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On the long‐time limit of the mean curvature flow in closed manifolds
Open accessAbstract
In this article, we show that generally almost regular flows, introduced by Bamler and Kleiner, in closed 3‐manifolds will either go extinct in finite time or flow to a collection of smooth embedded minimal surfaces, possibly with multiplicity. Using a perturbative argument, then we construct piecewise almost regular flows that either go extinct in finite time or flow to a stable minimal surface, possibly with multiplicity. We apply these results to construct minimal surfaces in 3‐manifolds in a variety of circumstances, mainly novel from the point of the view that the arguments are via parabolic methods.
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References 51 · Cited by 2
Research works printed but not linked: 58
- S. B.Angenent T.Ilmanen andJ. J. L.Velázquez Fattening from smooth initial data in mean curvature flow Preprint 2017.
- R. H.BamlerandB.Kleiner On the multiplicity one conjecture for mean curvature flows of surfaces arXiv preprint arXiv:2312.02106 2023. · arXiv
- Brakke K. A., 1978, The motion of a surface by its mean curvature. (MN‐20)
- 10.4007/annals.2016.183.2.6 · DOI
- 10.1007/s00222-015-0599-3 · DOI
- 10.4310/jdg/1622743139 · DOI
- 10.4310/jdg/1214446564 · DOI
- J.ChenandA.Sun Mean curvature flow with multiplicity2$\hskip.001pt 2$convergence in closed manifolds arXiv preprint arXiv:2402.04521 2024. · arXiv
- 10.1007/s00222-024-01258-0 · DOI · Paper page
- O.Chodosh K.Choi andF.Schulze Mean curvature flow with generic initial data ii arXiv preprint arXiv:2302.08409 2023. · DOI
- 10.4310/ACTA.2022.v228.n2.a1 · DOI
- Colding T. H., 2003, Surveys in differential geometry, vol. VIII (Boston, MA, 2002), 75
- 10.4007/annals.2012.175.2.7 · DOI
- 10.1007/s00222-015-0617-5 · DOI
- 10.1016/j.aim.2022.108715 · DOI
- 10.1007/s00222-006-0001-6 · DOI
- 10.2307/1971452 · DOI
- 10.4310/jdg/1214446559 · DOI
- 10.2307/1970227 · DOI
- 10.1007/BF02095997 · DOI
- 10.2307/1971486 · DOI
- 10.1215/00127094-2019-0009 · DOI
- 10.1215/00127094-0000008X · DOI
- 10.4310/jdg/1367438649 · DOI
- 10.1002/cpa.21852 · DOI
- 10.4310/CAG.2023.v31.n1.a2 · DOI
- 10.1512/iumj.1992.41.41036 · DOI
- 10.1090/memo/0520 · DOI
- T.Ilmanen Singularities of mean curvature flow of surfaces Preprint 1995.
- T.IlmanenandB.White Fattening in mean curvature flow arXiv preprint arXiv:2406.18703 2024. · arXiv
- D.Ketover Self‐shrinkers whose asymptotic cones fatten arXiv preprint arXiv:2407.01240 2024. · arXiv
- 10.4310/CAG.2013.v21.n2.a4 · DOI
- T.‐K.LeeandX.Zhao Closed mean curvature flows with asymptotically conical singularities arXiv preprint arXiv:2405.15577 2024. · arXiv
- 10.1112/jtopol/jtq031 · DOI
- 10.2307/2007026 · DOI
- 10.2307/1971088 · DOI
- 10.1007/s12220-017-9962-5 · DOI
- 10.1090/proc/13964 · DOI
- 10.2307/1970113 · DOI
- 10.2307/1971131 · DOI
- 10.2307/1971247 · DOI
- 10.2307/2006981 · DOI
- 10.4310/CAG.1993.v1.n2.a4 · DOI
- 10.1016/j.jfa.2024.110526 · DOI
- 10.4310/jdg/1685121322 · DOI
- A.SunandJ.Xue Initial perturbation of the mean curvature flow for asymptotical conical limit shrinker arXiv preprint arXiv:2107.05066 2021. · arXiv
- A.SunandJ.Xue Initial perturbation of the mean curvature flow for closed limit shrinker arXiv preprint arXiv:2104.03101 2021. · arXiv
- 10.4171/rmi/727 · DOI
- Z.WangandX.Zhou Existence of four minimal spheres inS3$ {S}^3$with a bumpy metric arXiv preprint arXiv:2305.08755 2023. · arXiv
- 10.1512/iumj.1991.40.40008 · DOI
- 10.4310/CAG.1995.v3.n2.a5 · DOI
- White B., 1997, Stratification of minimal surfaces, mean curvature flows, and harmonic maps, J. Reine Angew. Math., 488, 1
- B.White Evolution of curves and surfaces by mean curvature Proceedings of the International Congress of Mathematicians Vol. I (Beijing 2002) Higher Ed. Press Beijing 2002 pp.525–538.
- 10.1215/00127094-2009-019 · DOI
- 10.1353/ajm.2017.0029 · DOI
- 10.4310/CAG.2018.v26.n3.a7 · DOI
- 10.1007/s00526-024-02725-5 · DOI
- 10.4007/annals.2020.192.3.3 · DOI
Metadata
- Authors
- Mramor, A., & Sun, A.
- Affiliations
- University of Copenhagen (DK)
- Lehigh University (US)
- Corresponding author
- Mramor, Alexander
- Venue
- Journal of the London Mathematical Society
- Publisher
- Wiley
- Published
- 2026-01
- Volume
- 113
- Issue
- 1
- ISSN
- 0024-6107 (ISSN-L)
- 1469-7750
- OpenAlex
- W7125429916
- Cited by
- 2
- References
- 51
- Citations by year
- 2026: 1
- FWCI
- 2.20
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Licence
- Locations
- Journal of the London Mathematical Society (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Funders
- Danmarks Grundforskningsfond (CPH‐GEOTOP‐DNRF151)
- Carlsbergfondet (CF21‐0680)
- Topics
- Physical Sciences › Mathematics › Applied Mathematics › Geometric Analysis and Curvature Flows
- Physical Sciences › Mathematics › Geometry and Topology › Advanced Differential Geometry Research
- Physical Sciences › Mathematics › Geometry and Topology › Geometric and Algebraic Topology
- Keywords
- mean curvature flow, long-time limit, closed 3-manifolds, embedded minimal surfaces, multiplicity, finite-time extinction, stable minimal surfaces, parabolic method
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