Bubbling and quantitative stability for Alexandrov's Soap Bubble Theorem with $L^1$-type deviations

TL;DR

This paper advances the quantitative stability analysis of Alexandrov's Soap Bubble Theorem by weakening the deviation measure to an $L^1$-type integral, enabling proximity results to disjoint equal-radius balls.

Contribution

It introduces a new stability result using a weaker deviation measure involving the positive part of mean curvature differences, extending previous $L^{N-1}$-deviation results.

Findings

Established proximity to disjoint balls under $L^1$-type deviation.

Extended stability results to weaker curvature deviations.

Improved understanding of bubbling phenomena in mean curvature boundaries.

Abstract

The quantitative analysis of bubbling phenomena for almost constant mean curvature boundaries is an important question having significant applications in various fields including capillarity theory and the study of mean curvature flows. Such a quantitative analysis was initiated in [G. Ciraolo and F. Maggi, Comm. Pure Appl. Math. (2017)], where the first quantitative result of proximity to a set of disjoint balls of equal radii was obtained in terms of a uniform deviation of the mean curvature from being constant. Weakening the measure of the deviation in such a result is a delicate issue that is crucial in view of the applications for mean curvature flows. Some progress in this direction was recently made in [V. Julin and J. Niinikoski, Anal. PDE (2023)], where $L^{N-1}$-deviations are considered for domains in $\mathbb{R}^N$. In the present paper we significantly weaken the measure of the deviation, obtaining a quantitative result of proximity to a set of disjoint balls of equal radii for the following deviation $$ \int_{\partial \Omega } \left( H_0 - H \right)^+ dS_x, \quad \text{ where } \begin{cases} H \text{ is the mean curvature of } \partial \Omega , \\ H_0:=\frac{| \partial \Omega |}{N | \Omega |} , \\ \left( H_0 - H \right)^+:=\max\left\lbrace H_0 - H , 0 \right\rbrace , \end{cases} $$ which is clearly even weaker than $\Vert H_0-H \Vert_{L^1( \partial \Omega )}$.