几何与物理中若干非线性方程的存在性和奇性问题研究

In modern geometry and physics, many important problems can be expressed as nonlinear partial differential equations on manifolds, or they can be solved by using the theory of partial differential equations. The existence of solutions to these nonlinear equations is a fundamental problem in the analysis aspect, meanwhile, they can give direct or indirect solutions to the problems in geometry and physics. The solutions may develop singularities, this is crucial to the existence problem, furthermore, they often have rich geometric or topological properties, this kind of phenomema have appeared frequently in the study of harmonic maps and their heat flows, the mean curvature flows and the Ricci flows in recent years. We plan to study the existence problem and singularities for some nonlinear differential equations on manifolds. We will .study the convergence of the heat flow of Dirac-geodesics, the existence and convergence of global solutions, the blow up analysis for the heat flow of Dirac-harmonic maps in dimensions not less than two. We will study the existence problem and singularities and their geometric applications for generalized harmonic maps and their heat flows. We will also study singular solutions to some other related geometric differential equations, as well as nonlinear differential equations on manifolds with geometric singularities.

在现代几何与理论物理中，许多重要问题可以表示为流形上的非线性微分方程, 或者能够运用微分方程的理论方法加以解决。这些非线性方程解的存在性是分析学中的基本问题，同时，它们又直接或间接地给出几何与物理问题的解。这些方程的解可能会产生奇性，这不仅对于解的存在性问题至关重要，而且奇性本身往往具有丰富的几何拓扑性质，这类现象在近年来关于调和映照及其热流、平均曲率流和Ricci流的研究中不断地得到了印证。本项目计划对流形上若干非线性方程解的存在性和奇性问题展开研究。我们将研究拉克-测地线热流整体解的收敛性、二维以上情形狄拉克-调和映照热流的整体解的存在性、解的收敛性，以及解的Blow up性质等问题；研究广义调和映照及相应的广义调和映照热流的存在性、奇性等几何分析性质及其几何应用；研究其它几何非线性方程的奇性解与带几何奇性流形上的的非线性微分方程。

本项目研究了黎曼流形之间的广义调和映照、狄拉克-调和映照，相应于广义调和映照，研究了V-Laplace 算子的非线性方程、Ricci-Bourguinon流、Riemann 流形上的临界指数非线性方程等问题。 我们引入了一类新的广义调和映照，即VT-调和映照，它们有丰富的几何与物理背景。我们得到了VT-调和映照以及VT-调和映照热流方程解的存在性、唯一性定理，作为应用，得到Weyl流形、Hermitie流形、仿射流形之间调和映照，以及2维黎曼流形出发磁力调和映照的存在性定理；我们得到了完备非紧黎曼spin流形出发的带曲率项狄拉克-调和映照梯度估计和Liouville型定理；建立了V-Laplace 算子的非线性抛物型方程解的Li-Yau型估计，Harnack不等式，以及Souplet--Zhang型估计；得到了概Hermitie流形之间V-调和映照的 Schwarz 引理并给出其在全纯映照的应用；我们还得到了Ricci-Bourguinon流的拓展定理, Riemann 流形上临界指数非线性方程正解的存在性定理等结果。.