Function Theory and Minimal Surfaces

摘要 提案：DMS 9803144主要研究者：William P. Minicozzi 该项目有两个主要方向。 首先，我们正在研究函数理论，函数和部分的丛，流形上的曲率下限。 通常，我们感兴趣的调和函数或部分空间，满足一定的增长条件，完全非紧流形。 从我们以前的研究中可以看出，这个问题与紧流形上的函数和丛的截面的一致特征值估计密切相关。 这第二个问题有一个有趣的联系，研究渐近的半经典极限在数学物理和问题，其中来自复杂的微分几何。 第二，我们研究极小曲面的收敛性和紧性，重点是没有先验面积或全曲率界的情况。 如果极小曲面序列的面积是无界的，则经典极限不存在;然而，我们通常可以提取几何上有趣的广义极限。 有一些例子，由于拓扑的原因，这种序列的嵌入式极小曲面与控制拓扑结构可以构建。 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。 我们的主要兴趣是如何这些微分算子的基本性质取决于几何空间。 这类问题是基本的，并且在许多不同的数学领域（包括几何、拓扑、复几何和数学物理）都有相关性。 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。 极小曲面是面积的临界点，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.