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Local Convexity and Planarity Estimates for Mean Curvature Flow

  • Liang Cheng
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Abstract

We establish local convexity and cylindrical pinching estimates for hypersurface mean curvature flow under an intermediate convexity condition and integral bounds on the mean curvature. These estimates hold after a positive waiting time and do not require pointwise upper bounds for the mean curvature on the parabolic boundary. In arbitrary codimension, we also establish local planarity estimates for properly defined flows satisfying a uniform cn-pinching condition, without assuming bounded curvature. As applications, the hypersurface estimates yield convexity and cylindrical pinching for complete, spacetime-proper ancient hypersurface flows with uniform Euclidean area growth, without any bounded-curvature assumption. The planarity estimates imply that every complete, connected, spacetime-proper ancient flow satisfying the uniform cn-pinching condition is either a static affine n-plane or lies in a fixed affine (n+1)-plane for all time.

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Cite

BibTeXRIS
BibTeX
@misc{cheng2026local,
  title = {{Local Convexity and Planarity Estimates for Mean Curvature Flow}},
  author = {Liang Cheng},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.21903},
  eprint = {2610.21903},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.21903},
}
RIS
TY  - UNPB
TI  - Local Convexity and Planarity Estimates for Mean Curvature Flow
AU  - Liang Cheng
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2610.21903
UR  - https://arxiv.org/abs/2610.21903
AB  - We establish local convexity and cylindrical pinching estimates for hypersurface mean curvature flow under an intermediate convexity condition and integral bounds on the mean curvature. These estimates hold after a positive waiting time and do not require pointwise upper bounds for the mean curvature on the parabolic boundary. In arbitrary codimension, we also establish local planarity estimates for properly defined flows satisfying a uniform c_n-pinching condition, without assuming bounded curvature. As applications, the hypersurface estimates yield convexity and cylindrical pinching for complete, spacetime-proper ancient hypersurface flows with uniform Euclidean area growth, without any bounded-curvature assumption. The planarity estimates imply that every complete, connected, spacetime-proper ancient flow satisfying the uniform c_n-pinching condition is either a static affine n-plane or lies in a fixed affine (n+1)-plane for all time.
ER  -