次黎曼流形中的等周问题

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.

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。

项目组利用几何测度理论、变分法、非线性分析、黎曼几何等方法，围绕次黎曼流形中的等周不等式、赋予半对称联络的次黎曼空间中的几何分析问题等，系统深入地开展相关研究。通过Carnot群结构、几何测度论、Pansu导数以及dilation结构理论等技巧以及共形不变量、半调和微分算子等基本技术手段，本项目解决或部分解决了等周集的存在性、等周集边界的几何刻画、Sobolev不等式、部分Pansu猜想、半调和函数、渐近次黎曼空间中ADM质量猜想的证明等。..系统研究并给出几类权重空间中等周不等式的证明、等周集的刻画等；系统研究并给出赋予半对称联络次黎曼空间诸如共形几何的分析；较为系统研究了赋半对称联络次黎曼空间的半调和函数、证明了ADM正质量猜想问题。..初步建立了赋予半对称联络的次黎曼空间中的几何与分析框架。已发表或录用论文20余篇，组织了1场国际学术会议、参加国际数学家大会等会议6场。待出版专著一本《几何无套利分析》