  Given two curves bounding a region of area $A$ that evolve under curve
shortening flow, we propose the principle that the regularity of one should be
controllable in terms of the regularity of the other, starting from time
$A/\pi$. We prove several results of this form and demonstrate that no estimate
can hold before that time. As an example application, we construct solutions to
graphical curve shortening flow starting with initial data that is merely an
$L^1$ function.
