Monotonicity Formulae, Vanishing Theorems And Some Geometric Applications

In1980, Baird and Eells [BE] introduced the stress-energy tensor for maps between Riemannian manifolds, which unifies various results on harmonic maps. Stress-energy tensor is a useful tool for investigating the energy behaviour of the critical point of en-ergy functional, and also has important applications in geometric analysis problem, such as harmonic maps, Yang-Mills fields etc.. It is well known that harmonic maps are the nonlinear generalization of harmonic1-forms, and Yang-Mills fields are the nonlinear gen-eralization of harmonic2-forms. They all satisfy the conservation law. The critical points of geometric variation problem usually satisfy the conservation laws or its related proper-ties, so it is meaningful to study vector bundle-valued p-forms, especially those satifying the conservation laws.In this paper, we first establish some monotonicity formulae for vector bundle-valued p-forms by using stress-energy tensor. It is well known that the monotonicity formu-lae has many important applications in geometric variation problem, such as regularity theory, unique continuation problem, spectrum problem, vanishing theorem etc.. These applications involve global and local aspects. We focus on the global applications of the monotonicity formulae and try to get vanishing theorem under the energy growth con-dition, and finally we give some geometric applications. The applications in chapter2involve the Bernstein theorem for minimal submanifolds, the Ricci flatness theorem for Kahler manifolds and the uniformization type theorems. In chapter3, we use stress-energy tensor to study the L2-eigenforms for Laplacian acting on p-forms on complete manifolds and get some nonexistence results under various geometric conditions. And in chapter4, we use stress-energy tensor to study Liouville theorem for F-harmonic maps and its geometric applications on complete manifolds.Another main aspect of this paper is to investigate some global rigidity phenomenon in differential geometry. Global rigidity theorems are usually deduced by the vanishing of some geometric quantities, that is, expressed as vanishing theorems. Curvature pinching problem plays an important role in global differential geometry. In chapter5, we get a gap theorem for self-shrinkers in mean curvature flow. In chapter6, we deduce some Lp Ricci curvature pinching theorems for locally conformally flat Riemannian manifolds.