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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→−∞, and that these properties hold whenever the length is o(√(−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 L2 estimate for the height perturbation.

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Cite

BibTeXRIS
BibTeX
@misc{goh2026length,
  title = {{Length bounds and rigidity for ancient curve shortening flow on warped product surfaces}},
  author = {Beomseok Goh and Taehun Lee},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.34274},
  eprint = {2610.34274},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.34274},
}
RIS
TY  - UNPB
TI  - Length bounds and rigidity for ancient curve shortening flow on warped product surfaces
AU  - Beomseok Goh
AU  - Taehun Lee
PY  - 2026
DA  - 2026/10/25
PB  - arXiv
DO  - 10.48550/arxiv.2610.34274
UR  - https://arxiv.org/abs/2610.34274
AB  - 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→−∞, and that these properties hold whenever the length is o(√(−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.
ER  -