Ahmed Ellithy
We consider the Bartnik extension problem, which originates from Bartink's definition of quasi-local mass. In this problem, we consider Bartnik boundary data on a topological 2-sphere (induced metric and mean curvature, plus suitable stationary data), and we find a unique asymptotically flat initial data set \((M,g,K)\) that arises as a slice in a stationary spacetime \(\mathcal{M}\) solving Einstein's vacuum equations, with \( \partial M = S^2 \) realizing the boundary data. Furthermore, if the Bartnik data is time-symmetric, then \(K = 0\) and \(\mathcal{M}\) is static. In this talk, we will address the local theory near Schwarzschild spacetimes.
We present a new analytic framework for this extension problem. In this approach, we write the putative stationary spacetime in a double-geodesic gauge in which Einstein's equations reduce to a coupled elliptic system on the lapse function and the twist 1-form, together with transport equations for the second fundamental form of the geodesic leaves. In this gauge, the linearized static equations decouple and reduce to a non-local elliptic problem of Dirichlet-to-Neumann type on the boundary, while the genuinely stationary degrees of freedom reduce to an elliptic boundary value problem for the twist 1-form. To accommodate the mixed elliptic/transport structure, we work in Bochner-type spaces (specifically, continuous on the geodesic parameter with angular Sobolev regularity) instead of the classical Sobolev and Hölder spaces traditionally used for elliptic problems. These Bochner spaces we use are in fact traditionally used for hyperbolic and parabolic PDEs.
Within this framework, we prove the local well-posedness of the Bartnik extension problem for both static and stationary data near Schwarzschild spheres: existence, uniqueness, and smooth dependence on the prescribed boundary data. Along the way, we develop an elliptic solvability theory for boundary value problems in Bochner spaces, which has not to our knowledge been previously used for elliptic problems and maybe of independent interest as they provide an appropriate setting for systems with both elliptic and transport/hyperbolic structure.
