Local Well-Posedness for the Bartnik Stationary Extension Problem near Schwarzschild Spheres

Abstract

We investigate the Bartnik stationary extension conjecture, which arises from the definition of the spacetime Bartnik mass for a compact region in a general initial data set satisfying the dominant energy condition. This conjecture posits the existence and uniqueness (up to isometry) of an asymptotically flat stationary vacuum spacetime containing an initial data set $(M, \mathfrak{g}, \Pi)$ that realizes prescribed Bartnik boundary data on $\partial M$, consisting of the induced metric, mean curvature, and appropriate components of the spacetime extrinsic curvature $\Pi$. Building on the analytic framework developed in arXiv:2411.02801 for the static case, we show that, in a double geodesic gauge, the stationary vacuum Einstein equations reduce to a coupled system comprising elliptic and transport-type equations, with the genuinely stationary contributions encoded in an additional boundary value problem for a $1$-form $\theta$. We establish local well-posedness for the Bartnik stationary metric extension problem for Bartnik data sufficiently close to that of any coordinate sphere in any initial data set (possibly non time-symmetric) in Schwarzschild spacetime. This includes spheres arbitrary close to the apparent horizon in the initial data set. A key feature of our framework is that the linearized equations decouple: the equations for the metric and potential reduce to the previously solved static case, while the boundary value problem for $\theta$ is treated independently. We prove solvability of this boundary value problem in the Bochner-measurable function spaces adapted to the coupled system developed in arXiv:2411.02801, establishing uniform estimates for the vector spherical harmonic decomposition of $\theta$.