# On the differential structure of metric measure spaces and applications

created by gigli on 13 Mar 2012
modified on 21 May 2013

[BibTeX]

Preprint

Inserted: 13 mar 2012
Last Updated: 21 may 2013

Year: 2012

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.