Large isoperimetric regions in asymptotically hyperbolic manifolds

Abstract

We show the existence of isoperimetric regions of sufficiently large volumes in general asymptotically hyperbolic three manifolds. Furthermore, we show that large coordinate spheres in compact perturbations of Schwarzschild-anti-deSitter are uniquely isoperimetric. This is relevant in the context of the asymptotically hyperbolic Penrose inequality. Our results require that the scalar curvature of the metric satisfies $R_g\ge -6$, and we construct an example of a compact perturbation of Schwarzschild-anti-deSitter without $R_g\ge -6$ so that large centered coordinate spheres are not isoperimetric. The necessity of scalar curvature bounds is in contrast with the analogous uniqueness result proven by Bray for compact perturbations of Schwarzschild, where no such scalar curvature assumption is required. This demonstrates that from the point of view of the isoperimetric problem, mass behaves quite differently in the asymptotically hyperbolic setting compared to the asymptotically flat setting. In particular, in the asymptotically hyperbolic setting, there is an additional quantity, the "renormalized volume," which has a strong effect on the large-scale geometry of volume.

57 pages, 1 figure. Comments welcome!

References in corpus (5)

• Rigidity and Positivity of Mass for Asymptotically Hyperbolic Manifolds
• Present status of the Penrose inequality
• Effective versions of the positive mass theorem
• The isoperimetric inequality on asymptotically flat manifolds with nonnegative scalar curvature
• Existence and characterization of regions minimizing perimeter under a volume constraint inside Euclidean cones

Cited by in corpus (14)

• Effective versions of the positive mass theorem
• General Bounds on Holographic Complexity
• On the existence of isoperimetric regions in manifolds with nonnegative Ricci curvature and Euclidean volume growth
• Almost Rigidity of the Positive Mass Theorem for Asymptotically Hyperbolic Manifolds with Spherical Symmetry
• Isoperimetry, scalar curvature, and mass in asymptotically flat Riemannian $3$-manifolds
• The isoperimetric problem via direct method in noncompact metric measure spaces with lower Ricci bounds
• Rigidity of Hawking mass for surfaces in three manifolds
• Exhaustion of isoperimetric regions in asymptotically hyperbolic manifolds with scalar curvature $R\ge -6$
• Characterization of large isoperimetric regions in asymptotically hyperbolic initial data
• Existence and uniqueness of constant mean curvature foliations of general asymptotically hyperbolic 3-manifolds
• Regularity of inverse mean curvature flow in asymptotically hyperbolic manifolds with dimension $3$
• Existence of CMC-foliations in asymptotically cuspidal manifolds
• Geometric characterizations of asymptotically hyperbolic Riemannian 3-manifolds by the existence of a suitable CMC-foliation
• A Penrose-type inequality for static spacetimes