位势调和映照一般理论及其在物理与几何问题中的应用

The goal of this program is to systematically establish and develop a.general theory of harmonic map with potential (HMP), and give some applications of it. We established some basic properties of HMP: the first variation formula, Bochner type formula, basic gradient estimates and maximum principles; studied the affection of the potentials on the behavior of HMP, for HMP from compact manifolds with boundary and.complete noncompact manifolds into nonpositively curved manifolds, we clearly described how the analytic properties of the potential functions and the geometric properties of the manifolds influence the global existence, uniqueness and convergence of the heat flow, we first.discovered the relationship between the Hessian of the potential functions,.the first Dirichlet eigenvalue of the domain manifolds and the properties of HMP, finding the corresponding optimal condition; we established the second variation formula for HMP, and obtained some results on stability of these maps; for constant boundary value problems of HMP on geodesic balls in certain anifolds, we obtained a constancy theorem under considerable general conditions, it includes the corresponding results of the usual harmonic maps, and covers the important cases of the Euclidean space, the complex hyperbolic space and bounded.symmetric domains; we constructed multiple large smooth solutions of HMP on two dimensional discs; after establishing the basic frames for the general theory of HMP, we applied our results to the anisotropic Landau-Lifshitz equations on high dimensional domains in the theory of.continuous ferro-magnetic spin fields and got some results on maximum principles, existence and uniqueness of the solutions.

本项目研究黎曼流行间一类新的映照，即位势调和映照。它不仅包含通常的调和映照作为其重要特例，而且还有着非常丰富的物理和数学背景。如连续铁磁旋转场中著名的郎道—立弗希兹方程和超导理论的金兹伯格—郎道方程等物理问题，以及等参超曲面等几何问题，都可以作为其特殊情形。我们将系统地建立关于这类映照的一般理论，并给出物理与几何应用。...