Function Theory and Minimal Surfaces

函数论和最小曲面

• 批准号: 9803144
• 负责人: William Minicozzi
• 金额: $ 7.66万
• 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: 1998
• 资助国家: 美国
• 起止时间: 1998-07-15 至 2001-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9803144&HistoricalAwards=false
• 关键词: Function; Theory; Minimal; Surfaces

Abstract Proposal: DMS 9803144 Principal Investigator: William P. Minicozzi This project has two main directions. First, we are studying function theory, for functions and sections of bundles, on manifolds with a lower curvature bound. Typically, we are interested in spaces of harmonic functions or sections which satisfy certain growth conditions on complete noncompact manifolds. As is evident from our prior research, this problem is closely related to uniform eigenvalue estimates on compact manifolds for both functions and sections of bundles. This second problem has an interesting connection to the study of asymptotics for semiclassical limits in mathematical physics and to questions which come from complex differential geometry. Second, we are studying convergence and compactness of minimal surfaces with an emphasis on the case where there is no a priori area or total curvature bound. If the area of a sequence of minimal surfaces is unbounded, a classical limit cannot exist; however, often we can extract geometrically interesting generalized limits. There are examples where, for topological reasons, such sequences of embedded minimal surfaces with controlled topology can be constructed. This project has two main directions both of which involve the interplay between geometry and analysis. The first direction involves the study of Laplace operators on manifolds. Our primary interest is in how the basic properties of these differential operators depend on the geometry of the space. This sort of question is fundamental and has been relevant in many different areas of mathematics (including geometry, topology, complex geometry, and mathematical physics). The second main direction of this project is to study spaces of minimal surfaces in three manifolds. Minimal surfaces, which are critical points for area, are interesting physical, geometric, and analytic objects.

摘要提案：DMS 9803144 首席研究员：William P. Minicozzi 该项目有两个主要方向。 首先，我们正在研究具有较低曲率界的流形上的函数和丛的截面的函数论。 通常，我们对满足完全非紧流形上的某些增长条件的调和函数或截面空间感兴趣。 从我们之前的研究中可以明显看出，这个问题与函数和束部分的紧流形上的均匀特征值估计密切相关。 第二个问题与数学物理中半经典极限的渐近研究以及来自复杂微分几何的问题有着有趣的联系。 其次，我们正在研究最小曲面的收敛性和紧致性，重点是没有先验面积或总曲率界限的情况。 如果一系列最小曲面的面积是无界的，则经典极限不可能存在；然而，我们通常可以提取几何上有趣的广义极限。 在一些示例中，出于拓扑原因，可以构建具有受控拓扑的嵌入式最小表面序列。 该项目有两个主要方向，都涉及几何和分析之间的相互作用。 第一个方向涉及流形上拉普拉斯算子的研究。 我们的主要兴趣是这些微分算子的基本属性如何取决于空间的几何形状。 这类问题是基础性的，并且与数学的许多不同领域（包括几何、拓扑、复杂几何和数学物理）相关。 该项目的第二个主要方向是研究三个流形中的最小曲面空间。 最小曲面是面积的临界点，是有趣的物理、几何和分析对象。