Kaehler流形与CR流形中极小曲面及相关问题

本项目主要研究Kaehler流形和CR流形中极小曲面相关的几何问题，主要包括以下几个方面:.(1) 用变分法结合连续性方法来研究Kaehler曲面中全纯曲线的存在性问题。我们考虑一族新的泛函，希望用连续性方法将这一族泛函的临界点形变到全纯曲线；.(2) 研究辛平均曲率流和拉格朗日平均曲率流。我们希望一方面继续研究这些平均曲率流的奇点，另一方面寻找一些保证几何流延拓的充分条件；.(3) 研究CR流形的子流形几何。我们一方面希望考虑Sasaki流形的不变子流形的存在性问题，另一方面希望研究三维CR流形中的自由边界问题的分类与稳定性；.(4)研究欧氏空间超曲面的体积比较定理。我们希望通过利用几何流的方法来研究欧式空间超曲面与欧式空间超球面的体积比较定理。.这些问题涉及变分法，椭圆型、抛物型方程，子流形的几何等方面，它们的研究有趣而重要。

This project is mainly going to consider the problems relating to geometry of minimal submanifolds in Kaehler manifold and CR manifold. More precisely, we will work on:. (1) the existence of holomorphic curves in Kaehler surface via variational method combining with continuity method. We will consider a new family of functionals, and aim to use the continuity method to obtain a holomorphic curve by deformating the critical points of these functionals;. (2) the symplectic mean curvature flow and Lagrangian mean curvature flow. On the one hand, we would like to continue to study the singularities of such mean curvature flows; on the other hand, we want to find some sufficient conditions to extend the geometric flow; . (3) problems related to submanifolds in CR manifold. On the one hand, we hope to study the eixstence of invariant submanifold in a Sasaki manifold via the variational method and geometric flow mthod; on the other hand, we plan to study classification and stability problems of the free boundary problem in three dimensional CR manifold;. (4) the volume comparison theorem for hypdersurfaces in Euclidean space. We plan to use gemetric flow method to study the volume comparison theorem for hypdersurfaces in Euclidean space and round sphere in Euclidean space.. These problems involve variational method, the study of elliptic and parabolic eqautions, geometry of submanifolds, and so on. They are interesting and important.