Embedded Minimal Surfaces in Three Manifolds

三个流形中的嵌入式最小曲面

• 批准号: 0104187
• 负责人: William Minicozzi
• 金额: $ 13.37万
• 依托单位: Johns Hopkins University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-07-01 至 2004-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0104187&HistoricalAwards=false
• 关键词: Embedded; Minimal; Surfaces; Three; Manifolds

We propose various directions for studying embedded minimal surfacesin three-manifolds. This includes the study of the convergence of suchsurfaces, how they degenerate and whether or not their Morse index isbounded. There are many possible applications of results along these linesincluding to the spherical space-form problem, the topology ofthree-manifolds with positive scalar curvature etc. In addition, these results shouldyield new insight into minimal surfaces in Euclidean space.This project focuses on minimal surfaces in three-dimensionalmanifolds. Minimal surfaces are critical points for area (e.g.,soap films are least area surfaces) and arise naturally in manyproblems in mathematics and in the other sciences. Many of theclassical results in this area are on least area surfaces. In contrast,our main interest is in high index (i.e., highly unstable) embedded minimalsurfaces with bounded topology - a well-known example is given bythe standard helicoid. Roughly speaking, the point of some of our theoremsis that the behavior of the helicoid is typical.

我们提出了三流形中嵌入极小曲面的研究方向。这包括研究这些曲面的收敛性，它们如何退化以及它们的莫尔斯指数是否有界。沿着这条路线的结果有许多可能的应用，包括球面空间形式问题，具有正标量曲率的三流形拓扑等。此外，这些结果应该对欧几里得空间中的最小曲面产生新的见解。这个项目的重点是三维流形的最小表面。最小表面是面积的关键点（例如，肥皂膜是最小面积表面），在数学和其他科学的许多问题中自然出现。这方面的许多经典结果都是在最小面积表面上得到的。相比之下，我们的主要兴趣是具有有界拓扑的高指数（即高度不稳定）嵌入最小曲面-标准螺旋体给出了一个众所周知的例子。粗略地说，我们的一些定理表明，螺旋面的行为是典型的。