Convexity Problems in Submanifold Geometry and Topology

子流形几何和拓扑中的凸性问题

• 批准号: 0336455
• 负责人: Mohammad Ghomi
• 金额: $ 7.08万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-05-09 至 2005-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0336455&HistoricalAwards=false
• 关键词: Convexity; Problems; Submanifold; Geometry; Topology

ABSTRACT DMS - 0204190.The principal investigator is interested in concrete problems in classical differential geometry and topology of curves and surfaces in Euclidean space, specially those which involve some notion of convexity. The proposed investigations include: (i) Certain nodal domains (shadows) cast on a surface by vectorfields which correspond to natural transformations, and developing the applications of these for surfaces of constant mean curvature, and surfaces whose gauss map satisfies a two-piece-property; (ii) Closed curves without parallel tangent lines(skew loops) and their relation to quadric surfaces; (iii) Global properties oflocally convex surfaces with boundary, including connections with Monge-Ampereequations, and a convex hull property which is dual to that of minimal surfaces;(iv) Existence and regularity of certain deformations of space curves (unfoldings)to study extremals of knot energies and distortion.The study of curves and surfaces has been the primary motivation for the development of much of differential geometry and geometric topology, which in turn has found significant applications in physical sciences. Notions of convexity have often proved fruitful for solving problems in this area, specially those which involve optimizing various quantities. Those aspects of the principal investigator's work dealing with shadows on illuminated surfaces is motivated in part by a study of soap films andmay lead to applications for computer vision. Further, the investigations on knotenergies may be of interest in studying the DNA. The primary motivation of theinvestigator, however, is based on aesthetic considerations and the intuitivevisual appeal of low dimensional geometric problems.

摘要 DMS - 0204190。首席研究员对经典微分几何以及欧几里得空间中曲线和曲面拓扑的具体问题感兴趣，特别是那些涉及凸性概念的问题。拟议的研究包括：（i）通过与自然变换相对应的矢量场投射在表面上的某些节点域（阴影），并开发这些节点域在平均曲率恒定的表面以及高斯图满足两部分的表面上的应用财产; (ii) 没有平行切线（斜环）的闭合曲线及其与二次曲面的关系； (iii) 有边界的局部凸曲面的全局性质，包括与蒙日-安培方程的联系，以及与极小曲面对偶的凸包性质；(iv) 研究极值的空间曲线（展开）的某些变形的存在性和规律性曲线和曲面的研究一直是许多微分几何和几何拓扑发展的主要动力，而这些又在物理科学中找到了重要的应用。 凸性的概念通常被证明对于解决该领域的问题非常有效，特别是那些涉及优化各种数量的问题。 首席研究员处理照明表面阴影的工作部分是受到肥皂膜研究的启发，并可能导致计算机视觉的应用。此外，对结能的研究可能对研究 DNA 很有意义。然而，研究者的主要动机是基于美学考虑和低维几何问题的直观视觉吸引力。