Asymptotically Hyperbolic 3-Metric with Ricci Flow Foliation

Abstract

In general relativity, there have been a number of successful constructions for asymptotically flat metrics with a certain background foliation. In particular, Lin (Calc Var 49(3–4):1309–1335, 2014) used a foliation by the Ricci flow on 2-spheres to establish an asymptotically flat extension, and Sormani and Lin (Ann Henri Poincaré 17(10):2783–2800, 2016) proved useful results with this extension. In this paper, we construct asymptotically hyperbolic 3-metrics with the Ricci flow foliation inspired by Lin’s work. We also study the rigid case when the Hawking mass of the inner surface of the manifold agrees with its total mass as in Sormani and Lin (2016).