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arXiv · 2610.34274

Length bounds and rigidity for ancient curve shortening flow on warped product surfaces

Beomseok Goh·Taehun Lee
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Abstract

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.

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Beomseok Goh, Taehun Lee. 2026-10-25. Length bounds and rigidity for ancient curve shortening flow on warped product surfaces. https://arxiv.org/abs/2610.34274

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