A distance comparison principle for evolving curves

. A lower bound for the ratio of extrinsic and intrinsic distance is proved for embedded curves evolving by curve shortening flow in the plane. The estimates yield a new approach to longtime existence results and can be applied to related evolution equations. Mathematische Fakultat Universitat Tubingen Auf der Morgenstelle 10 72076 Tubingen Germany AMS No.: 58G11, 35K20 Keywords: Curve shortening, Mean curvature flow 1 1. Introduction Let F : # [0, T ] # R 2 be a smooth family of embedded curves, where # is either S 1 or an interval. We say that # moves by the curve shortening flow (or mean curvature flow) if (1.1) d dt F (p, t) = -# #(p, t), p # #, t # [0, T ], where # and # are a choice of unit normal and corresponding geodesic curvature respectively, such that -## = -# # is the curvature vector. If we let s = s t be the arclength parameter on # t = F (, t)(#), then -# # = (d 2 /ds 2 )F and equation (1.1) can be written in the form (1.2) d dt - d 2 d...