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

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