On the Bartnik mass of non-negatively curved CMC spheres

TL;DR

This paper derives upper bounds for the Bartnik mass of non-negatively curved constant mean curvature spheres, relating it to the Hawking mass and analyzing its behavior under various geometric conditions.

Contribution

It provides new upper bounds for the Bartnik mass of CMC spheres with non-negative curvature, connecting it to the Hawking mass and exploring limiting cases.

Findings

Upper bound approaches Hawking mass as g becomes round or H→0

Bound is zero for sufficiently large H

Bound is not more than half the radius, equal to Hawking mass at H=0

Abstract

Let $g$ be a smooth Riemannian metric on $\mathbb{S}^2$ and $H>0$ a constant. We establish an upper bound for the corresponding Bartnik mass $\mathfrak m_B(\mathbb{S}^2, g, H)$ assuming that the Gauss curvature $K_g$ is non-negative. Our upper bound approaches the Hawking mass $\mathfrak m_H(\mathbb{S}^2, g, H)$ when either $g$ becomes round or else $H\to 0$, the bound is zero for $H$ sufficiently large, and in any case the bound is not more than $r/2=\mathfrak m_H(\mathbb{S}^2, g, 0)$. We obtain upper bounds on $\mathfrak m_B(\mathbb{S}^2, g, H)$ as well in the case when $g$ is arbitrary and $H$ is sufficiently large depending on $g$.