UC Santa Cruz

In this thesis we study the stability of the Ricci flow. The stability problem of Ricci flow in different settings have been considered by Ye \cite{Ye}, Li-Yin \cite{LY}, Schn\"urer-Schulze-Simon \cite{SSS} and Bamler \cite{Bam} etc. We consider a more general case and extend the results to the general case, that is, in the setting of asymptotically hyperbolic Einstein (AHE) manifolds with rough initial data. First we introduce the background of the problem and results on the long time behavior of Ricci flow in detail. Then we compare the difference in methodology of theses results and extend to the AHE case. We consider the normalized Ricci flow on a AHE manifold with initial metrics which are perturbations of a non-degenerate AHE metric $h_0$. The key step is to obtain exponential decay of certain geometric quantities. Then we prove that the normalized Ricci flow converges exponentially fast to $h_0$, if the perturbation is $L^2$-bounded and $C^0$-small.