UC Santa Cruz

This thesis presents a comprehensive investigation into the properties of asymptoti- cally hyperbolic manifolds and provides an exact definition for asymptotically symmetric manifolds.Chapter 1 begins with a thorough classification of symmetric spaces of non-compact type, as detailed in Section 1.1. Utilizing parabolic geometry, we then explore the boundary geometry of symmetric spaces of non-compact type, aiming to precisely define asymptotically symmetric manifolds in Section 1.2. Chapter 2 focuses on the perturbation existence of asymptotically hyperbolic Einstein manifolds. Following the methodology proposed by O. Biquard, we present the concep- tual proof of perturbation existence for general asymptotically symmetric manifolds, as outlined in their work [5]. In Chapter 3, we examine the stability of asymptotically hyperbolic Einstein manifolds under normalized Ricci flow. Drawing on R. Bamler’s research [1], we establish a reduction of the stability problem to estimating the heat kernel for the Lichnerowicz operator (refer to Lemma 3.2.2). Furthermore, we discuss the underlying ideas behind proving these heat kernel estimates. Finally, in the last chapter, we introduce our improved result on long-time existence, building upon the work presented in [42]. This enhancement in long-time existence demonstrates the significant contributions made by this thesis.