Some recent developments in differential geometry

Abstract

Until recently differential geometry was the s tudy of fixed curves or surfaces in space and of abstract manifolds with fixed Riemannian metrics. Now geometers have begun to s tudy curves and surfaces that are subjected to various forces and that flow or evolve with time in response to those forces. Perhaps the simplest example (but already a very subtle one) is the curve-shortening flow. Consider a simple closed curve in the plane, and suppose that it moves so that the velocity at each point on the curve is equal to the curvature vector of the curve at that point. Thus on a convex curve, every point moves into the region bounded by the curve, whereas on a general curve, points on the portions that bend inward will move outward. This motion of curves arises naturally from thinking of the space of all smooth embedded curves as an infinite dimensional manifold M: a curve moves so that its velocity is minus the gradient of the length function on M. What happens to a curve as it flows in this way? Some facts are rather straightforward to establish: (1) Disjoint curves remain disjoint. This fact is an example of the maximum principle for parabolic partial differential equations. To see w h y it is true, consider two closed curves, one inside the other, that are initially disjoint. Suppose that at some later time they intersect each other. At the first such time, the curves must be tangent at the point p where they meet, and the curvature of the inner curve at p must be greater than or equal to the curvature of the outer curve at p. If strict inequality holds, then (at p) the inner curve is moving inward faster than the outer curve. But that implies that a moment earlier a portion of the " inner" curve was outside of the "outer" curve, a contradiction. If the curvatures at p are equal, a more subtle argument is required (Cf. [21]).