Conformal Fundamental Forms and the Asymptotically Poincare-Einstein Condition

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216224%3A14310%2F23%3A00133850" target="_blank" >RIV/00216224:14310/23:00133850 - isvavai.cz</a>

Result on the web

<a href="https://www.iumj.indiana.edu/oai/2023/72/9518/9518.xml" target="_blank" >https://www.iumj.indiana.edu/oai/2023/72/9518/9518.xml</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1512/iumj.2023.72.9518" target="_blank" >10.1512/iumj.2023.72.9518</a>

Result language

angličtina

Original language name

Conformal Fundamental Forms and the Asymptotically Poincare-Einstein Condition

Original language description

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.

Czech name

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Czech description

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Type

J_imp - Article in a specialist periodical, which is included in the Web of Science database

CEP classification

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OECD FORD branch

10101 - Pure mathematics

Project

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Continuities

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year

2023

Confidentiality

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Name of the periodical

Indiana University Mathematics Journal

ISSN

0022-2518

e-ISSN

1943-5258

Volume of the periodical

72

Issue of the periodical within the volume

6

Country of publishing house

US - UNITED STATES

Number of pages

70

Pages from-to

2215-2284

UT code for WoS article

001166610900002

EID of the result in the Scopus database

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