Abstract
In this thesis, we consider nonlinear evolution equations on manifolds. In chapter 1, we introduce some problems of the deformation of Riemannian metrics and the motion of submanifolds which my research concerned in the past few years.In chapter 2, based on Bochner formula, mass decay estimates and elliptic Moser iteration, we first show the global existence of the 3-dimensional Calabi flow on any closed 3-manifold with an arbitrary background metric g0. Second, we show the asymptotic convergence of a subsequence of solutions of the Calabi flow on a closed 3-manifold. With its application, we prove the existence of extremal metrics for quadratic functional of scalar curvature on a closed 3-manifold which is served as an extension of the Yamabe problem on closed manifolds.In chapter 3, we prove the Harnack estimate for the evoloved curvature R of the modified Ricci flow on complete R2 under some curvature assumptions.In chapter 4, we show that, under suitable assumptions for a initial Lagrangian torus T02, the shape of Lagrangian tori Tt2 in R4 under the mean curvature flow approaches the shape of a product torus very rapidly. In particular, no singularities will develop before the Lagrangian tori Tt2 collapse to a point in a finite time.