Lagrangean Isotopy Problem and Special Lagrangian Submanifolds in Symplectic Geometry

格罗莫夫关于伪全纯曲线的学术论文开创了辛几何的新纪元。在那里发展起来的理论，以及Donaldson和Seiber-Witten理论，可能是现代微分几何中下列原理最壮观的表现：研究与问题有关的某些非线性椭圆型偏微分方程整体解的性质，可以深刻地洞察潜在流形的几何。该研究项目的目的是将这一原理应用于辛流形中的拉格朗日子流形理论，其中相应的偏微分方程组是由特殊的拉格朗日条件施加的，并将所发展的技术应用于拉格朗日同位问题的研究。

The seminar paper of M. Gromov on pseudoholomorphic curves had started a new era in the symplectic geometry. The theory developed there, as also the Donaldson and Seiber-Witten theories, are probably the most spectacular manifestation of the following principle of modern differential geometry: The study of properties of global solutions of certain non-linear elliptic PDEs related to the problem gives deep insight into the geometry of the underlying manifold. The aim of the proposed research project is to apply this principle to the theory of Lagrangean submanifolds in symplectic manifolds, in which the corresponding PDE is imposed by the special Lagrangean condition, and then use the developed techniques in the study of the Lagrangean isotopy problem.