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PreprintCC BY 4.0arXiv

Ancient mean curvature flow asymptotic to a minimal quadratic cone

  • Junyoung Park
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Abstract

In this paper, we prove that a smooth, properly embedded ancient mean curvature flow that is asymptotic to an O(n)× O(m) symmetric minimal quadratic cone for n +m ≥ 10, and lies on one side of the cone has to have unique asymptotics in the parabolic region. When additionally assuming mean convexity, we upgrade unique asymptotics to full uniqueness, and show that such flow has to be a stationary flow given by one of the leaves of the Hardt-Simon foliation. This is the first rigidity/unique asymptotics theory for ancient mean curvature flow with a singular minimal cone as the asymptotic model.

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Cite

BibTeXRIS
BibTeX
@misc{park2026ancient,
  title = {{Ancient mean curvature flow asymptotic to a minimal quadratic cone}},
  author = {Junyoung Park},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.11011},
  eprint = {2608.11011},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.11011},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Ancient mean curvature flow asymptotic to a minimal quadratic cone
AU  - Junyoung Park
PY  - 2026
DA  - 2026/09/19
PB  - arXiv
DO  - 10.48550/arxiv.2608.11011
UR  - https://arxiv.org/abs/2608.11011
AB  - In this paper, we prove that a smooth, properly embedded ancient mean curvature flow that is asymptotic to an O(n)× O(m) symmetric minimal quadratic cone for n +m ≥ 10, and lies on one side of the cone has to have unique asymptotics in the parabolic region. When additionally assuming mean convexity, we upgrade unique asymptotics to full uniqueness, and show that such flow has to be a stationary flow given by one of the leaves of the Hardt-Simon foliation. This is the first rigidity/unique asymptotics theory for ancient mean curvature flow with a singular minimal cone as the asymptotic model.
ER  -