Mathematical Sciences: Nonlinear Elliptic Equations in Differential Geometry

数学科学：微分几何中的非线性椭圆方程

• 批准号: 9704861
• 负责人: Wenxiong Chen
• 金额: $ 5.47万
• 依托单位: Missouri State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-06-01 至 2000-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9704861&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Nonlinear; Elliptic; Equations

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.

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。