Stability of quermassintegral inequalities along inverse curvature flows

TL;DR

This paper investigates the stability of quermassintegral inequalities using inverse curvature flows, demonstrating that the flow accelerates convergence to spherical shapes and establishing stability for nearly spherical sets.

Contribution

It introduces a rescaled inverse curvature flow that preserves certain quermassintegrals and proves a stability inequality for nearly spherical domains using this flow method.

Findings

The flow accelerates the decrease of the $k$-th quermassintegral towards the sphere.

The rate of decrease relates to the Fraenkel asymmetry of the domain.

Stability inequalities are established for nearly spherical sets.

Abstract

In this paper, we consider the stability of quermassintegral inequalities along a inverse curvature flow. We choose a special rescaling of the flow such that the $k$-th quermassintegral is decreasing and the $k-1$-th quermassintegral is preserved. Along this rescaled flow, we prove that the decreasing rate of the $k$-th quermassintegral is faster than the Fraenkel asymmetry of the domain when approaching to the sphere. This leads to the stability inequality of quermassintegral inequalities for nearly spherical sets using the flow method.