arXiv research

Paper provides an example showing CD inequality doesn't imply CDE' inequality.

The study estimates curvature on graphs with large girth.

Sharp log-Sobolev inequalities proved for ${\sf CD}(0,N)$ spaces.

Graphs satisfy Li-Yau inequality under $CD(0,n)$ curvature condition.

New inequalities link probability density norms to Sobolev norms and Kantorovich distances.

The paper proves a Moser-Trudinger inequality on metric spaces with curvature-dimension conditions.

The CD equalities were introduced to imply the gradient estimate of laplace operator on graphs. This article is based on the unbounded Laplacians, and finally concludes some equivalent properties of the CD(K,$\infty$)and CD(K,n).

We study some equivalent properties of the curvature-dimension conditions $CD(n,K)$ inequality on infinite, but locally finite graph. These equivalences are gradient estimate, Poincaré type inequalities and reverse Poincaré inequalities. And we also obtain one equivalent property of gradient estimate for a new notion o…

We show a connection between the $CDE'$ inequality and the $CDψ$ inequality. In particular, we introduce a $CD_ψ^\varphi$ inequality as a slight generalization of $CDψ$ which turns out to be equivalent to $CDE'$ with appropriate choices of $\varphi$ and $ψ$. We use this to prove that the $CDE'$ inequality implies the c…

Researchers prove inequalities for reaction-diffusion systems using a new curvature-dimension condition.

The paper proves inequalities and entropy formulas for Witten Laplacian on manifolds.

Extends gradient estimates for heat equation under Finsler geometric flows.

The article examines entropy-information inequalities for continuous-time Markov chains under curvature-dimension conditions.

The study establishes a curvature-dimension condition for discrete Markov chains.

Paper proves Brunn-Minkowski inequality and curvature dimension condition are equivalent in weighted Riemannian manifolds.

Paper improves eigenvalue bounds for Laplacian in Alexandrov spaces.

The paper characterizes curvature-dimension conditions and related inequalities on Riemannian manifolds.

Abstract shows entropy and convexity definitions of very strict $CD(K,N)$ spaces are equivalent.

Paper proposes a new confidence dimension to measure DNN generalization.

We prove that the Abresch-Gromoll inequality holds on infinitesimally Hilbertian CD(K,N) spaces in the same form as the one available on smooth Riemannian manifolds.

The following inequality \cat X\le \cat Y+\lceil\frac{hd(X)-r}{r+1}\rceil holds for every locally trivial fibration between $ANE$ spaces $f:X\to Y$ which admits a section and has the $r$-connected fiber where $hd(X)$ is the homotopical dimension of $X$. We apply this inequality to prove that \cat X\le \lceil\frac{\dim …

We extend the range of $N$ to negative values in the $(K,N)$-convexity (in the sense of Erbar--Kuwada--Sturm), the weighted Ricci curvature $Ric_N$ and the curvature-dimension condition $CD(K,N)$. We generalize a number of results in the case of $N>0$ to this setting, including Bochner's inequality, the Brunn--Minkowsk…

The paper defines curvature dimension inequalities on directed graphs and evaluates them.

Paper investigates rigidity of synthetic Ricci curvature bounds on manifolds.

Graphs with nonnegative Bakry-Émery curvature have volume doubling and Poincaré inequalities.

Quantitative estimates for inequalities on sub-Riemannian manifolds.

The paper proves properties of Renyi entropy power on Riemannian manifolds.

In this note we continue the analysis of metric measure space with variable ricci curvature bounds. First, we study $(κ,N)$-convex functions on metric spaces where $κ$ is a lower semi-continuous function, and gradient flow curves in the sense of a new evolution variational inequality that captures the information that …

In this paper, we study Li-Yau gradient estimates for the solutions $u$ to the heat equation $\partial_tu=Δu$ on graphs under the curvature condition $CD(n,-K)$ introduced by Bauer et al. in \cite{BHLLMY}. As applications, we derive Harnack inequalities and heat kernel estimates on graphs. Also we present a type of Ham…

Study confirms equivalence in Heisenberg groups between curvature-dimension conditions and strong Brunn-Minkowski inequalities.

We prove a new generalization of the Cheeger-Gromoll splitting theorem where we obtain a warped product splitting under the existence of a line. The curvature condition in our splitting is a curvature dimension inequality of the form $CD(0,1)$. Even though we have to allow warping in our splitting, we are able to recov…

Rigidity and almost rigidity of Sobolev inequalities on compact spaces with lower Ricci curvature bounds.

In this note, we concentrate on the sub-Laplace on the nilpotent Lie group of rank two, which is the infinitesimal generator of the diffusion generated by $n$ Brownian motions and their $\frac{n(n-1)}2$ Lévy area processes, which is the simple extension of the sub-Laplace on the Heisenberg group $\mathbb{H}$. In order …

In this note we show that in metric measure spaces satisfying the reduced curvature-dimension condition CD*(K,N) we always have geodesics in the Wasserstein space of probability measures that satisfy the critical convexity inequality of CD*(K,N) also for intermediate times and in addition the measures along these geode…

Uniform Poincaré inequalities established for various metric spaces.

Paper proves a sharp weighted Isoperimetric inequality for substatic manifolds.

The paper characterizes $(K, m)$-Ricci solitons and flows using $W$-entropy.

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

The paper examines stability of the Sobolev inequality in metric spaces with curvature dimension conditions.

It is shown that curvature-dimension bounds CD(N, k) for a metric measure space (X,d,m) in the sense of Sturm imply a weak L^1- Poincare-inequality under some symmetry assumption on the choice of transport rays in the cut locus of (X,d). This condition is satisfied if (X,d) has m-almost surely no branching points.

Sharp inequalities in nonnegative Ricci curvature spaces using mass transport.

The goal of this paper is twofold: we study metric measure spaces $(X,d,m)$ with variable lower bounds for the Ricci curvature and we study pathwise coupling of Brownian motions. Given any lower semicontinuous function $k:X\to \mathbb R$ we introduce the curvature-dimension condition $CD(k,\infty)$ which canonically ex…

We prove local Poincaré inequalities under various curvature-dimension conditions which are stable under the measured Gromov-Hausdorff convergence. The first class of spaces we consider is that of weak CD(K,N) spaces as defined by Lott and Villani. The second class of spaces we study consists of spaces where we have a …

Optimal maps exist in very strict $CD(K,\infty)$ spaces despite plan uniqueness issues.

The study presents examples of $CD(0,N)$ spaces with varying dimensions and discusses the limitations of the $CD(0,N)$ condition.

Klartag recently gave a beautiful alternative proof of the isoperimetric inequalities of Levy-Gromov, Bakry-Ledoux, Bayle and E. Milman on weighted Riemannian manifolds. Klartag's approach is based on a generalization of the localization method (so-called needle decompositions) in convex geometry, inspired also by opti…

In this paper, we prove logarithmic Sobolev inequalities and derive the Hamilton Harnack inequality for the heat semigroup of the Witten Laplacian on complete Riemannian manifolds equipped with $K$-super Perelman Ricci flow. We establish the $W$-entropy formula for the heat equation of the Witten Laplacian and prove a …

Study finds solutions to nonlinear Schrödinger equation on finite graphs.