UC Irvine

This dissertation contains two parts. The first part considers related problems of Laplace operator on Kähler 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 Kähler manifold. Together with Hezari, Kelleher and Seto [9], we give an alternative proof of the asymptotic expansion.