Minimal Submanifolds in Riemannian Geometry

本研究计划涉及最小曲面理论的解析方法在几何和拓扑问题上的应用。最小化问题在数学和科学的许多分支中自然出现。例如，导航问题涉及在地球表面寻找最小长度的路径（“测地线”）。最小曲面是测地线的二维类似物，是面积函数的最小值（或简单的临界点），在材料科学中自然出现；例如流体界面问题和弹性问题。最小表面的一个简单的物理例子是将钢丝架浸入肥皂溶液后形成的肥皂膜。根据表面张力规律，这种肥皂膜具有稳定的性质，即在轻微变形下变大。在现代几何中，最小曲面（和子流形）有着惊人的应用，例如在广义相对论和低维拓扑中。本文研究的对象是几何变分问题的临界点。在某些几何条件下（如曲率条件），这些物体通常可以被证明满足非常特殊的性质。然后可以使用这些信息来获取所考虑的整个几何类的信息。这个项目的结果旨在增加我们对空间曲率和拓扑之间基本关系的理解。

This research proposal deals with applications of analytic methods of minimal surface theory to geometric and topological problems. Minimization problems arise naturally in many branches of mathematics and science. For example, problems in navigation involve finding paths of least length (`geodesics') on the earth's surface. Minimal surfaces, which are two-dimensional analogs of geodesics, are minimizers (or simply critical points) of the area function, and arise naturally in material science; for example in fluid interface problems and elasticity problems. A simple physical example of a minimal surface is the soap film that forms after dipping a wire frame into a soap solution. By the laws of surface tension this soap film has the property that it is stable, that is it becomes larger under slight deformations. In modern geometry, minimal surfaces (and submanifolds) have had striking applications, for example to general relativity and low dimensional topology. The objects under investigation in this proposal are critical points for geometric variational problems. Such objects can often, under certain geometric conditions (such as a curvature condition) be shown to satisfy very special properties. This information can then be used to obtain information about the whole geometric class under consideration. The results of this project are aimed at increasing our understanding of the fundamental relationships between the curvature and topology of spaces.