Shortening Curves on Surfaces

METHODS of shortening a curve in a manifold have been used to establish the existence of closed geodesics, and in particular of simple closed geodesics on 2-spheres. For this purpose, a curve evolution process should (a) not increase the number of self-intersections of a curve, (b) exist for all time or until a curve collapses to a point, (c) shorten curves sufficiently fast so that curves which exist for all time converge to a geodesic, and (d) depend continuously on the choice of initial curve. Birkhoff originated what is now known as the Birkhoff curve shortening process, where midpoints of polygonal approximations to a curve are successively connected by geodesic segments [4]. This type of shortening has the advantage that (b), (c) and (d) are easy to establish, but the disadvantage that (a) seems difficult to arrange. A process of evolving a curve on a surface by its curvature is perhaps the most natural flow. Short term existence is easy to establish for this flow, but long term existence involves deep questions in PDEs and geometry. This flow has recently been studied with considerable success in a series of papers [7,8,9, 11. All four of the desired properties have been shown to hold for the flow by curvature of an embedded curve on a Riemannian surface. For non-embedded curves in Riemannian surfaces, some open questions remain about the types of singularities which may develop in the curvature flow. In particular, it is not known whether arcs of double points can be created. In this paper we introduce a new curve shortening flow. Like the Birkhoff process, this flow involves replacing arcs of a curve with geodesic segments. Unlike the Birkhoff process, it picks out its piecewise-geodesic structure purely from the geometry of the image manifold rather than from a parametrization of the curve. This flow, which we call the disk Jlow, is developed in $1. In $2 we use the flow to solve a purely topological problem concerning intersections of curves on surfaces. Turaev [17] has posed the problem in the following form: