Stability for Serrin's problem and Alexandroff's theorem in warped product manifolds

TL;DR

This paper establishes quantitative stability results for classical geometric PDE problems, including Serrin's problem and constant mean curvature classification, within warped product manifolds, advancing understanding of geometric stability in these spaces.

Contribution

It introduces new stability theorems for Serrin's problem, Brendle's Heintze-Karcher inequality, and constant mean curvature classification in warped product spaces, utilizing recent developments in stability analysis of level sets.

Findings

Double stability theorem for Serrin's problem in spaceforms

Stability results for Heintze-Karcher inequality

Classification stability for constant mean curvature surfaces

Abstract

We prove quantitative versions for several results from geometric partial differential equations. Firstly, we obtain a double stability theorem for Serrin's overdetermined problem in spaceforms. Secondly, we prove stability theorems for Brendle's Heintze-Karcher inequality respectively constant mean curvature classification in a class of warped product spaces. The key tool is the first author's recent development of stability for level sets of a function under smallness of the traceless Hessian thereof.