Research in the Geometry of Minimal and Constant Mean Curvature Surfaces

格兰特命题摘要本文研究三维欧氏空间和一般单连通齐次三维流形X中适当嵌入的极小和常平均曲率曲面的几何、渐近性态、共形结构和拓扑。 （1）在三维欧氏空间中，更一般地，在一般X空间中，对有限拓扑的完全嵌入常平均曲率曲面的渐近性质进行分类; (2)证明了X中常平均曲率球面的模空间M（X）是一个由平均曲率函数参数化的区间，并且完全由指数为1的球面组成（在X是一个圆球与R的乘积的情况下，这个区间的端点是指数为0的球面）。(3)当X同胚于3维欧氏空间时，则M（X）中的球面被嵌入，并且可以被选择以产生空间X的在单点穿孔的叶理。(4)当X同胚于3维欧氏空间时，则X中的等周问题的解是唯一的，且有界于M（X）中的球面。本文还将研究局部有界亏格的嵌入极小和常平均曲率曲面的紧性、可去奇点结果、正则性和收敛性等相关理论技巧。 另一个主要目标是理解欧氏空间和其他三维流形中完全的、有界的嵌入的极小和常平均曲率曲面的存在性和不存在性。极小曲面是第一个重要的二维变分法的例子，首先由欧拉在1735年左右描述;常平均曲率曲面首先由德劳内在1835年研究，也是变分法的重要例子。 物理上最小的表面可以局部地建模为电线上的肥皂膜或相对于其局部边界的最小面积的表面;物理上恒定的平均曲率表面可以局部地建模为电线上的肥皂膜，每侧具有恒定的压力差。 极小曲面在三维拓扑学和黎曼几何的研究中起着重要的作用。 在这个建议中的研究涉及嵌入的最小和常平均曲率曲面的全局和局部性质，以及这些结果在三维拓扑和几何基础研究中的可能应用。部分原因是与其他数学领域的重要联系，以及可以轻松地制作美丽的经典例子的计算机图形图片，最小和恒定的平均曲率曲面仍然是科普文章和公共科学展览的主要主题之一。 因此，间接地，令人兴奋的研究问题概述了这一建议有助于使许多年轻的科学家和数学家的前沿研究。

Abstract of Grant ProposalIn this proposal the researcher will study the geometry, asymptotic behavior, conformal structure and topology of properly embedded minimal and constant mean curvature surfaces in the 3-dimensional Euclidean space and in a general simply-connected homogeneous 3-manifolds X. There are several goals of the proposal which include: (1) To classify the asymptotic behavior of complete embedded constant mean curvature surfaces of finite topology in 3-dimensional Euclidean space, and more generally, in a general X. (2) To prove that the moduli space M(X) of constant mean curvature spheres in X is an interval parametrized by the mean curvature function and consists entirely of index-1 spheres (this interval has an end point with an index-0 sphere in the case X the the product of a round sphere with R). (3) When X is homeomorphic to 3-dimensional Euclidean space, then the spheres in M(X) are embedded and can be chosen to give rise to a foliation of the space X punctured in a single point. (4) When X is homeomorphic to 3-dimensional Euclidean space, then solutions to the isoperimetric problems in X are unique and bounded by the spheres in M(X). Related theoretical techniques concerning compactness, removable singularities results, regularity and convergence of embedded minimal and constant mean curvature surfaces of locally bounded genus will be investigated as well. Another main goal is to understand the existence and nonexistence of complete, bounded embedded minimal and constant mean curvature surfaces in Euclidean 3-space and in other 3-dimensional manifolds.Classical minimal and constant mean curvature surface theory has its roots in18-th and 19-{th century mathematics. Minimal surfaces are the first important two dimensional examples of what is called the calculus of variations, first described by Euler around 1735; constant mean curvature surfaces were first studied by Delaunay in 1835 also represent important examples in the calculus of variations. Physically minimal surfaces can be modeled locally as soap films on wires or by surfaces of least-area relative to their local boundaries; physically constant mean curvature surfaces can be modeled locally as soap films on wires with a constant pressure difference on each side. Minimal surfaces play an important role as a tool in the study of 3-dimensional topology and Riemannian geometry. The research in this proposal concerns global and local properties of embedded minimal and constant mean curvature surfaces and possible applications of these results to basic research in three-dimensional topology and geometry. In part because of the important connections with other areas of mathematics and the ease in which it is possible to make computer graphics pictures of beautiful classical examples, minimal and constant mean curvature surfaces continue to be one of the principal topics for popular science articles and public science exhibits. Thus, indirectly, the exciting research problems outlined in this proposal help bring many young scientists and mathematicians to the frontiers of research.