Laplacian comparison for Alexandrov spaces

TL;DR

This paper extends Laplacian comparison theorems to Alexandrov spaces with a generalized Ricci curvature condition, leading to a topological splitting result, thus broadening geometric analysis tools in non-smooth spaces.

Contribution

It introduces an infinitesimal Bishop-Gromov volume comparison condition for Alexandrov spaces and proves a Laplacian comparison theorem under this condition.

Findings

Established a Laplacian comparison theorem for Alexandrov spaces

Proved a topological splitting theorem based on the new curvature condition

Extended geometric analysis techniques to non-smooth Alexandrov spaces

Abstract

We consider an infinitesimal version of the Bishop-Gromov relative volume comparison condition as generalized notion of Ricci curvature bounded below for Alexandrov spaces. We prove a Laplacian comparison theorem for Alexandrov spaces under the condition. As an application we prove a topological splitting theorem.