Characterizations of monotonicity of vector fields on metric measure spaces

TL;DR

This paper characterizes the convexity of functions and the monotonicity of vector fields on metric measure spaces with Ricci curvature bounds, providing new tools for rigidity theorems in non-smooth geometry.

Contribution

It introduces a novel approach to analyze convexity and monotonicity in non-smooth spaces, aiding in the proof of classical rigidity theorems.

Findings

Characterization of convexity and monotonicity in metric measure spaces

Application to splitting theorem and volume cone implies metric cone theorem

New methods for rigidity results in non-smooth geometry

Abstract

We characterize the convexity of functions and the monotonicity of vector fields on metric measure spaces with Riemannian Ricci curvature bounded from below. Our result offers a new approach to deal with some rigidity theorems such as `splitting theorem' and `volume cone implies metric cone theorem' in non-smooth context.