Well-posed geometric boundary data in General Relativity, III: conformal-volume boundary data

TL;DR

This paper proves the local-in-time well-posedness of the vacuum Einstein equations with specific boundary conditions involving conformal class and volume forms, advancing mathematical understanding of boundary value problems in general relativity.

Contribution

It introduces a novel boundary condition framework combining conformal class and volume form data, establishing well-posedness for the Einstein equations with these boundary conditions.

Findings

Proves local-in-time well-posedness of IBVP for Einstein equations

Defines boundary conditions involving conformal class and scalar density

Establishes mathematical foundation for boundary data in GR

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.