Bartnik Mass of CMC surfaces under a Spectral non-negativity condition

Abstract

Let $g$ be a smooth Riemannian metric and $H$ a positive function on ${\mathbb{S}}^2$. We prove that the Bartnik mass of the triple $({\mathbb{S}}^2,g,H)$ is bounded above by $\sqrt{|{\mathbb{S}}^2{|}_g/16\pi }$ provided the first eigenvalue ${\lambda }_1(g)$ of the operator $(-{\Delta }_g+K_g)$ is non-negative. We prove a similar result assuming $H$ is only nonnegative and ${\lambda }_1(g)>0$. These eigenvalue conditions, in particular, impose no lower bound on $K_g$ (even under an area constraint \cite{Mantoulidis_2015} \S3) and thereby extend previous results which assume $K_g\ge 0$.

Theorem 1.2 has been added. Reference [6] has been added