On boundary value problems for Einstein metrics

TL;DR

This paper analyzes boundary value problems for Einstein metrics on manifolds with boundary, proving that the Einstein moduli space ${\mathcal{E}}$ is a smooth infinite-dimensional Banach manifold under the condition $\pi_{1}(M,\partial M)=0$, i.e., deformations are unobstructed. It shows that standard Dirichlet/Neumann boundary data are not Fredholm, but natural mixed boundary-value problems yield Fredholm boundary maps of index 0, including a conformal-class boundary map. The results extend to complete manifolds with locally asymptotically flat ends and to the Einstein equations coupled to various matter fields, such as scalars, sigma-models, and gauge fields. The approach combines the Bianchi gauge, implicit-function theorem, and careful elliptic boundary analysis to reveal robust local moduli-space structure with broad applicability in geometric analysis and mathematical relativity.

Abstract

On any given compact (n+1)-manifold M with non-empty boundary, it is proved that the moduli space of Einstein metrics on M is a smooth, infinite dimensional Banach manifold under a mild condition on the fundamental group. Thus, the Einstein moduli space is unobstructed. The Dirichlet and Neumann boundary maps to data on the boundary are smooth, but not Fredholm. Instead, one has natural mixed boundary-value problems which give Fredholm boundary maps. These results also hold for manifolds with compact boundary which have a finite number of locally asymptotically flat ends, as well as for the Einstein equations coupled to many other fields.

Theorems & Definitions (20)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• Lemma 2.2
• Corollary 2.3
• Proposition 2.4
• Proposition 2.5
• Remark 2.6
• Corollary 2.7