arXiv research

Researchers develop weighted GJMS operators for smooth metric measure spaces.

Proves existence and uniqueness of weighted metrics for smooth spaces.

Paper finds lower bounds for eigenvalues of Bi-drifted Laplacian on smooth metric measure spaces.

The goal of the paper is to study the angle between two curves in the framework of metric (and metric measure) spaces. More precisely, we give a new notion of angle between two curves in a metric space. Such a notion has a natural interplay with optimal transportation and is particularly well suited for metric measure …

We introduce an universum of the Polish (=complete separable metric) space - the convex cone of distance matrices and study its geometry. It happened that the generic Polish spaces in this sense of this universum is so called Urysohn spaces defined by P.S.Urysohn in 20-th, and generic metric triple (= metric space with…

In this short note we compare the weighted Laplacians on real and complex (Kähler) metric measure spaces. In the compact case Kähler metric measure spaces are considered on Fano manifolds for the study of Kähler-Einstein metrics while real metric measure spaces are considered with Bakry-Émery Ricci tensor. There are tw…

Metric measure boundary vanishes on certain spaces without boundary.

Study stability of curvature-dimension condition for negative dimensions.

The study shows finite measure-preserving isometry groups for certain metric measure spaces.

Sharp inequality in spaces with non-negative Ricci curvature.

The aim of this note is to study the measure-valued Ricci tensor on smooth metric measure space with boundary, which is a generalization of Bakry-Emery's modified Ricci tensor on weighted Riemannian manifold. As an application, we offer a new approach to study curvature-dimension condition of smooth metric measure spac…

Extended Rank-One Theorem to special metric spaces.

We provide a quick overview of various calculus tools and of the main results concerning the heat flow on compact metric measure spaces, with applications to spaces with lower Ricci curvature bounds. Topics include the Hopf-Lax semigroup and the Hamilton-Jacobi equation in metric spaces, a new approach to differentiati…

Study Bochner formula on metric measure spaces for vanishing Betti numbers.

Develops new synthetic Ricci flow concepts for metric measure 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…

Corrects errors in previous studies on metric measure spaces.

A mass-type invariant for smooth metric measure spaces and its relation with the fractional Yamabe problem

The paper extends entropy formulas to super Ricci flows on metric measure spaces.

Study group actions in metric spaces, proving convergence of lens spaces.

Study shows volume constraints lead to isoperimetric constant bounds in specific metric spaces.

Proves sufficiency of countable test plans for BV functions on metric spaces.

Implementing $k$-NN classification using Gromov--Wasserstein distances

Study of geometric analysis on asymmetric metric spaces, including heat flow and Sobolev spaces.

The study examines stability of metric measure spaces with integral Ricci curvature bounds.

The paper proves gradient estimates for nonlinear parabolic equations on smooth metric measure spaces.

Locally homogeneous RCD spaces are shown to be smooth manifolds.

We survey work of Lott-Villani and Sturm on lower Ricci curvature bounds for metric-measure spaces.

Study Finsler metric measure manifolds' concentration properties.

In this paper, we will study the (linear) geometric analysis on metric measure spaces. We will establish a local Li-Yau's estimate for weak solutions of the heat equation and prove a sharp Yau's gradient gradient for harmonic functions on metric measure spaces, under the Riemannian curvature-dimension condition $RCD^*(…

We introduce a weak notion of barycenter of a probability measure $μ$ on a metric measure space $(X, d, {\bf m})$, with the metric $d$ and reference measure ${\bf m}$. Under the assumption that optimal transport plans are given by mappings, we prove that our barycenter $B(μ)$ is well defined; it is a probability measur…

The main result of this article states that the (K;N)-cone over some metric measure space satisfies the reduced Riemannian curvature-dimension condition RCD^*(KN;N+1) if and only if the underlying space satisfies RCD^*(N-1;N). The proof uses a characterization of reduced Riemannian curvature-dimension bounds by Bochner…

Study solves Yamabe problems on metric measure spaces with or without boundary.

In this paper, we study the asymptotic behavior of the volume of spheres in metric measure spaces. We first introduce a general setting adapted to the study of asymptotic isoperimetry in a general class of metric measure spaces. We then introduce a notion of "being asymptotically isoperimetric" for a family of finite a…

In this paper, we prove that a metric measure space which has at least one open set isometric to an interval, and for which the (possibly non-unique) optimal transport map exists from any absolutely continuous measure to an arbitrary measure, is a one-dimensional manifold (possibly with boundary). As an immediate corol…

We introduce and study the notion of the energy of a smooth metric measure space, which includes as special cases the Yamabe constant and Perelman's $ν$-entropy. We then investigate some properties the energy shares with these constants, in particular its relationship with the $κ$-noncollapsing property. Finally, we us…

This is an exposition of the theory of differentiable structures on metric measures spaces, in the sense of Cheeger and Keith.

We prove that a metric measure space equipped with a Dirichlet form admitting an Euclidean heat kernel is necessarily isometric to the Euclidean space. This helps us providing an alternative proof of Colding's celebrated almost rigidity volume theorem via a quantitative version of our main result. We also discuss the c…

In this note we prove the Heintze-Karcher inequality in the context of essentially non-branching metric measure spaces satisfying a lower Ricci curvature bound in the sense of Lott-Sturm-Villani. The proof is based on the the needle decomposition technique for metric measure spaces introduced by Cavalletti-Mondino. Mor…

We prove the differentiability of Lipschitz maps X-->V, where X is a complete metric measure space satisfying a doubling condition and a Poincaré inequality, and V is a Banach space with the Radon Nikodym Property (RNP). The proof depends on a new characterization of the differentiable structure on such metric measure …

For smooth metric measure spaces $(M, g, e^{-f} dvol)$ we prove a Liuoville-type theorem when the Bakry-Emery Ricci tensor is nonnegative. This generalizes a result of Yau, which is recovered in the case $f$ is constant. This result follows from a gradient estimate for f-harmonic functions on smooth metric measure spac…

Paper investigates conditions for independence of weak gradients on metric spaces.

New formulations for comparing metric measure spaces with arbitrary positive measures.

Sharp estimates derived for quasilinear equations on metric measure spaces.

New metrics avoid high-dimensional analysis challenges, proving convergence without 'curse of dimensionality'.

The paper derives gradient estimates for solutions of certain equations on metric measure spaces.

Global gradient estimates for Fisher-KPP equation on Finsler metric measure spaces.

A new metric for comparing probability measures on graphs, scalable and negative definite.