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

César Rosales

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arXiv · 2211.16400

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

César Rosales

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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.

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César Rosales. 2022-11-29. Stable capillary hypersurfaces and the partitioning problem in balls with radial weights. https://arxiv.org/abs/2211.16400

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