Estimates of the Bartnik mass

Given a metric $\gamma$ of nonnegative Gauss curvature and a positive function $H$ on a $2$-sphere $\Sigma$, we estimate the Bartnik quasi-local mass of $(\Sigma, \gamma, H)$ in terms of the area, the total mean curvature, and a quantity depending only on $\gamma$, measuring the roundness of the metric. If $\gamma$ has positive Gauss curvature, the roundness of $\gamma$ in the estimate is controlled by the ratio $\kappa$ between the maximum and the minimum of the Gauss curvature. As $\kappa \to 1$, the estimate approaches a sharp estimate for round spheres with arbitrary, positive mean curvature functions. Enroute we observe an estimate of the supremum of the total mean curvature among nonnegative scalar curvature fill-ins of a closed manifold with positive scalar curvature.