On the Bartnik mass of non-negatively curved CMC spheres

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 $g$.