共形几何中的偏微分方程探究

The problem of prescribing certain curvatures, which is a generalization of the well-known Yamabe problem, is one of central topics in conformal geometry. To study this problem is equivalent to exploring the existence of solutions of certain nonlinear partial differential equations. The aim of this project is to use the method of conformal geometric flow to find solutions of those kind of PDEs. More precisely, we expect to study the two problems as follow:.1. Prescribed Gaussian and boundary geodesic curvatures on surfaces: Let $(M, g_0)$ be a compact surface with boundary. Assume that $f(x)$ and $h(x)$ are two smooth functions defined, respectively, in $M$ and on the boundary of $M$. Does there exist a conformal metric $g$ such that $f(x)$ and $h(x)$ can be realized, respectively, as the Gaussian curvature and the boundary geodesic curvature of the metric $g$? .2. Prescribed Webster curvature on Cauchy-Riemann manifolds: Let $(M,T_0)$ be a $2n+1$-dimensional strictly pseudo-convex compact Cauchy Riemann manifold without boundary. Does there exist a contact form $T$ conformally related to $T_0$ such that $f(x)$ is the Webster curvature of the Webster metric $g_T$?

预定曲率问题是共形几何研究中的一个中心课题，它是著名的Yamabe问题的一种推广。 研究预定曲率问题等价于解一类非线性偏微分方程。 本项目的目标就是要利用共形几何流的方法去研究这类偏微分方程的解的存在性。我们期望研究如下两个问题：.1. 曲面上预定高斯曲率与测地曲率问题: 设(M, g_0)是一个带有黎曼度量g_0的紧致带边曲面，f(x)与h(x)是分别定义在M的内部和边界上的光滑函数。问是否存在一个与g_0共形的度量g使得在度量g下的高斯曲率为f(x)，而测地曲率为h(x)？.2. 柯西-黎曼流形上预定Webster曲率问题：设(M, T_0)是一个实维数为2n+1的带切触形式T_0的严格伪凸紧致无边柯西-黎曼流形。问给定M上的光滑函数f(x)，是否存在一个与T_0共形的切触形式T使得在Webster度量g_T下的Webster曲率为f(x)？