Convexity Problems in Submanifold Geometry and Topology

Georgia Tech Research Corporation, Atlanta GA

Investigators

Abstract

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 of locally convex surfaces with boundary, including connections with Monge-Ampere equations, 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 and may lead to applications for computer vision. Further, the investigations on knot energies may be of interest in studying the DNA. The primary motivation of the investigator, however, is based on aesthetic considerations and the intuitive visual appeal of low dimensional geometric problems.