Positive Ricci curvature through Cheeger deformations

TL;DR

This paper analyzes Cheeger deformations on manifolds with group actions, providing new curvature estimates and applications, including lifting positive Ricci curvature from quotients and streamlining classical results.

Contribution

It offers new curvature estimates near singular orbits and answers a key question about lifting positive Ricci curvature, extending classical results with new techniques.

Findings

New curvature estimates near singular orbits

Answer to Ricci curvature lifting question

Streamlined proof of classical results

Abstract

This paper is devoted to a deep analysis of the process known as Cheeger deformation, applied to manifolds with isometric group actions. Here, we provide new curvature estimates near singular orbits and present several applications. As the main result, we answer a question raised by a seminal result of Searle--Wilhelm about lifting positive Ricci curvature from the quotient of an isometric action. To answer this question, we develop techniques that can be used to provide a substantially streamlined version of a classical result of Lawson and Yau, generalize a curvature condition of Chav\'ez, Derdzinski, and Rigas, as well as, give an alternative proof of a result of Grove and Ziller.