调和映射热流中的若干问题

Geometric flow is doubtlessly one of the most active research fields in geometric analysis in these years. Our project concerns mainly various aspects of the harmonic map heat flow, including the regularity, convergence and its application to minimal surface. For general targets, it's a difficult problem to get the convergence of 2-dimenional harmonic map heat flow. We'll attempt to get some new results, and hopefully, this will be helpful to find minimal surfaces in general manifolds, as well as application to the symplectic isotopy problem. This can also be regarded as the reduced model of our general harmonic-conformal flow when the genus of the domain is zero. We also plan to study the harmonic map heat flow coupled with other geometric flows, mainly the long time existence and convergence of the harmonic-conformal flow for domains with higher genus. The usual method nowadays to find minimal surface in general manifolds via heat flow is the mean curvature flow. The harmonic-conformal flow is a new approach to this problem, and the analysis of the singular set may be simpler, together with more information to be given. We are also interested in the regularity issue of the higher dimenional harmonic map heat flow, especially the existence of the solution with minimal initial data blowing up in finite time, which has positive answer in the Navier-Stokes equation. To the best of our knowledge, little achievement has been made in this field.

几何流无疑是当今几何分析领域中最前沿的方向之一。本项目主要研究调和映射热流的正则性、收敛性及其在极小曲面中的应用。对一般的目标流形，如何得到二维调和映射热流的一致收敛性是一个困难的问题，我们预期能得到一些新的结果，并将其应用于寻找一般流形中的极小球面，甚至希望用于研究著名的辛同痕问题。我们同时研究调和映射热流与其他几何流的耦合，重点是高亏格曲面上调和共形流的长时间存在性和收敛性，特别是在负曲率目标流形的情形。目前以热方程来寻找一般流形上的极小曲面的主要方法是平均曲率流，而我们所研究的调和共形流是不同于平均曲率流的新的方法。相对来说，奇点分析更为简单，也能给出更多的信息。我们也将用Navier-Stokes方程中的技巧来研究高维调和映射热流的正则性，特别是极小爆破解的存在性问题。目前为止，用这种技巧来讨论调和映射热流正则性的结果还非常之少。

调和映射，极小曲面及其相关的几何流是几何分析中重要的研究方向。本项目主要研究了二维调和映射热流的一致收敛性以及调和共形流的短时存在性和长时间存在性。对于从二维球面出发到具有非负双全纯截面曲率的Kahler流形的调和映射热流，我们证明：对于能量密度一致有界且反全纯能量足够小的初值映射，其相应的调和映射热流长时间光滑存在，且一致（指数）收敛到一个全纯映射。我们也研究了从一般高亏格曲面出发的调和共形流。对于非正曲率的目标流形，借助于源流形面元不变的特性，我们得到了映射和度量的各阶导数在有限时刻的有界性，进而得到该流的长时间存在性。以上结果对于用热流方法寻找极小曲面有一定的意义。