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 $K_g\geq 0$.