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 $\sqrt{|\mathbb{S}^2|_g/16\pi}$ provided the first eigenvalue of the operator $(-\Delta_g+K_g)$ 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 $K_g\geq 0$.