Comptes Rendus
no. 2
Differential geometry
The GBC mass for asymptotically hyperbolic manifolds
[La masse de Gauss–Bonnet–Chern sur des variétés asymptotiquement hyperboliques]

Présenté par : the Editorial Board

Yuxin Ge1 ; Guofang Wang2 ; Jie Wu2, 3
1 Laboratoire dʼanalyse et de mathématiques appliquées, CNRS UMR 8050, Département de mathématiques, Université Paris-Est–Créteil–Val-de-Marne, 61, avenue du Général-de-Gaulle, 94010 Créteil cedex, France
2 Albert-Ludwigs-Universität Freiburg, Mathematisches Institut, Eckerstr. 1, 79104 Freiburg, Germany
3 School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, PR China
Comptes Rendus. Mathématique, Volume 352 (2014) no. 2, pp. 147-151.

En utilisant la courbure de Gauss–Bonnet, on introduit une nouvelle masse dʼordre supérieur – la masse de Gauss–Bonnet–Chern –, sur des variétés asymptotiquement hyperboliques. On montre quʼil sʼagit dʼun invariant géométrique. On démontre également le théorème de masse positive sur des graphes sur lʼespace hyperbolique Hn et des inégalités dʼAlexandrov–Fenchel à poids dans Hn pour toute hypersurface convexe de type horosphérique. Ainsi, on obtient une inégalité de type Penrose optimale pour cette masse sur toute variété asymptotiquement hyperbolique qui est graphe sur Hn avec un horizon au bord, à condition que la condition dʼénergie dominante L˜k⩾0 soit satisfaite.

By using the Gauss–Bonnet curvature, we introduce a higher-order mass, the Gauss–Bonnet–Chern mass, for asymptotically hyperbolic manifolds and show that it is a geometric invariant. Moreover, we prove a positive mass theorem for this new mass for asymptotically hyperbolic graphs. Then, we prove the weighted Alexandrov–Fenchel inequalities in the hyperbolic space Hn for any horospherical convex hypersurface Σ. As an application, we obtain an optimal Penrose-type inequality for this new mass for asymptotically hyperbolic graphs with a horizon type boundary Σ, provided that a dominant energy condition L˜k⩾0 holds.

Reçu le :
Accepté le :
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DOI : 10.1016/j.crma.2013.11.019
Yuxin Ge 1 ; Guofang Wang 2 ; Jie Wu 2, 3

1 Laboratoire dʼanalyse et de mathématiques appliquées, CNRS UMR 8050, Département de mathématiques, Université Paris-Est–Créteil–Val-de-Marne, 61, avenue du Général-de-Gaulle, 94010 Créteil cedex, France
2 Albert-Ludwigs-Universität Freiburg, Mathematisches Institut, Eckerstr. 1, 79104 Freiburg, Germany
3 School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, PR China 
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Yuxin Ge; Guofang Wang; Jie Wu. The GBC mass for asymptotically hyperbolic manifolds. Comptes Rendus. Mathématique, Volume 352 (2014) no. 2, pp. 147-151. doi : 10.1016/j.crma.2013.11.019. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.11.019/

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☆ This project is partly supported by SFB/TR71 “Geometric partial differential equations” of DFG.

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