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BLITZ, Samuel Harris, A Rod GOVER a Andrew WALDRON. Conformal Fundamental Forms and the Asymptotically Poincare-Einstein Condition. Indiana University Mathematics Journal. INDIANA UNIV MATH JOURNAL, 2023, roč. 72, č. 6, s. 2215-2284. ISSN 0022-2518. Dostupné z: https://dx.doi.org/10.1512/iumj.2023.72.9518.

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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  -

Základní údaje
Originální název	Conformal Fundamental Forms and the Asymptotically Poincare-Einstein Condition
Autoři	BLITZ, Samuel Harris (840 Spojené státy, domácí), A Rod GOVER a Andrew WALDRON.
Vydání	Indiana University Mathematics Journal, INDIANA UNIV MATH JOURNAL, 2023, 0022-2518.

Další údaje
Originální jazyk	angličtina
Typ výsledku	Článek v odborném periodiku
Obor	10101 Pure mathematics
Stát vydavatele	Spojené státy
Utajení	není předmětem státního či obchodního tajemství
WWW	URL
Impakt faktor	Impact factor: 1.100 v roce 2022
Kód RIV	RIV/00216224:14310/23:00133850
Organizační jednotka	Přírodovědecká fakulta
Doi	http://dx.doi.org/10.1512/iumj.2023.72.9518
UT WoS	001166610900002
Klíčová slova anglicky	Extrinsic conformal geometry; hypersurface embeddings; Poincare-Einstein metrics; Yamabe problem
Štítky	rivok
Příznaky	Mezinárodní význam, Recenzováno
Změnil	Změnila: Mgr. Marie Šípková, DiS., učo 437722. Změněno: 21. 3. 2024 10:29.

Anotace
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.

Anotace

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.