Analysis of Some PDEs over Manifolds

Abstract

In this dissertation I discuss and investigate the analytic aspect of several elliptic and parabolic partial differential equations arising from Rimannian 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. The new ingredient of this thesis is the first introduction to the vector field flow, a joint work with Professor Kefeng Liu at UCLA, which may lead to a new way to prove Hopf conjectures, which state that any positively curved manifold of even dimension has positive Euler characteristic and \(\mathbb{S}^2 x \mathbb{S}\) does not admit any Riemannian metric of positive sectional curvature, and give affirmative answer to Yau's question on the existence of an effective \(\mathbb{S}^1\)-action on positively curved manifolds. Furthermore, our flow is naturally related to Navier-Stokes equations. The vector field flow is defined as \(\partial_t(X_t)^i=\Delta_{LB}(X_t)^i+\nabla^idiv(X_t)+R^i_j(X_t)^j, X_0=X,\) where X is a fixed vector field on \(\mathcal{M}, \partial_t \doteqdot \frac {\partial} {\partial t}\) is the time derivative, and \(\Delta_{LB}\)denotes the Laplace-Beltrami operator on \(\mathcal{M}\). If we define \(Ric^\#\), the (1, 1)-tensor field associated to Ric, by \(g(Ric^\#(X),Y) \doteqdot Ric(X,Y)\), where X,Y are two vector fields, then \(Ric^\#\) is an operator on the space of vector fields, denoted by \(C^\infty (\mathcal{M},T\mathcal{M})\), and the above flow can be rewritten as \(\partial_tX_t=\Delta_{LB}X_t+\nabla div(X_t)+Ric^\#(X_t)\). We establish some basic properties, including long time existence and convergence, and formulate a conjecture on the vector field flow on positively curved manifold. This conjecture immediately implies Hopf conjectures and Yau's question.