Skip to content
DraconiticResearch
Search Sources API
203,234 studies31 sourcesUpdated 42 min ago
PreprintarXiv licencearXiv

Generic mean curvature flow has finitely many singularities

  • J. M. Daniels-Holgate
  • Or Hershkovits
  • Gábor Székelyhidi
Open at sourcePDFDOICite

Abstract

We show that for a generic compact initial surface in ℝ3, and any generic compact initial mean convex hypersurface in higher dimensions, the mean curvature flow encounters only finitely many singularities. This resolves the full generic Morse regularity conjectures of Ilmanen and Colding-Minicozzi-Pedersen in the above two cases. More generally, we show that if a mean curvature flow of a compact hypersurface has only multiplicity one spherical and cylindrical singularities, then after a small perturbation of the initial condition only non-degenerate cylindrical and spherical singularities appear.

Related

  1. The mean curvature blows up at nondegenerate neck pinch singularities of Lagrangian mean curvature flow
    Jason D. Lotay, Goncalo Oliveira
    arXivPreprintCC BY 4.0Full textGeometry & topologyarXiv
  2. Generic Mean Curvature Flow With Rough Weights
    Fabius Krämer, Tim Laux, Keisuke Takasao
    arXivPreprintCC BY 4.0Full textAnalysis & PDEsGeometry & topologyarXiv
  3. Singularities of Mean Curvature Flow with Bounded Total Curvature
    Yongheng Han, Zhen Wang
    arXivPreprintarXiv licenceFull textGeometry & topologyarXiv
  4. Existence of Lagrangian Mean Curvature Flow Singularities with Bounded Mean Curvature
    Maxwell Stolarski, Wei-Bo Su
    arXivPreprintCC BY 4.0Full textAnalysis & PDEsGeometry & topologyarXiv
  5. Mean curvature flows with cylindrical singularities II: stability and genericity
    Ao Sun, Jinxin Xue
    arXivPreprintCC BY 4.0Full textAnalysis & PDEsGeometry & topologyarXiv

Cite

BibTeXRIS
BibTeX
@misc{danielsholgate2026generic,
  title = {{Generic mean curvature flow has finitely many singularities}},
  author = {J. M. Daniels-Holgate and Or Hershkovits and Gábor Székelyhidi},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.12763},
  eprint = {2610.12763},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.12763},
}
RIS
TY  - UNPB
TI  - Generic mean curvature flow has finitely many singularities
AU  - J. M. Daniels-Holgate
AU  - Or Hershkovits
AU  - Gábor Székelyhidi
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2610.12763
UR  - https://arxiv.org/abs/2610.12763
AB  - We show that for a generic compact initial surface in ℝ^3, and any generic compact initial mean convex hypersurface in higher dimensions, the mean curvature flow encounters only finitely many singularities. This resolves the full generic Morse regularity conjectures of Ilmanen and Colding-Minicozzi-Pedersen in the above two cases. More generally, we show that if a mean curvature flow of a compact hypersurface has only multiplicity one spherical and cylindrical singularities, then after a small perturbation of the initial condition only non-degenerate cylindrical and spherical singularities appear.
ER  -