Formal theory of cornered asymptotically hyperbolic Einstein metrics

TL;DR

This paper develops a formal framework for analyzing asymptotically hyperbolic Einstein metrics with corners, extending previous work, studying eigenfunction expansions, and establishing existence and obstructions related to boundary conditions.

Contribution

It introduces a formal expansion method for cornered asymptotically hyperbolic Einstein metrics, generalizes prior results, and identifies invariants and obstructions at the boundary corners.

Findings

No smooth compactification exists for generic corners.

Formal existence of Einstein metrics up to boundary dimension order.

Obstruction invariant for totally geodesic finite boundary.

Abstract

This paper makes a formal study of asymptotically hyperbolic Einstein metrics given, as conformal infinity, a conformal manifold with boundary. The space on which such an Einstein metric exists thus has a finite boundary in addition to the usual infinite boundary and a corner where the two meet. On the finite boundary a constant mean curvature umbilic condition is imposed. First, recent work of Nozaki, Takayanagi, and Ugajin is generalized and extended showing that such metrics cannot have smooth compactifications for generic corners embedded in the infinite boundary. A model linear problem is then studied: a formal expansion at the corner is derived for eigenfunctions of the scalar Laplacian subject to certain boundary conditions. In doing so, scalar ODEs are studied that are of relevance for a broader class of boundary value problems and also for the Einstein problem. Next, unique formal existence at the corner, up to order at least equal to the boundary dimension, of Einstein metrics in a cornered asymptotically hyperbolic normal form which are polyhomogeneous in polar coordinates is demonstrated for arbitrary smooth conformal infinity. Finally it is shown that, in the special case that the finite boundary is taken to be totally geodesic, there is an obstruction to existence beyond this order, which defines a conformal hypersurface invariant.