Convexity Problems in Submanifold Geometry and Topology

DMS -0204190.主要研究者对经典微分几何中的具体问题和欧几里德空间中曲线曲面的拓扑学感兴趣，特别是那些涉及凸性概念的问题。拟研究的内容包括：（i）由对应于自然变换的向量场投射在曲面上的某些节点域（阴影），并发展这些节点域在常平均曲率曲面和高斯映射满足两段性质的曲面上的应用;（ii）没有平行切线的闭曲线（3）有边界局部凸曲面的整体性质，包括与Monge-Amper方程的联系，以及与极小曲面对偶的凸船体性质;（iv）空间曲线的某些变形（展开）的存在性和规律性，以研究纽结能量和畸变的极值曲线和曲面的研究一直是微分几何和几何拓扑学发展的主要动力，这反过来又在物理科学中找到了重要的应用。 凸性的概念在解决这一领域的问题上经常被证明是富有成效的，特别是那些涉及优化各种数量的问题。 主要研究者处理照明表面上阴影的工作的部分动机是肥皂膜的研究，并可能导致计算机视觉的应用。此外，knotenergies的调查可能是在研究DNA的兴趣。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吸引力。

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.