Well-posed geometric boundary data in General Relativity, III: Conformal-mean curvature boundary data

Abstract

This is the third work in a series on the (local in time) well-posedness of the initial boundary value problem (IBVP) for the vacuum Einstein equations in general relativity with geometric boundary conditions. Here we study the conformal-mean curvature boundary conditions, consisting of the conformal class of the boundary metric and mean curvature of the boundary. We prove that at metrics of uniformly bounded geometry to all orders, the linearized problem has a solution space with dense range in $C^{\infty }$ and establish a Holmgren-type uniqueness theorem valid for general smooth linearized solutions. These results require the addition of an arbitrary corner angle term at the intersection of the Cauchy surface and the timelike boundary.

Substantial revision of previous version, v1, due to a gap in the proof of the main linearized existence theorem of v1. Statement of main linearized existence theorem weakened. This version is now Part III of the series, in place of prior Part I. 27 pages