On the Spectrum of Laplace Operator and Asymptotic Expansion of Bergman Kernel on Kahler Manifolds

This dissertation contains two parts. The first part considers related problems of Laplace operator on Kahler manifolds. Together with my advisor Zhiqin Lu, we generalized the spectrum relation in [5] to any Hermitian manifolds. And we proved the closure of Laplace operator on the moduli space of polarized Calabi-Yau manifolds is self-adjoint. The second part considers the asymptotic expansion of the Bergman kernel on a polarized Kahler manifold. Together with Hezari, Kelleher and Seto [9], we give an alternative proof of the asymptotic expansion.