Convexity Problems in Submanifold Geometry and Topology

DMS-0204190主要研究欧氏空间中经典微分几何和曲线曲面拓扑中的具体问题，特别是涉及某些凸性概念的问题。所提出的研究包括：(I)向量场投射到曲面上的与自然变换相对应的某些节域(阴影)，以及这些区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)没有平行切线的闭合曲线(斜环)及其与二次曲面的关系；(Iii)有边界的局部凸曲面的整体性质，包括与Monge-Ampere方程的联系，以及与极小曲面的凸壳性质是对偶的；(Iv)空间曲线(展开)的某些变形的存在和规律，以研究结点能量的极值和变形。对曲线和曲面的研究一直是许多微分几何和几何拓扑学发展的主要动机，而这反过来又在物理科学中得到了重要的应用。凸性的概念通常被证明在解决这一领域的问题方面是卓有成效的，特别是那些涉及优化各种量的问题。首席研究人员处理照明表面阴影的工作的这些方面，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，对结能的研究可能会对研究DNA感兴趣。然而，研究人员的主要动机是基于美学考虑和低维几何问题的直觉吸引力。

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.