具有相对论效应的Lagrange系统的整体变分方法

本项目将通过整体变分方法，研究class A 时空中一些基本而有趣的问题。具体来讲，在class A 环面上，我们将完善Aubry-Mather理论；我们将研究相应的Hamilton-Jacobi方程的粘性解的性质，从而建立具有相对论效应的弱KAM理论；我们将研究位于上同调锥内的那些上同调对应的调和形式的性质；我们还将研究systolic不等式及Hopf猜测等刚性问题。

In this program we will study some fundamental and interesting problems in class A spacetimes by the method of globally variational method. More precisely, on class A tori， we will make Aubry-Mather theory more complete; we will study the properties of viscosity solutions to the associated Hamilton-Jacobi equations, and thus build a foundation for weak KAM theory for relativistic Lagrangians; we will study the properties of harmonic forms associated to the cohomology in the interior of the cohomological cone; we will also study some rigidity problems, such as systolic inequalities, Hopf conjecture etc.