Mathematical Sciences: Nonlinear Elliptic Equations in Differential Geometry

9704861 Chen本课题属于黎曼几何和非线性椭圆偏微分方程领域。研究者希望研究在球上规定标量曲率的问题：在球上给定一个函数，能否找到一个与标准度规共形的度规，其标量曲率是给定函数？这个问题可以用拟线性椭圆型偏微分方程来表述；当前的项目是研究这个基本微分方程的各个方面。黎曼流形是曲面的高维推广。这种流形在理论物理中有着众所周知的应用。黎曼流形具有距离或度规的概念。通过对度规求导我们可以测量它的曲率。规定曲率的问题是看看是否可以逆转这个过程：首先规定曲率性质，然后寻找合适的度规。

9704861 Chen This project lies in the area of Riemannian geometry and nonlinear elliptic partial differential equations. The investigator wishes to pursue the problem of prescribing scalar curvature on spheres: given a function on a sphere, can one find a metric conformal to the standard metric whose scalar curvature is the given function? This problem can be stated in terms of a quasi-linear elliptic partial differential equation; the current project is to study various aspects of this underlying differential equation. Rimannian manifolds are higher dimensional generalizations of curved surfaces. Such manifolds have well-known applications in theoretical physics. A Riemannian manifold possesses a notion of distance, or a metric. And by taking the derivative of the metric one can measure its curvature. The problem of prescribing curvature then is to see if one can reverse this process: one specifies the curvature property first, and then look for a suitable metric.