On the Bartnik mass of non-negatively curved CMC spheres

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 $g$.

10 pages; Reference to [13] corrected and reference to [2] expanded upon in introduction, Corrections to bibliography