Conformal Fundamental Forms and the Asymptotically Poincar\'e--Einstein Condition

TL;DR

This paper introduces higher fundamental forms as conformally invariant tensors that determine when a conformally compact metric can be transformed into an asymptotically Poincaré--Einstein metric, advancing the understanding of conformal hypersurface invariants.

Contribution

It constructs a sequence of conformally invariant higher fundamental forms and proves their vanishing characterizes conformally Einstein metrics, extending conformal hypersurface theory with tractor calculus.

Findings

Higher fundamental forms are conformally invariant and determine Einstein conditions.

Vanishing of these forms is necessary and sufficient for conformally Einstein metrics.

Develops tractor calculus tools for conformal hypersurface embeddings.

Abstract

An important problem is to determine under which circumstances a metric on a conformally compact manifold is conformal to a Poincar\'e--Einstein metric. Such conformal rescalings are in general obstructed by conformal invariants of the boundary hypersurface embedding, the first of which is the trace-free second fundamental form and then, at the next order, the trace-free Fialkow tensor. We show that these tensors are the lowest order examples in a sequence of conformally invariant higher fundamental forms determined by the data of a conformal hypersurface embedding. We give a construction of these canonical extrinsic curvatures. Our main result is that the vanishing of these fundamental forms is a necessary and sufficient condition for a conformally compact metric to be conformally related to an asymptotically Poincar\'e--Einstein metric. More generally, these higher fundamental forms are basic to the study of conformal hypersurface invariants. Because Einstein metrics necessarily have constant scalar curvature, our method employs asymptotic solutions of the singular Yamabe problem to select an asymptotically distinguished conformally compact metric. Our approach relies on conformal tractor calculus as this is key for an extension of the general theory of conformal hypersurface embeddings that we further develop here. In particular, we give in full detail tractor analogs of the classical Gauss Formula and Gauss Theorem for Riemannian hypersurface embeddings.