The Ricci Flow of Asymptotically Hyperbolic Mass

Abstract

In this thesis, we generalize the notion of asymptotically hyperbolic mass (first introduced by Wang in 2001) to manifolds with toroidal ends. Using this generalized definition, we show that under a normalized Ricci flow with asymptotically hyperbolic, conformally compact initial data with a well-defined mass, the mass will decay exponentially in time to zero, in contradistinction to the constant behaviour of asymptotically flat mass under Ricci flow. We then use this result for the evolution of asymptotically hyperbolic mass to prove that there does not exist a breather solution to the normalized Ricci flow with non-zero mass. Further, we provide a proof of the rigidity case of the Positive Mass Theorem in the asymptotically hyperbolic setting, using Ricci flow. We note that this result for the exponential behaviour of asymptotically hyperbolic mass provides support for a conjecture in general relativity stated by Horowitz and Myers.