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.

摘要建议：DMS 9803144主要研究人员：William P. Minicozzi这个项目有两个主要方向。 首先，我们正在研究函数理论，即束的功能和部分，在曲率结合较低的流形上。 通常，我们对谐波函数或部分的谐波空间感兴趣，这些空间满足了完全不相容的歧管上的某些生长条件。 从我们先前的研究中可以明显看出，此问题与束束的功能和部分的紧凑型歧管均匀特征值估计密切相关。 第二个问题与对数学物理学中半经典限制的渐近学研究以及来自复杂差异几何形状的问题的研究有着有趣的联系。 其次，我们正在研究最小表面的收敛性和紧凑性，重点是没有先验区域或总曲率结合的情况。 如果一系列最小表面的面积是无限的，则不可能存在经典的限制。但是，我们通常可以提取几何有趣的广义极限。 有一些例子，出于拓扑原因，可以构建具有控制拓扑的嵌入式最小表面的序列。 该项目有两个主要方向，涉及几何和分析之间的相互作用。 第一个方向涉及将拉普拉斯操作员研究的研究。 我们的主要兴趣是这些差分运算符的基本特性如何依赖于空间的几何形状。 这种问题是基本的，并且在数学的许多不同领域（包括几何学，拓扑，复杂的几何和数学物理学）都具有相关性。 该项目的第二个主要方向是研究三个歧管中最小表面的空间。 最小的表面是区域的关键点，是有趣的物理，几何和分析对象。