关于带边流形上的k-Yamabe问题的研究

In this project, we shall research into the k-Yamabe problem on the manifolds with boundary which is a hot issue in differential geometry. The problem, in essence, is to solve a fully nonlinear Partial Differential Equation with Neumann boundary condition. We would like to develop our research work based on our preparatory work in the following four aspects. At first, the Estimation Theorems for the k-Yamabe Equation and the corresponding parabolic equation will be set up. Secondly, we shall look for a suitable functional such that the k-Yamabe Equation is its Euler-Lagrange Equation. Thirdly, we would like to determinate the initial values of the corresponding parabolic equation and get the Existence Theorem of the solutions. Fourthly, we shall consider the relationship between the variational structure and the conformal invariant. The key research problems of the project are the two previous ones. Although it is a fully nonlinear Partial Differential Equation, the geometry background would be quite helpful in solving it. Furthermore, it is interesting that the variational structure of the equation may imply some geometrical properties of the manifolds in the conformal class.

本项目将深入研究带边流形上的k-Yamabe问题。该问题是微分几何中的一个热点，在本质上就是要解一个Neumann边值条件的完全非线性的偏微分方程。本项目将立足于已有的前期工作，将带边流形上的k-Yamabe问题分成如下四个问题展开研究：（1）给出带边流形上的k-Yamabe方程及相应的抛物方程的解的一阶、二阶估计。（2）寻找适当的泛函，使得带边流形上的k-Yamabe 方程为该泛函的Euler-lagrange方程。（3）确定抛物方程的初值，并得到解的存在性定理。（4）考察方程的变分结构与共形不变量的关系。我们认为这个问题解决的关键在于合适的估计定理的建立以及恰当的泛函的选择。同时我们也认识到该问题丰富的几何背景对于解方程是有着积极意义的。此外，共形类中方程的变分结构也有望揭示出流形本身的某些性质，这不能不说是个有趣的现象。