Convexity Problems in Submanifold Geometry and Topology

摘要DMS - 0204190。主要研究欧几里得空间中经典微分几何和曲线曲面拓扑的具体问题，特别是涉及到一些凸性概念的问题。建议的研究包括：(i)与自然变换相对应的向量场在表面上投射的某些节点域（阴影），并开发这些应用于恒定平均曲率的表面和高斯映射满足两段性质的表面；没有平行切线的闭合曲线（斜环）及其与二次曲面的关系；（iii）具有边界的局部凸曲面的整体性质，包括与monge - ampere方程组的连接，以及与极小曲面对偶的凸壳性质；（iv）空间曲线某些变形（展开）的存在性和规律性，以研究结能和畸变的极值。对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，而微分几何和几何拓扑学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，特别是那些涉及优化各种量的问题上，经常被证明是卓有成效的。这位首席研究员研究被照亮表面上阴影的部分原因是对肥皂膜的研究，这可能会导致计算机视觉的应用。此外，对节能的研究可能会对DNA的研究产生兴趣。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视觉吸引力。

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.