Stable capillary hypersurfaces and the partitioning problem in balls with radial weights

TL;DR

This paper studies stable capillary hypersurfaces in weighted balls with radial symmetry, showing they are topologically disks and characterizing isoperimetric regions under Gaussian weights.

Contribution

It proves that stable capillary hypersurfaces with symmetry are topologically disks and characterizes isoperimetric regions in weighted balls, extending known results to weighted and symmetric settings.

Findings

Stable capillary hypersurfaces are homeomorphic to disks.

Interior boundaries of isoperimetric regions are disks of revolution.

For n=2, stable capillary surfaces of genus 0 are disks of revolution.

Abstract

In a round ball $B\subset\mathbb{R}^{n+1}$ endowed with an $O(n+1)$-invariant metric we consider a radial function that weights volume and area. We prove that a compact two-sided hypersurface in $B$ which is stable capillary in weighted sense and symmetric about some line containing the center of $B$ is homeomorphic to a closed $n$-dimensional disk. When combined with Hsiang symmetrization and other stability results this allows to deduce that the interior boundary of any isoperimetric region in $B$ for the Gaussian weight is a closed $n$-disk of revolution. For $n=2$ we also show that a compact weighted stable capillary surface in $B$ of genus 0 is a closed disk of revolution.