Minimal Submanifolds in Riemannian Geometry

黎曼几何中的最小子流形

基本信息

批准号：
RGPIN-2015-04436
负责人：
Fraser, Ailana
金额：
$ 1.82万
依托单位：
University of British Columbia
依托单位国家：
加拿大
项目类别：
Discovery Grants Program - Individual
财政年份：
2018
资助国家：
加拿大
起止时间：
2018-01-01 至 2019-12-31
项目状态：
已结题

来源：
https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=651095
关键词：
Minimal Submanifolds Riemannian Geometry

项目摘要

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.**

本研究提案涉及最小曲面理论的分析方法在几何和拓扑问题中的应用。最小化问题自然出现在数学和科学的许多分支中。例如，导航问题涉及寻找地球表面上最短长度的路径（“测地线”）。最小曲面是测地线的二维类似物，是面积函数的极小值（或简称临界点），并且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物理示例是将线框浸入肥皂溶液后形成的肥皂膜。根据表面张力定律，该肥皂膜具有稳定的特性，即在轻微变形的情况下会变大。在现代几何中，最小曲面（和子流形）具有引人注目的应用，例如广义相对论和低维拓扑。该提案中所研究的对象是几何变分问题的关键点。在某些几何条件（例如曲率条件）下，此类对象通常可以满足非常特殊的属性。然后可以使用该信息来获取有关所考虑的整个几何类的信息。该项目的结果旨在加深我们对空间曲率和拓扑之间基本关系的理解。**