The Hilbert manifold of asymptotically flat metric extensions

TL;DR

This paper extends Bartnik's phase space framework for Einstein equations to manifolds with boundary, showing that solutions form a Hilbert manifold and linking critical points of the mass functional to stationary and black hole solutions.

Contribution

It generalizes the phase space construction to manifolds with boundary and establishes the correspondence between mass functional critical points and stationary solutions.

Findings

Solutions form a Hilbert submanifold with boundary conditions

Critical points of the mass functional are stationary solutions

In vacuum, critical points are only Schwarzschild solutions

Abstract

In [Comm. Anal. Geom., 13(5):845-885, 2005.], Bartnik described the phase space for the Einstein equations, modelled on weighted Sobolev spaces with local regularity $(g,\pi)\in H^2\times H^1$. In particular, it was established that the space of solutions to the contraints form a Hilbert submanifold of this phase space. The motivation for this work was to study the quasilocal mass functional now bearing his name. However, the phase space considered there was over a manifold without boundary. Here we demonstrate that analogous results hold in the case where the manifold has an interior compact boundary, where Dirichlet boundary conditions are imposed on the metric. Then, still following Bartnik's work, we demonstrate the critical points of the mass functional over this space of extensions correspond to stationary solutions. Furthermore, if this solution is sufficiently regular then it is in fact a static black hole solution. In particular, in the vacuum case, critical points only occur at exterior Schwarzschild solutions; that is, critical points of the mass over this space do not exist generically. Finally, we briefly discuss the case when the boundary data is Bartnik's geometric data.