Local structure theory of Einstein manifolds with boundary

TL;DR

This paper investigates the local structure of the moduli space of Einstein manifolds with boundary, confirming conjectures in three dimensions and extending results to higher dimensions for special Einstein metrics.

Contribution

It proves that the boundary data map is a local diffeomorphism in three dimensions and extends similar results to Ricci flat and negative Einstein metrics in higher dimensions.

Findings

Confirmed Anderson's conjecture in 3D for Einstein metrics.

Established local diffeomorphism property of boundary data map.

Extended results to higher dimensions for special Einstein metrics.

Abstract

We study local structure of the moduli space of compact Einstein metrics with respect to the boundary conformal metric and mean curvature. In dimension three, we confirm M. Anderson's conjecture in a strong sense, showing that the map from Einstein metrics to such boundary data is generically a local diffeomorphism. In dimensions greater than three, we obtain similar results for Ricci flat metrics and negative Einstein metrics under new non-degenerate boundary conditions.