Date of Completion

7-10-2020

Embargo Period

7-7-2020

Keywords

Asymptotically hyperbolic manifolds, Positive mass theorem, Ricci flow, Warped product

Major Advisor

Lan-Hsuan Huang

Associate Advisor

Ovidiu Munteanu

Associate Advisor

Damin Wu

Associate Advisor

Guozhen Lu

Field of Study

Mathematics

Degree

Doctor of Philosophy

Open Access

Open Access

Abstract

Asymptotically hyperbolic manifolds are natural objects to be considered in certain physical circumstances. This dissertation particularly focuses on construction and rigidity of such manifolds, which includes the following three main results.

Firstly, we obtain a new construction of a 3-dimensional asymptotically hyperbolic manifold from a 2-sphere by using a solution of the Ricci flow as a foliation. These asymptotically hyperbolic manifolds provide examples of `admissible extensions' in the context of an asymptotically hyperbolic analogue of the Bartnik mass.

Secondly, we prove the equality case of the positive mass theorem for asymptotically hyperbolic manifolds without a spin assumption. This is the last piece necessary to complete the proof of the positive mass theorem in the asymptotically hyperbolic setting, which was a long-standing open problem in the area.

Lastly, we establish some warped product splitting theorems with a scalar curvature lower bound. The imposed conditions are motivated by the notion of outer-trapped surfaces, which is used to study black holes using local geometry. Moreover, the resulting warped product serves as a model space for asymptotically locally hyperbolic manifolds.

Download
COinS