Some Nice Results About Anistropic Mean Curvature Flow

Abstract

<p>Imagine stretching out a rubber band on a flat surface and letting go suddenly. Picture the way the rubber band contracts in slow motion and that should give you a good idea of how mean curvature flow dictates the evolution of plane curves. The more stretched out the rubber band, the faster it snaps back. Just like the rubber band returns to it's original round shape no matter how it is stretched, any smooth plane curve will evolve under mean curvature flow to a circle. Suppose that you try to kink the rubber band, try to force a sharp corner into it. As soon as you let go those kinks disappear. Similarly a piecewise smooth curve will smooth out instantaneously under mean curvature flow. Now suppose that you stretch out the rubber band and put kinks in it, but instead of letting go completely, you hold those kinks in place. The rest of the rubber band will still try to shrink back to it's original circular shape. This is the major topic of this paper-how do piecewise smooth curves behave under mean curvature flow if their kinks are held fast? It turns out that the initial evolution of a curve in such a situation depends completely on the number and precise angles of those kinks.</p> <p>One of the earliest references on mean curvature flow is a 1956 paper [15] which explored a specific case of piecewise smooth curves evolving by mean curvature and found that by counting the number of sides one could determine how the enclosed area would change (Theorem 4.1). This was a surprising result because in the smooth case, the area enclosed is always shrinking, but by adding some sharp corners it became possible that the area would increase initially. Little attention seems to have been paid to piecewise smooth curves and mean curvature flow since then, with one notable exception being a paper by L. Bronsard and F. Reitich [5] which proved that the curves analyzed in the 1956 paper could really exist!</p> <p>The main result of this paper is Theorem 4.4 which is a generalization of the aforementioned Theorem 4.1. The new result generalizes the original in two ways: first it is non-specific with respect to the angles at the corners, and second, it allows for the flow to be anisotropic; the evolution of the curve may depend on it's orientation in the plane. Two proofs of this result are presented. One uses ideas from the 1956 paper and is fairly intuitive. The other proof follows the strategy of a more recent paper [10] and proves the result as an intrinsic property of the curve. The final section of the paper mentions some other questions and topics related to mean curvature flow and includes a new result about the behavior of curves evolving on the unit sphere according to a generalized version of mean curvature flow.</p>