Conformal Fundamental Forms and the Asymptotically
Poincare-Einstein Condition

J 2023

Conformal Fundamental Forms and the Asymptotically Poincare-Einstein Condition

BLITZ, Samuel Harris, A Rod GOVER and Andrew WALDRON

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

Language

English

Type of outcome

Článek v odborném periodiku

Field of Study

10101 Pure mathematics

Country of publisher

United States of America

Confidentiality degree

není předmětem státního či obchodního tajemství

References:

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

Abstract

V originále

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.

Citovat

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.

@article{2386463,
   author = {Blitz, Samuel Harris and Gover, A Rod and Waldron, Andrew},
   article_number = {6},
   doi = {http://dx.doi.org/10.1512/iumj.2023.72.9518},
   keywords = {Extrinsic conformal geometry; hypersurface embeddings; Poincare-Einstein metrics; Yamabe problem},
   language = {eng},
   issn = {0022-2518},
   journal = {Indiana University Mathematics Journal},
   title = {Conformal Fundamental Forms and the Asymptotically Poincare-Einstein Condition},
   url = {https://www.iumj.indiana.edu/oai/2023/72/9518/9518.xml},
   volume = {72},
   year = {2023}
}

TY  - JOUR
ID  - 2386463
AU  - Blitz, Samuel Harris - Gover, A Rod - Waldron, Andrew
PY  - 2023
TI  - Conformal Fundamental Forms and the Asymptotically Poincare-Einstein Condition
JF  - Indiana University Mathematics Journal
VL  - 72
IS  - 6
SP  - 2215-2284
EP  - 2215-2284
PB  - INDIANA UNIV MATH JOURNAL
SN  - 00222518
KW  - Extrinsic conformal geometry
KW  - hypersurface embeddings
KW  - Poincare-Einstein metrics
KW  - Yamabe problem
UR  - https://www.iumj.indiana.edu/oai/2023/72/9518/9518.xml
N2  - 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.
ER  -

BLITZ, Samuel Harris, A Rod GOVER and Andrew WALDRON. Conformal Fundamental Forms and the Asymptotically Poincare-Einstein Condition. \textit{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.

Detailed Information on Publication Record