Conformal Fundamental Forms and the Asymptotically Poincare-Einstein Condition

Kód výsledku v 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>

Výsledek na webu

<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>

Jazyk výsledku

angličtina

Název v původním jazyce

Conformal Fundamental Forms and the Asymptotically Poincare-Einstein Condition

Popis výsledku v původním jazyce

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.

Název v anglickém jazyce

Conformal Fundamental Forms and the Asymptotically Poincare-Einstein Condition

Popis výsledku anglicky

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.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

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OECD FORD obor

10101 - Pure mathematics

Projekt

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Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2023

Kód důvěrnosti údajů

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

Název periodika

Indiana University Mathematics Journal

ISSN

0022-2518

e-ISSN

1943-5258

Svazek periodika

72

Číslo periodika v rámci svazku

6

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

70

Strana od-do

2215-2284

Kód UT WoS článku

001166610900002

EID výsledku v databázi Scopus

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