Characterization of large isoperimetric regions in asymptotically hyperbolic initial data

TL;DR

This paper characterizes large isoperimetric regions in asymptotically hyperbolic 3-manifolds, showing they are uniquely given by the canonical foliation, without requiring rotational symmetry at infinity.

Contribution

It provides the first characterization of large isoperimetric regions in asymptotically hyperbolic manifolds without assuming rotational symmetry.

Findings

Canonical foliation leaves are the unique large isoperimetric regions.

The scalar curvature condition R ≥ -6 is essential.

First such characterization without symmetry assumptions.

Abstract

Let $(M,g)$ be a complete Riemannian $3$-manifold asymptotic to Schwarzschild-anti-deSitter and with scalar curvature $R \geq - 6$. Building on work of A.~Neves and G.~Tian and of the first-named author, we show that the leaves of the canonical foliation of $(M, g)$ are the unique solutions of the isoperimetric problem for their area. The assumption $R \geq -6$ is necessary. This is the first characterization result for large isoperimetric regions in the asymptotically hyperbolic setting that does not assume exact rotational symmetry at infinity.