Beijing Journal of Pure and Applied Mathematics

Volume 1 (2024)

Number 2

Estimates of the Bartnik mass

Pages: 489 – 513

DOI: https://dx.doi.org/10.4310/BPAM.2024.v1.n2.a3

Authors

Pengzi Miao (Department of Mathematics, University of Miami, Coral Gables, Florida, USA)

Annachiara Piubello (Department of Mathematics, University of Miami, Coral Gables, Florida, USA)

Abstract

Given a metric $\gamma$ of nonnegative Gauss curvature and a positive function $H$ on a 2-sphere $\Sigma$, we estimate the Bartnik quasi-local mass of $(\Sigma, \gamma, H)$ in terms of the area, the total mean curvature, and a quantity depending only on $\gamma$, measuring the roundness of the metric. If $\gamma$ has positive Gauss curvature, the roundness of $\gamma$ in the estimate is controlled by the ratio $\kappa$ between the maximum and the minimum of the Gauss curvature. As $\kappa \to 1$, the estimate approaches a sharp estimate for round spheres with arbitrary, positive mean curvature functions.

Enroute we observe an estimate of the supremum of the total mean curvature among nonnegative scalar curvature fill-ins of a closed manifold with positive scalar curvature.

Keywords

quasi-local mass, scalar curvature

2010 Mathematics Subject Classification

Primary 58J32. Secondary 53C20.

Received 3 August 2023

Accepted 12 January 2024

Published 17 July 2024