The GBC mass for asymptotically hyperbolic manifolds

Ge, Yuxin; Wang, Guofang; Wu, Jie

Comptes Rendus. Mathématique, Tome 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 ${\mathbb{H}}^n$ et des inégalités dʼAlexandrov–Fenchel à poids dans ${\mathbb{H}}^n$ 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 ${\mathbb{H}}^n$ avec un horizon au bord, à condition que la condition dʼénergie dominante ${\tilde{L}}_k⩾0$ 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 ${\mathbb{H}}^n$ 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 ${\tilde{L}}_k⩾0$ holds.

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