次黎曼流形中的等周问题

本课题拟用几何测度、非线性分析等方法，深入研究次黎曼流形中的等周问题。主要研究等周集的存在性、等周集边界的几何刻画、Sobolev不等式与局部等周不等式的等价性等。对次黎曼Carnot群之间Lipschitz映射，构建面积、余面积公式，研究Carnot群中常平均曲率曲面的稳定性、极小化子的正则性，给出基于Current的第一、第二基本形式以及Varifold的变分公式；构建Current的形变定理和闭包定理，证明等周不等式和Pansu猜想。这些研究可望进一步弄清次黎曼Carnot群的本质结构，建立Carnot群中关于Current的几何分析框架，进一步发展次黎曼Carnot群中几何测度论和次椭圆算子理论。由于次黎曼Carnot群在控制论、经典力学、规范场论、次椭圆算子等纯粹数学和应用数学领域都有重要的应用，它在过去的二十年来引起了广泛的关注，所以开展这一领域的研究非常必要。

This project will focus on studying deeply the isoperimetric problem in sub-Riemannian manifolds by virtue of the methods of Geometric Measure Theory and nonlinear analysis,etc. We will study mainly the existence of isoperimetric sets, the geometric descriptions of the boundary of isoperimetric sets and the equivalence of Sobolev inequalities and local isoperimetric inequalities. We first construct the area formula and coarea formula for the Lipschitz map between sub-Riemannian Carnot groups, and show the stability of surfaces with constant man curvatures and the regularity of minimizers. We will give out the first and second fundamental forms with respect to Currents, and the variational formulae on Varifolds. We also construct the deformation theorem and closure theorem for Currents, and derive isoperimetric inequalities, the Pansu's conjecture on sub- Riemannian spaces. These resarches on the project will be conducive to knowing clearly the essential structure of Sub-Riemannian Carnot groups, and form a framework of geometry and analysis on Currents in Carnot groups, and admit further Geometric Measure Theory and sub-elliptic operator theories. Sub-Riemannian Carnot group plays a central role in the general problem of analysis and geometry in metric spaces, with applications ranging from differential geometry to holonomic mechanics, getting through control theory, classical mechanics, gauge fields and sub-elliptic operator, etc., it has provoked the extensive interests in the past two decades, thus it is indispensable to carry out the geometry and analysis in sub-Riemannian Carnot groups.