Penrose type inequalities for asymptotically hyperbolic graphs

Mattias Dahl (1) , Romain Gicquaud (2) , Anna Sakovich (1)

1 KTH STOCKHOLM - Institutionen för Matematik - KTH
2 LMPT - Laboratoire de Mathématiques et Physique Théorique

In this paper we study asymptotically hyperbolic manifolds given as graphs of asymptotically constant functions over hyperbolic space $\bH^n$. The graphs are considered as subsets of $\bH^{n+1}$ and carry the induced metric. For such manifolds the scalar curvature appears in the divergence of a 1-form involving the integrand for the asymptotically hyperbolic mass. Integrating this divergence we estimate the mass by an integral over an inner boundary. In case the inner boundary satisfies a convexity condition this can in turn be estimated in terms of the area of the inner boundary. The resulting estimates are similar to the conjectured Penrose inequality for asymptotically hyperbolic manifolds. The work presented here is inspired by Lam's article concerning the asymptotically Euclidean case.

Mots clés: Asymptotically hyperbolic manifold; Riemannian Penrose inequality

Domaines: Géométrie différentielle [math.DG]; Relativité Générale et Cosmologie Quantique [gr-qc]

Mattias Dahl, Romain Gicquaud, Anna Sakovich. Penrose type inequalities for asymptotically hyperbolic graphs. Annales Henri Poincaré, 2012, pp.ISBN: 1424-0637. ⟨10.1007/s00023-012-0218-4⟩. ⟨hal-00783623⟩