Conformal Fundamental Forms and the Asymptotically Poincare-Einstein Condition

BLITZ, Samuel Harris, A Rod GOVER and Andrew WALDRON. Conformal Fundamental Forms and the Asymptotically Poincare-Einstein Condition. Indiana University Mathematics Journal. INDIANA UNIV MATH JOURNAL, 2023, vol. 72, No 6, p. 2215-2284. ISSN 0022-2518. Available from: https://dx.doi.org/10.1512/iumj.2023.72.9518.

Basic information

• Original name: Conformal Fundamental Forms and the Asymptotically Poincare-Einstein Condition
• Authors: BLITZ, Samuel Harris (840 United States of America, belonging to the institution), A Rod GOVER and Andrew WALDRON.
• Edition: Indiana University Mathematics Journal, INDIANA UNIV MATH JOURNAL, 2023, 0022-2518.

Other information

• Original language: English
• Type of outcome: Article in a journal
• Field of Study: 10101 Pure mathematics
• Country of publisher: United States of America
• Confidentiality degree: is not subject to a state or trade secret
• WWW: URL
• Impact factor: Impact factor: 1.100 in 2022
• RIV identification code: RIV/00216224:14310/23:00133850
• Organization unit: Faculty of Science
• Doi: http://dx.doi.org/10.1512/iumj.2023.72.9518
• UT WoS: 001166610900002
• Keywords in English: Extrinsic conformal geometry; hypersurface embeddings; Poincare-Einstein metrics; Yamabe problem
• Tags: rivok
• Tags: International impact, Reviewed

Abstract

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