A distance comparison principle for curve shortening flow with free boundary

Abstract

We introduce a reflected chord-arc profile for curves with orthogonal boundary condition and obtain a chord-arc estimate for embedded free boundary curve shortening flows in a convex planar domain. As a consequence, we are able to prove that any such flow either converges in infinite time to a (unique) ``critical chord'', or contracts in finite time to a ``round half-point'' on the boundary.

arXiv comment32 pages, 2 figures; adjusted to a more precise barrier function in Theorem 5.3