On the problem of filling by a Poincaré-Einstein metric in dimension 4

2025/09/22 by Sun‐Yung A. Chang, Chang, Sun-Yung Alice, Yuxin Ge +1 · 2 citations
Mathematics · #Geometry and complex manifolds #Nonlinear Partial Differential Equations #Geometric Analysis and Curvature Flows

paper · pdf · doi:10.48550/arxiv.2509.18430

Abstract

Given a metric defined on a manifold of dimension three, we study the problem of finding a conformal filling by a Poincaré-Einstein metric on a manifold of dimension four. We establish a compactness result for classes of conformally compact Einstein 4-manifolds under conformally invariant conditions. A key step in the proof is a result of rigidity for the hyperbolic metric on \mathbb B⁴ or S¹ × \mathbbB³. As an application, we also derive some existence results of conformal filling in for metrics in a definite size neighborhood of the canonical metric; when the conformal infinity is either S³ or S¹ × S².

Cited by

• Poincaré-Einstein 4-manifolds with conformally Kähler geometry
• On the Poincaré-Einstein manifolds with cylindrical conformal infinity

Related

• Compactness of conformally compact Einstein manifolds in dimension 4
• Einstein metrics with prescribed conformal infinity on 4-manifolds
• Self-dual and even Poincaré-Einstein metrics in dimension four
• On the structure of conformally compact Einstein metrics
• On compactness conformally compact Einstein manifolds and uniqueness of Graham-Lee metrics, III