Function Theory and Minimal Surfaces

摘要建议：DMS 9803144首席研究员：William P.Minicozzi该项目有两个主要方向。首先，我们正在研究具有下曲率下界的流形上的函数和丛的截面的函数论。通常，我们对完全非紧流形上满足某些增长条件的调和函数或截面的空间感兴趣。从我们以前的研究中可以看出，这个问题与紧致流形上函数和丛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复杂微分几何中的问题有一个有趣的联系。其次，我们正在研究极小曲面的收敛和紧性，重点是没有先验面积或全曲率界限的情况。如果极小曲面序列的面积是无界的，则不可能存在经典极限；然而，我们通常可以提取几何上有趣的广义极限。由于拓扑原因，可以构造具有受控拓扑的这种嵌入极小曲面序列的例子。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的拉普拉斯算子的研究。我们主要感兴趣的是这些微分算子的基本性质如何依赖于空间的几何。这类问题是基本的，在许多不同的数学领域(包括几何学、拓扑学、复杂几何学和数学物理)都是相关的。这个项目的第二个主要方向是研究三个流形中的极小曲面空间。极小曲面是面积的关键点，是有趣的物理、几何和分析对象。

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.