Research in the Geometry of Minimal and Constant Mean Curvature Surfaces

摘要奖：DMS 1309236，主要研究者：William H. Meeks研究人员将研究三维欧氏空间中适当嵌入的最小和常平均曲率曲面的几何，渐近行为，共形结构和拓扑，更一般地说，在单连通齐次三维流形X中。一个目标是在这样的空间X中对常平均曲率球进行分类，并研究它们何时被嵌入，以及它们是否总是代表X中等周问题的解。另一个目标的建议是分类的渐近行为完全嵌入常平均曲率曲面的有限拓扑在三维欧氏空间，更一般地说，在齐次三维流形。理论技术有关的可去除的奇异性，紧性，规则性和收敛性的嵌入最小和常平均曲率曲面的序列将被调查，以及。最后一个目标是理解欧氏空间和其他三维空间中的完全的、有界的嵌入的极小和常平均曲率曲面的存在/不存在问题。经典的极小和常平均曲率曲面理论起源于18和19世纪的数学。极小曲面是所谓的变分法的第一个重要的二维例子，这一主题首先由欧拉在1735年左右描述;常平均曲率曲面首先由德劳内在1835年研究，它们也是变分法的重要例子。物理上最小的表面可以被建模为电线上的肥皂膜或相对于其边界的最小面积的表面;物理上恒定的平均曲率表面可以被建模为电线上的肥皂膜，每侧具有恒定的压力差。极小曲面作为一种工具在三维拓扑和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。

AbstractAward: DMS 1309236, Principal Investigator: William H. Meeks The researcher will study the geometry, asymptotic behavior, conformal structure and topology of properly embedded minimal and constant mean curvature surfaces in three-dimensional Euclidean space, and more generally, in simply connected homogeneous three-manifolds X. One goal is to classify constant mean curvature spheres in such spaces X and to investigate when they are embedded and if they always represent solutions to isoperimetric problems in X. Another goal of the proposal is to classify the asymptotic behavior of complete embedded constant mean curvature surfaces of finite topology in three-dimensional Euclidean space, and more generally, in homogeneous three-manifolds. Theoretical techniques concerning removable singularities, compactness, regularity and convergence of sequences of embedded minimal and constant mean curvature surfaces will be investigated as well. A final goal is to understand the existence/nonexistence problem for complete, bounded embedded minimal and constant mean curvature surfaces in Euclidean three-space and in other three-dimensional spaces.Classical minimal and constant mean curvature surface theory has its roots in 18 th and 19 th century mathematics. Minimal surfaces are the first important two-dimensional examples of what is called the calculus of variations, a subject first described by Euler around 1735; constant mean curvature surfaces were first studied by Delaunay in 1835 and they also represent important examples in the calculus of variations. Physically minimal surfaces can be modeled as soap films on wires or by surfaces of least-area relative to their boundaries; physically constant mean curvature surfaces can be modeled 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 three-dimensional topology and 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.