On the differential structure of metric measure spaces and applications

TL;DR

This paper develops an abstract differential calculus for metric measure spaces, enabling the definition of Laplacians and analyzing their properties, especially in spaces with Ricci curvature bounds, without relying on traditional chart-based analysis.

Contribution

It introduces a chart-free differential calculus on metric measure spaces and applies it to define and study the Laplacian, extending analysis to non-smooth spaces with curvature bounds.

Findings

Laplacian of the distance function is a measure with sharp comparison properties.

The calculus applies to spaces with Ricci curvature bounded below and dimension bounded above.

Results include conditions under which the Laplacian measure exhibits standard properties.

Abstract

The main goals of this paper are: i) To develop an abstract differential calculus on metric measure spaces by investigating the duality relations between differentials and gradients of Sobolev functions. This will be achieved without calling into play any sort of analysis in charts, our assumptions being: the metric space is complete and separable and the measure is Borel, non negative and locally finite. ii) To employ these notions of calculus to provide, via integration by parts, a general definition of distributional Laplacian, thus giving a meaning to an expression like $\Delta g=\mu$, where $g$ is a function and $\mu$ is a measure. iii) To show that on spaces with Ricci curvature bounded from below and dimension bounded from above, the Laplacian of the distance function is always a measure and that this measure has the standard sharp comparison properties. This result requires an additional assumption on the space, which reduces to strict convexity of the norm in the case of smooth Finsler structures and is always satisfied on spaces with linear Laplacian, a situation which is analyzed in detail.