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

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 $\sqrt{\mathrm{\mid }{\mathbb{S}}^2{\mathrm{\mid }}_g\mathrm{/}16\pi }$ provided the first eigenvalue of the operator $(-{\mathrm{\Delta }}_g+K_g)$ is non-negative. This eigenvalue condition, in particular, imposes no lower bound on $K_g$ (even under an area constraint) and thereby extends previous results which assume $K_g\ge 0$.