UC Santa Barbara

In this thesis, we study linear stability of Einstein metrics and develop the theory of Perelman's $\lambda$-functional on compact manifolds with isolated conical singularities. The thesis consists of two parts. In the first part, inspired by works in \cite{DWW05}, \cite{GHP03}, and \cite{Wan91}, by using a Bochner type argument, we prove that complete Riemannian manifolds with non-zero imaginary Killing spinors are stable, and provide a stability condition for Riemannian manifolds with non-zero real Killing spinors in terms of a twisted Dirac operator. Regular Sasaki-Einstein manifolds are essentially principal circle bundles over K\"{a}hler-Einstein manifolds. We prove that if the base space of a regular Sasaki-Einstein manifold is a product of at least two K\"{a}hler-Einstein manifolds, then the regular Sasaki-Einstein manifold is unstable. More generally, we show that Einstein metrics on principal torus bundles constructed in \cite{WZ90} are unstable, if the base spaces are products of at least two K\"{a}hler-Einstein manifolds.

In the second part, we prove that the spectrum of $-4\Delta+R$ consists of discrete eigenvalues with finite multiplicities on a compact Riemannian manifold of dimension $n$ with a single conical singularity, if the scalar curvature of cross section of conical neighborhood is greater than $n-2$. Moreover, we obtain an asymptotic behavior for eigenfunctions near the singularity. As a consequence of these spectrum properties, we extend the theory of Perelman's $\lambda$-functional on smooth compact manifolds to compact manifolds with isolated conical singularities.