Length bounds and rigidity for ancient curve shortening flow on warped product surfaces
We study ancient solutions to curve shortening flow on a class of warped product surfaces. For closed embedded ancient solutions whose curves are not null-homotopic, we prove that graphicality at every time is equivalent to a uniform length bound as $t\to-\infty$, and that these properties hold whenever the length is $o(\sqrt{-t})$. Under an additional positivity condition on the linearized height equation, every such solution with bounded length is a family of slices. We derive backward convergence to a limiting circle from the length bound and prove slice rigidity by an $L^2$ estimate for the height perturbation.
math.DG↗