Conformal Fundamental Forms and the Asymptotically Poincaré--Einstein Condition

Abstract

An important problem is to determine under which circumstances a metric on a conformally compact manifold is conformal to a Poincaré-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 Poincaré-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.

A Sharp Characterization of the Willmore Invariant

“…Studying higher transverse order jet coefficients requires more finesse: indeed, we require a conformally-invariant extension of II. For the purposes of the calculations to be carried out in the remainder of this article, we use the canonical choice [5]…”

“…Computing the conformally-invariant higher-order transverse jet coefficients amounts to applying conformally-invariant normal derivative operators of sequentially higher transverse order to II e . This procedure was carried out in detail in [5], where such jet coefficients were called (conformal) fundamental forms. In particular, an mth conformal fundamental form is a symmetric rank-2 tensor-valued hypersurface density with weight 3 − m and transverse order m − 1.…”

“…Much of what follows in this section is a brief summary of explanations and results given in [15,16,5].…”

“…There are quite severe restrictions on conformal hypersurface embeddings that belong to either of these families [5]. Indeed, the vast majority of such embeddings do not belong to either family.…”

“…A consequence of this theorem is that, for d even, the Willmore invariant can always be written in terms of a specific finite set of fundamental conformally-invariant tensors known as conformal fundamental forms; this result is captured in Corollary 3.8. These tensors, initially characterized in [5], are described in more detail in Section 2.2. The critical tool used to prove this corollary is known as the ambient construction and is summarized in Section 4, along with other technical results required for the proof of Corollary 3.8.…”