Mathematical Sciences: Geodesics and Minimal Surfaces in Manifolds with Non-negative Curvature

首席研究员打算继续研究非负曲率流形中的测地线和最小曲面，特别强调四维流形。最小曲面理论将用于研究流形的截面曲率和拓扑之间的关系。最小曲面是可以被表征为对表面上的任何边界曲线具有最小表面积的曲面。它也可以被描述为使“表面张力”能量最小化或平均曲率为零。在某种意义上，最小曲面是一个二维的测地线类比，是一条连接两点的最小长度曲线。首席研究员将使用最小曲面理论来研究空间的全局属性和局部属性（如曲率）之间的关系。

The principal investigator intends to continue his study of geodesics and minimal surfaces in manifolds of non-negative curvature with particular emphasis on manifolds of dimension four. Minimal surface theory will be used to investigate the relationships between the sectional curvature and topology of a manifold. A minimal surface is a surface which may be characterized as one which has minimal surface area for any bounding curve on the surface. It may also be characterized as one which minimizes the "surface tension" energy or which has mean curvature zero. In some sense, a minimal surface is a two dimensional analog of a geodesic, a curve of least length joining two points. The principal investigator will use the theory of minimal surfaces to study relationships between the global properties of a space and local properties such as curvature.