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

TL;DR

This work addresses local-in-time well-posedness for the vacuum Einstein equations with geometric boundary data, focusing on conformal-mean curvature data on a timelike boundary. By working in harmonic (wave) gauge and linearizing about a smooth vacuum metric, the authors establish that a linearized IBVP is well-posed only after including corner data, specifically the corner angle $\alpha$ at $\Sigma$, and they prove that the linearized map is an isomorphism in $C^{\infty}$. The analysis combines localization near the corner with a boundary Dirichlet–Neumann–Dirichlet mix for boundary data, and an iterative scheme yields local existence; uniqueness follows from a self-adjoint variational structure. These results lay the groundwork for a nonlinear theory (via Nash–Moser) and clarify how corner data influence the geometric parametrization of vacuum solutions, with Appendix describing explicit flat-cylinder examples that demonstrate the corner-angle dependence.

Abstract

We study 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. For conformal-mean curvature boundary conditions, consisting of the conformal class of the boundary metric and mean curvature of the boundary, well-posedness does not hold without imposing additional angle data at the corner. When the corner angle is included as corner data, we prove well-posedness of the linearized problem in $C^{\infty}$, where the linearization is taken at any smooth vacuum Einstein metric.

Theorems & Definitions (37)

• Conjecture 1
• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Lemma 2.1
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3