Rigidity of Conformally Compact Manifolds with the Round Sphere as the Conformal Infinity

TL;DR

This paper proves that complete conformally compact manifolds with a pole, a lower Ricci curvature bound, and conformal infinity in the round sphere class must be hyperbolic space, under certain scalar curvature conditions.

Contribution

It establishes a rigidity result linking geometric and conformal boundary conditions to hyperbolic space, extending previous rigidity theorems.

Findings

Such manifolds are hyperbolic space under the given conditions.

The result applies to manifolds with a pole and specific curvature bounds.

It connects conformal infinity properties to the global geometry of the manifold.

Abstract

In this paper we prove that under a lower bound on the Ricci curvature and an asymptotic assumption on the scalar curvature, a complete conformally compact manifold $(M^{n+1},g)$, with a pole $p$ and with the conformal infinity in the conformal class of the round sphere, has to be the hyperbolic space.