Second-order mass estimates for static vacuum metrics with small Bartnik data

Given on the $2$-sphere Bartnik data (prescribed metric and mean curvature) that is a small perturbation of the corresponding data for the standard unit sphere in Euclidean space, we estimate to second order, in the size of the perturbation, the mass of the asymptotically flat static vacuum extension (unique up to diffeomorphism) which is a small perturbation of the flat metric on the exterior of the unit ball in Euclidean space and induces the prescribed data on the boundary sphere. As an application we obtain a new upper bound on the Bartnik mass of small metric spheres to fifth order in the radius.