Analysis of some PDEs over manifolds

In this dissertation I discuss and investigate the analytic aspect of several elliptic and parabolic partial differential equations arising from Riemannian and complex geometry, including the generalized Ricci flow, Gaussian curvature flow of negative power, mean curvature flow of positive power, harmonic-Ricci flow, vector field flow, and also Mabuchi-Yau functionals and Donaldson equation over complex manifolds.