Conformal geometry of embedded manifolds with boundary from universal holographic formulæ

authored by

Cesar Arias, A. Rod Gover, Andrew Waldron

Abstract

For an embedded conformal hypersurface with boundary, we construct critical order local invariants and their canonically associated differential operators. These are obtained holographically in a construction that uses a singular Yamabe problem and a corresponding minimal hypersurface with boundary. They include an extrinsic Q-curvature for the boundary of the embedded conformal manifold and, for its interior, the Q-curvature and accompanying boundary transgression curvatures. This gives universal formulæ for extrinsic analogs of Branson Q-curvatures that simultaneously generalize the Willmore energy density, including the boundary transgression terms required for conformal invariance. It also gives extrinsic conformal Laplacian power type operators associated with all these curvatures. The construction also gives formulæ for the divergent terms and anomalies in the volume and hyper-area asymptotics determined by minimal hypersurfaces having boundary at the conformal infinity. A main feature is the development of a universal, distribution-based, boundary calculus for the treatment of these and related problems.

Organisation(s)

Riemann Center for Geometry and Physics

External Organisation(s)

Universidad Andres Bello
University of Auckland
University of California at Davis

Type

Artikel

Journal

Advances in mathematics

Volume

384

No. of pages

59

ISSN

0001-8708

Publication date

25.06.2021

Publication status

Veröffentlicht

Peer reviewed

Yes

ASJC Scopus Sachgebiete

Mathematik (insg.)

Electronic version(s)

https://doi.org/10.48550/arXiv.1906.01731 (Access: Offen)
https://doi.org/10.1016/j.aim.2021.107700 (Access: Geschlossen)