拉格朗日子流形的若干问题

Lagrangian submanifolds appear naturally in the context of classical mechanics and mathematical physics. Also,they have rich geometric properties which appear naturally in string theory, therefore, many mathematician and physic scientists are interested in the research of the Lagrangian submanifold geometry. This project focuses on some problems of Lagrangian submanifold,which are closed related to the Partial differential equation, Complex geometry and Sympletic geometry, and are also the important projects of global differential geometry. We will mainly study the rigidity problems of special Lagrangian submanifolds and submanifolds with higher codimension as well as the construction and classification of some Lagrangian submanifolds with special properties. The problems of rigidity are the Bernstein type problems of special Lagrangian submanifolds and submanifolds with higher codimension. Construction of special Lagrangian submanifolds is beneficial to understanding the geometric property of special Lagrangian fibration and constructing of special Lagrangian fibration.

拉格朗日子流形是经典力学和数学物理中自然出现的研究对象，其本身有丰富的几何性质，这些性质很自然地出现在玄理论的研究中，因此拉格朗日子流形受到了数学家和物理学家的关注。本项目主要研究拉格朗日子流形的若干问题，这些问题与偏微分方程、复几何和辛几何等数学分支密切相关,是整体微分几何的重要课题之一。我们拟研究特殊拉格朗日子流形以及一般高余维极小子流形的刚性问题和具有某些特殊性质的拉格朗日子流形的显示构造、分类以及一些重要性质。刚性问题主要是特殊拉格朗日子流形及高余维极小子流形的伯恩斯坦问题。显示构造有益于我们理解特殊拉格朗日纤维化的几何结构和奇性构造。

项目成员在国家自然科学基金项目的资助下，做了如下工作：给出p-双调和子流形是极小子流形的判定条件，同时得到截面曲率为非正常数的空间形式的弱凸p-双调和超曲面一定是极小超曲面的结果；得到从Hermitian流形到凯勒流形的多重调和映射的一些刚性结果；利用应力能量张量方法和Hessian比较定理，得到广义调和映射（包含f-调和映射、F-稳态映射、弱f-稳态映射等）能量单调公式，并得到系列刘维尔型结果；将Green和Wu的比较定理进一步完善，使得Hessian比较定理和拉普拉斯比较定理在更广区域上成立；得到关于双曲空间形式中超曲面的一些刚性结果。