Area, Volume and Capacity in non--compact $3$--manifolds with non--negative scalar curvature

TL;DR

This work studies the interaction between curvature, capacity, and geometry in noncompact 3-manifolds with nonnegative scalar curvature by deriving gradient-based area and volume inequalities for level sets of capacitary potentials. It adapts Colding–Minicozzi gradient techniques to manifolds with a minimal boundary and to the boundaryless Green's-function setting, obtaining sharp area and volume inequalities and a sharp area–capacity inequality that extends Bray–Miao, with rigidity identified by Schwarzschild geometry. In the boundaryless case, a Green's-function construction yields a geometric proof of the 3D positive mass inequality. Overall, the paper connects scalar curvature, capacity, and Schwarzschild geometry through monotone, level-set analyses, offering precise inequalities and rigidity statements that sharpen our understanding of mass and capacity in geometric analysis.

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 a 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_Maz_Oro}.

Theorems & Definitions (9)

• Theorem 1.1
• Definition 2.1
• Definition 2.2
• Theorem 2.3: Riemannian positive mass theorem
• proof
• Proposition 2.4
• proof
• proof : Proof of Theorem \ref{['mainTh']}
• Remark 2.5