Mass, Capacitary Functions, and the Mass-to-Capacity Ratio

Abstract

We study connections among the ADM mass, positive harmonic functions, and capacity of the boundary on asymptotically flat 3-manifolds of nonnegative scalar curvature. We start with new formulae detecting the mass via positive harmonic functions. Then we derive a family of monotone quantities and geometric inequalities assuming the manifold has simple topology. As a first application, we observe several additional proofs of the 3-dimensional Riemannian positive mass theorem. One proof leads to new, sufficient conditions implying positivity of the mass via

C^0

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$C^0$$\end{document}

-geometry of regions separating the boundary and

\infty

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\infty $$\end{document}

. A special case of such conditions shows if a region enclosing the boundary has relative small volume, then the mass is positive. As further applications, we obtain integral identities for the mass-to-capacity ratio. We also promote the inequalities to become equality on Schwarzschild manifolds outside rotationally symmetric spheres. Among other things, we show the mass-to-capacity ratio is always bounded below by one minus the square root of the normalized Willmore functional of the boundary. Prompted by these findings, we carry out a study of manifolds satisfying a constraint on the mass-to-capacity ratio in the context of the Bartnik quasi-local mass.

Keywords

Harmonic functions / Mass / Capacity / Scalar curvature / Mathematical Sciences / Pure Mathematics

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Funding

National Science Foundation(DMS-1906423)

RIGHTS & PERMISSIONS

Peking University