Preprint

Inserted: 19 apr 2025

Year: 2025

Abstract:

Let $(M,g)$ be a $3$--dimensional, complete, one--ended Riemannian manifold, with a minimal, compact and connected boundary. We assume that $M$ has a simple topology and that the scalar curvature of $(M,g)$ is non--negative. Moreover, we suppose that $(M,g)$ admits a $2$--capacitary potential $v$ with $v,\,\vert \nabla v\vert\to 0$ at infinity. In this note, we provide a gradient integral estimate for the level sets of the function $u=1-v$. This estimate leads to a sharp volume comparison for the sub--level sets of $u$, and a sharp area comparison of the level sets of $u$. From this last comparison it follows a sharp area--capacity inequality, originally derived by Bray and Miao, thereby extending its cases of validity. This work is based on the recent paper by Colding and Minicozzi. Finally, for completeness, we also show the same type of area and volume comparison, in the case where $(M,g)$ has no boundary, replacing the function $u$ with one related to the minimal positive Green's function. This volume comparison leads to a more geometric proof of the positive mass inequality than the one given in \cite{Ago_MazOro}.