Rigidity of Poincaré-Einstein manifolds with flat Euclidean conformal infinity
Sanghoon Lee, Fang Wang
TL;DR
This work proves a rigidity result for complete Poincaré-Einstein manifolds with flat Euclidean conformal infinity, showing that under an adapted boundary defining function and nonnegative scalar curvature of the compactified metric, together with quadratic decay of the full curvature, the manifold must be the standard hyperbolic space. The approach centers on propagating curvature information along level sets of the adapted defining function, leveraging monotonicity formulas, Liouville-type arguments for Schrödinger-type operators, and a curved Hardy inequality to manage regularity issues. The authors extend the rigidity to generalized adapted compactifications with a parameter $\gamma>\max(1,\tfrac{n}{2}-1)$ and provide a robust framework that unifies 4D and higher-dimensional cases via a hierarchy of curvature functionals $F_{k,l}$. Beyond the rigidity results, the paper constructs explicit noncompact-boundary examples through stereographic projections from smooth, non-smooth, and singular conformally compact Einstein manifolds, demonstrating quadratic curvature decay and enriching the landscape of PE geometries with noncompact infinities. These insights have potential implications for moduli questions and blow-up analyses in PE geometry and related AdS/CFT contexts.
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.