time asymtotic of the h…

Develops optimal transport in Lorentzian spaces with synthetic 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…

We introduce the notions of `super-Ricci flows' and `Ricci flows' for time-dependent families of metric measure spaces $(X,d_t,m_t)_{t\in I}$. 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 study proves timelike Ricci bounds for low regularity spacetimes using optimal transport.

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…

Synthetic Ricci flows defined for metric measure spaces.

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

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

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

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

Survey on gluing constructions under lower curvature bounds.

Study fine Pólya-Szegő inequalities in metric spaces with applications.

Develops new synthetic Ricci flow concepts for metric measure spaces.

Compactness theory for super Ricci flows provides convergence results.

Compact gravity models yield tiny spin-two field masses.

We prove that if $(X,\mathsf d,\mathfrak m)$ is an essentially non-branching metric measure space with $\mathfrak m(X)=1$, having Ricci curvature bounded from below by $K$ and dimension bounded from above by $N \in (1,\infty)$, understood as a synthetic condition called Measure-Contraction property, then a sharp isoper…

We study decreasing rearrangements of functions defined on (possibly non-smooth) metric measure spaces with Ricci curvature bounded below by $K>0$ and dimension bounded above by $N\in (1,\infty)$ in a synthetic sense, the so called $CD(K,N)$ spaces. We first establish a Polya-Szego type inequality stating that the $W^{…

The goal of the paper is to sharpen and generalise bounds involving the Cheeger's isoperimetric constant $h$ and the first eigenvalue $λ_{1}$ of the Laplacian. A celebrated lower bound of $λ_{1}$ in terms of $h$, $λ_{1}\geq h^{2}/4$, was proved by Cheeger in 1970 for smooth Riemannian manifolds. An upper bound on $λ_{1…

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

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

Study on cones over metric spaces with curvature bounds.

New rigidity theorem for sharp spectral gap in nonnegatively curved spaces.

Sharp ABP estimate on metric spaces via optimal transport.

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…

New inequality for eigenfunctions on curved spaces.

We derive precise transformation formulas for synthetic lower Ricci bounds under time change. More precisely, for local Dirichlet forms we study how the curvature-dimension condition in the sense of Bakry-Emery will transform under time change. Similarly, for metric measure spaces we study how the curvature-dimension c…