Rigidity of Poincaré-Einstein manifolds with flat Euclidean conformal infinity

Abstract

In this paper, we prove a rigidity theorem for Poincaré-Einstein manifolds whose conformal infinity is a flat Euclidean space. The proof relies on analyzing the propagation of curvature tensors over the level sets of an adapted boundary defining function. Additionally, we provide examples of Poincaré-Einstein manifolds with non-compact conformal infinities. Furthermore, we draw analogies with Ricci-flat manifolds exhibiting Euclidean volume growth, particularly when the compactified metric has non-negative scalar curvature.

50 pages, all comments are welcome!