Well-posed geometric boundary data in General Relativity, III: conformal-volume boundary data
Zhongshan An, Michael T. Anderson
TL;DR
This work proves the local-in-time well-posedness of the vacuum Einstein IBVP on a spacetime with a finite timelike boundary under conformal-volume boundary data, uniting initial data on a Cauchy surface with boundary data that include the conformal class of the boundary metric and a volume-density scalar. The authors construct a gauge-fixed, harmonic-coordinate framework by introducing the gauged map $\Phi^H$ from the space of vacuum metrics in harmonic gauge ${\mathbb E}^H$ to a compatible boundary-data space, and they establish this map as a smooth tame diffeomorphism (locally in time) with a tame inverse, via a Nash-Moser argument. The results yield the existence, uniqueness (up to diffeomorphism in the bulk), and continuous dependence of smooth vacuum solutions on smooth initial and boundary data, and they clarify gauge-invariance issues and relationships to earlier Dirichlet boundary data results and ill-posedness phenomena in geometric boundary data. Moreover, the framework demonstrates that the marked moduli space ${\mathcal E}_*$ and the moduli space ${\mathcal E}$ are smooth tame Fréchet manifolds, providing a robust geometric parametrization of vacuum solutions and a platform for comparing different boundary-data formulations.
Abstract
In this third work in a series, we prove the local-in-time well-posedness of the IBVP for the vacuum Einstein equations in general relativity with twisted DIrichlet boundary conditions on a finite timelike boundary. The boundary conditions consist of specification of the pointwise conformal class of the boundary metric, together with a scalar density involving a combination of the volume form of the bulk metric restricted to the boundary together with the volume form of the boundary metric itself.