Research in the Geometry of Minimal and Constant Mean Curvature Surfaces

本课题将研究三维欧几里德空间和一般单连通齐次3流形x中适当嵌入的最小和常数平均曲率曲面的几何、渐近行为、共形结构和拓扑结构。(1)对三维欧几里德空间中有限拓扑的完全嵌入常平均曲率曲面的渐近行为进行分类，更一般地说，一般X(2)证明M (X)的模空间X是一个常数平均曲率空间间隔平均曲率函数,由完全参数化的索引1球(这间隔有一个终点,一个索引0球在X R)的圆球体的产物。(3)当X是同胚的三维欧几里得空间,那么M (X)的球体是嵌入式的,可以选择产生空间X的绿叶刺穿在一个单一的点。(4)当X同胚于三维欧几里德空间时，则X中等周问题的解是唯一的，且被M(X)中的球有界。本文还研究了局部有界属的嵌入最小和常平均曲率曲面的紧性、可移奇异性结果、正则性和收敛性等相关理论技术。另一个主要目标是理解欧几里得三维空间和其他三维流形中完整的、有界嵌入的最小曲率曲面和常数平均曲率曲面的存在性和不存在性。经典的最小和常平均曲率曲面理论起源于18世纪和19世纪的数学。最小曲面是所谓变分法的第一个重要二维例子，最早由欧拉在1735年左右描述；1835年，德劳内首次研究了常平均曲率曲面，它也代表了变分学中的重要例子。物理上最小的表面可以局部建模为电线上的肥皂膜或相对于其局部边界的最小面积表面；物理上恒定的平均曲率表面可以局部地建模为每边有恒定压差的电线上的肥皂膜。极小曲面在三维拓扑和黎曼几何的研究中起着重要的作用。本提案的研究涉及嵌入最小和恒定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。在某种程度上，由于与其他数学领域的重要联系，以及很容易用计算机图形化的方式描绘美丽的经典例子，最小和常数平均曲率曲面继续成为科普文章和公共科学展览的主要主题之一。因此，间接地，本提案中概述的令人兴奋的研究问题帮助许多年轻的科学家和数学家走向研究的前沿。

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.