Function Theory and Minimal Surfaces

摘要提案：DMS 9803144首席研究员：William P. Minicozzi本项目主要有两个方向。首先，我们研究了具有下曲率界的流形上的函数和束的截面的函数理论。通常，我们对完全非紧流形上满足一定生长条件的调和函数空间或截面感兴趣。从我们先前的研究中可以看出，这个问题与紧流形上的函数和束的截面的一致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性研究和来自复杂微分几何的问题有一个有趣的联系。其次，我们研究了最小曲面的收敛性和紧性，重点研究了没有先验面积和总曲率界的情况。如果最小曲面序列的面积无界，则经典极限不存在；然而，通常我们可以提取几何上有趣的广义极限。有一些例子，其中，由于拓扑的原因，这样的嵌入极小表面序列与控制拓扑结构可以被构造。这个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性质是如何依赖于空间的几何形状的。这类问题是基本的，并且与许多不同的数学领域（包括几何、拓扑、复杂几何和数学物理）相关。本课题的第二个主要方向是研究三流形中最小曲面的空间。最小曲面是面积的关键点，是有趣的物理、几何和解析对象。

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.