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Geometric Flows

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Type-II singularities of two-convex immersed mean curvature flow

Theodora Bourni / Mat Langford
Published Online: 2016-10-17 | DOI: https://doi.org/10.1515/geofl-2016-0001


We show that any strictly mean convex translator of dimension n ≥ 3 which admits a cylindrical estimate and a corresponding gradient estimate is rotationally symmetric. As a consequence, we deduce that any translating solution of the mean curvature flow which arises as a blow-up limit of a two-convex mean curvature flow of compact immersed hypersurfaces of dimension n ≥ 3 is rotationally symmetric. The proof is rather robust, and applies to a more general class of translator equations. As a particular application, we prove an analogous result for a class of flows of embedded hypersurfaces which includes the flow of twoconvex hypersurfaces by the two-harmonic mean curvature.


  • [1] U. Abresch and J. Langer. The normalized curve shortening flow and homothetic solutions. J. Differential Geom., 23(2):175- 196, 1986.Google Scholar

  • [2] Steven J. Altschuler and Lang F.Wu. Translating surfaces of the non-parametric mean curvature flow with prescribed contact angle. Calc. Var. Partial Differ. Equ., 2(1):101-111, 1994.Google Scholar

  • [3] Ben Andrews. Contraction of convex hypersurfaces in Euclidean space. Calc. Var. Partial Differential Equations, 2(2):151-171, 1994.Google Scholar

  • [4] Ben Andrews. Harnack inequalities for evolving hypersurfaces. Math. Z., 217(2):179-197, 1994.Google Scholar

  • [5] Ben Andrews. Noncollapsing in mean-convex mean curvature flow. Geom. Topol., 16(3):1413-1418, 2012.CrossrefWeb of ScienceGoogle Scholar

  • [6] Ben Andrews and Mat Langford. Cylindrical estimates for hypersurfaces moving by convex curvature functions. Anal. PDE, 7(5):1091-1107, 2014.Google Scholar

  • [7] Ben Andrews, Mat Langford, and James McCoy. Non-collapsing in fully non-linear curvature flows. Ann. Inst. H. Poincaré Anal. Non Linéaire, 30(1):23-32, 2013.Google Scholar

  • [8] Simon Brendle. A sharp bound for the inscribed radius under mean curvature flow. Invent. Math., 202(1):217-237, 2015.Web of ScienceGoogle Scholar

  • [9] Simon Brendle and Gerhard Huisken. A fully nonlinear flow for two-convex hypersurfaces. Preprint, arXiv:1507.04651 [math.DG].Google Scholar

  • [10] Julie Clutterbuck, Oliver C. Schnürer, and Felix Schulze. Stability of translating solutions to mean curvature flow. Calc. Var. Partial Differ. Equ., 29(3):281-293, 2007.Google Scholar

  • [11] Klaus Ecker. Regularity theory for mean curvature flow. Boston, MA: Birkhäuser, 2004.Web of ScienceGoogle Scholar

  • [12] Klaus Ecker and Gerhard Huisken. Interior estimates for hypersurfaces moving by mean curvature. Invent. Math., 105(3):547-569, 1991.Google Scholar

  • [13] David Gilbarg and Neil S. Trudinger. Elliptic partial differential equations of second order. Reprint of the 1998 ed. Berlin: Springer, reprint of the 1998 ed. edition, 2001.Google Scholar

  • [14] Richard S. Hamilton. Convex hypersurfaces with pinched second fundamental form. Commun. Anal. Geom., 2(1):167-172, 1994.Google Scholar

  • [15] Richard S. Hamilton. Harnack estimate for the mean curvature flow. J. Differential Geom., 41(1):215-226, 1995.Google Scholar

  • [16] Robert Haslhofer. Uniqueness of the bowl soliton. Preprint, arXiv:math/1408.3145 [math.DG].Google Scholar

  • [17] Robert Haslhofer and Bruce Kleiner. Mean curvature flow of mean convex hypersurfaces. To appear in Comm. Pure Appl. Math. Preprint available at arXiv:1304.0926 [math.DG].Google Scholar

  • [18] Robert Haslhofer and Bruce Kleiner. On Brendle’s estimate for the inscribed radius under mean curvature flow. Int. Math. Res. Not., 2015(15):6558-6561, 2015. Google Scholar

  • [19] Gerhard Huisken. Local and global behaviour of hypersurfaces moving by mean curvature. In Differential geometry. Part 1: Partial differential equations on manifolds. Proceedings of a summer research institute, held at the University of California, Los Angeles, CA, USA, July 8-28, 1990, pages 175-191. Providence, RI: American Mathematical Society, 1993.Google Scholar

  • [20] Gerhard Huisken and Carlo Sinestrari. Convexity estimates for mean curvature flow and singularities of mean convex surfaces. Acta Math., 183(1):45-70, 1999.Google Scholar

  • [21] Gerhard Huisken and Carlo Sinestrari. Mean curvature flow singularities for mean convex surfaces. Calc. Var. Partial Differential Equations, 8(1):1-14, 1999.Google Scholar

  • [22] Gerhard Huisken and Carlo Sinestrari. Mean curvature flow with surgeries of two-convex hypersurfaces. Invent. Math., 175(1):137-221, 2009.Google Scholar

  • [23] Gerhard Huisken and Carlo Sinestrari. Convex ancient solutions of the mean curvature flow. J. Differential Geom., 101(2):267-287, 2015.Google Scholar

  • [24] Mat Langford. Motion of hypersurfaces by curvature. PhD thesis, 8 2014.Google Scholar

  • [25] Mat Langford and Stephen Lynch. Inscribed and exscribed curvature pinching for fully non-linear curvature flows. In preparation.Google Scholar

  • [26] Francisco Martín, Andreas Savas-Halilaj, and Knut Smoczyk. On the topology of translating solitons of the mean curvature flow. Calc. Var. Partial Differ. Equ., 54(3):2853-2882, 2015.Google Scholar

  • [27] Richard Sacksteder. On hypersurfaces with no negative sectional curvatures. Am. J. Math., 82:609-630, 1960.CrossrefGoogle Scholar

  • [28] Xu-Jia Wang. Convex solutions to the mean curvature flow. Ann. of Math. (2), 173(3):1185-1239, 2011.Web of ScienceGoogle Scholar

About the article

Received: 2016-05-09

Accepted: 2016-09-02

Published Online: 2016-10-17

Published in Print: 2016-10-01

Citation Information: Geometric Flows, Volume 2, Issue 1, Pages 1–17, ISSN (Online) 2353-3382, DOI: https://doi.org/10.1515/geofl-2016-0001.

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© 2017. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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