Abstract
We prove a sharp pinching estimate for immersed mean convex solutions of mean curvature flow which unifies and improves all previously known pinching estimates, including the umbilic estimate of Huisken (J Differ Geom 20(1):237–266, 1984), the convexity estimates of Huisken–Sinestrari (Acta Math 183(1):45–70, 1999) and the cylindrical estimate of Huisken–Sinestrari (Invent Math 175(1):137–221, 2009; see also Andrews and Langford in Anal PDE 7(5):1091–1107, 2014; Huisken and Sinestrari in J Differ Geom 101(2):267–287, 2015). Namely, we show that the curvature of the solution pinches onto the convex cone generated by the curvatures of any shrinking cylinder solutions admitted by the initial data. For example, if the initial data is \((m+1)\)-convex, then the curvature of the solution pinches onto the convex hull of the curvatures of the shrinking cylinders \(\mathbb {R}^m\times S^{n-m}_{\sqrt{2(n-m)(1-t)}}\), \(t<1\). In particular, this yields a sharp estimate for the largest principal curvature, which we use to obtain a new proof of a sharp estimate for the inscribed curvature for embedded solutions (Brendle in Invent Math 202(1):217–237, 2015; Haslhofer and Kleiner in Int Math Res Not 15:6558–6561, 2015; Langford in Proc Am Math Soc 143(12):5395–5398, 2015). Making use of a recent idea of Huisken–Sinestrari (2015), we then obtain a series of sharp estimates for ancient solutions. In particular, we obtain a convexity estimate for ancient solutions which allows us to strengthen recent characterizations of the shrinking sphere due to Huisken–Sinestrari (2015) and Haslhofer–Hershkovits (Commun Anal Geom 24(3):593–604, 2016).
Similar content being viewed by others
Notes
If \(M^n=\partial K\) for some convex body \(K\subset \mathbb {R}^{n+1}\), we will say, explicitly, that \(M^n\) bounds a convex body.
Defined on time-dependent vector fields u by \(\nabla _tu:=[\partial _t,u]-HA(u)\) and extended to the tensor algebra in the usual way.
For a mean convex hypersurface, we will always define \(\overline{k}\) and \(\underline{k}\) with respect to the normal whose mean curvature is non-negative. This agrees with the outward pointing normal if \(M^n\) is connected.
It is tempting to conjecture that all ancient solutions should have bounded mean curvature as \(t\rightarrow -\infty \); however, recent numerical evidence suggests that this is false [11].
That is, invariant under permutation of components.
Note that the cylindrical point \(\mathrm {diag}(0,\dots ,0,1)\) is ruled out by the initial pinching condition \(\kappa _1+\cdots +\kappa _{m+1}\ge \alpha H\), \(m\in \{0,1,\dots ,n-2\}\).
References
Alexandroff, A.D.: Almost everywhere existence of the second differential of a convex function and some properties of convex surfaces connected with it. Leningrad State Univ. Ann. [Uchenye Zapiski] Math. Ser. 6, 3–35 (1939)
Andrews, B.: Pinching estimates and motion of hypersurfaces by curvature functions. J. Reine Angew. Math. 608, 17–33 (2007)
Andrews, B.: Moving surfaces by non-concave curvature functions. Calc. Var. Partial Differ. Equ. 39(3–4), 649–657 (2010)
Andrews, B.: Noncollapsing in mean-convex mean curvature flow. Geom. Topol. 16(3), 1413–1418 (2012)
Andrews, B., Hopper, C.: The Ricci Flow in Riemannian Geometry: A Complete Proof of the Differentiable 1/4-Pinching Sphere Theorem, Volume 2011 of Lecture Notes in Mathematics. Springer, Heidelberg (2011)
Andrews, B., Langford, M.: Cylindrical estimates for hypersurfaces moving by convex curvature functions. Anal. PDE 7(5), 1091–1107 (2014)
Andrews, B., Langford, M.: Two-sided non-collapsing curvature flows. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 15, 543–560 (2016)
Andrews, B., Langford, M., McCoy, J.: Non-collapsing in fully non-linear curvature flows. Ann. Inst. H. Poincaré Anal. Non Linéaire 30(1), 23–32 (2013)
Andrews, B., Langford, M., McCoy, J.: Convexity estimates for hypersurfaces moving by convex curvature functions. Anal. PDE 7(2), 407–433 (2014)
Andrews, B., Langford, M., McCoy, J.: Convexity estimates for surfaces moving by curvature functions. J. Differ. Geom. 99(1), 47–75 (2015)
Angenent, S.: An ancient compact solution to curve shortening. https://www.youtube.com/watch?v=8Ez0QoJ3XG8. Accessed 14 Aug 2016
Bourni, T., Langford, M.: Type-II singularities of two-convex immersed mean curvature flow. Geom. Flows 2, 1–17 (2016)
Brendle, S.: A sharp bound for the inscribed radius under mean curvature flow. Invent. Math. 202(1), 217–237 (2015)
Brendle, S., Huisken, G.: Mean curvature flow with surgery of mean convex surfaces in \({\mathbb{R}}^3\). Invent. Math. 203(2), 615–654 (2016)
Bryan, P., Ivaki, M.N., Scheuer, J.: On the classification of ancient solutions to curvature flows on the sphere. Preprint, arXiv:1604.01694 [math.DG]
Da Lio, F.: Remarks on the strong maximum principle for viscosity solutions to fully nonlinear parabolic equations. Commun. Pure Appl. Anal. 3(3), 395–415 (2004)
Daskalopoulos, P., Hamilton, R., Sesum, N.: Classification of compact ancient solutions to the curve shortening flow. J. Differ. Geom. 84(3), 455–464 (2010)
Gage, M., Hamilton, R.S.: The heat equation shrinking convex plane curves. J. Differ. Geom. 23, 69–96 (1986)
Hamilton, R.S.: Four-manifolds with positive curvature operator. J. Differ. Geom. 24(2), 153–179 (1986)
Haslhofer, R.: Uniqueness of the bowl soliton. Geom. Topol. 19(4), 2393–2406 (2015)
Haslhofer, R., Hershkovits, O.: Ancient solutions of the mean curvature flow. Commun. Anal. Geom. 24(3), 593–604 (2016)
Haslhofer, R., Kleiner, B.: Mean curvature flow of mean convex hypersurfaces. Commun. Pure Appl. Math. (to appear)
Haslhofer, R., Kleiner, B.: On Brendle’s estimate for the inscribed radius under mean curvature flow. Int. Math. Res. Not. 15, 6558–6561 (2015)
Huisken, G.: Flow by mean curvature of convex surfaces into spheres. J. Differ. Geom. 20(1), 237–266 (1984)
Huisken, G., Sinestrari, C.: Convexity estimates for mean curvature flow and singularities of mean convex surfaces. Acta Math. 183(1), 45–70 (1999)
Huisken, G., Sinestrari, C.: Mean curvature flow singularities for mean convex surfaces. Calc. Var. Partial Differ. Equ. 8(1), 1–14 (1999)
Huisken, G., Sinestrari, C.: Mean curvature flow with surgeries of two-convex hypersurfaces. Invent. Math. 175(1), 137–221 (2009)
Huisken, G., Sinestrari, C.: Convex ancient solutions of the mean curvature flow. J. Differ. Geom. 101(2), 267–287 (2015)
Langford, M.: Motion of hypersurfaces by curvature. Ph.D. Thesis, Australian National University, 8 (2014)
Langford, M.: The optimal interior ball estimate for a \(k\)-convex mean curvature flow. Proc. Am. Math. Soc. 143(12), 5395–5398 (2015)
Mantegazza, C.: Lecture notes on mean curvature flow. Progress in Mathematics, vol. 290. Birkhäuser/Springer Basel AG, Basel (2011)
Sheng, W., Wang, X.-J.: Singularity profile in the mean curvature flow. Methods Appl. Anal. 16(2), 139–155 (2009)
White, B.: The nature of singularities in mean curvature flow of mean-convex sets. J. Am. Math. Soc. 16(1), 123–138 (2003). (electronic)
Acknowledgements
This work has been discussed in the geometric analysis research seminar directed by Klaus Ecker at the Freie Universität Berlin. I am grateful to the members of the geometric analysis group and to the students who attended this seminar for many useful comments. I am particularly indebted to Stephen Lynch for providing helpful comments on a draft of this paper. I am also grateful to Carlo Sinestrari for helpful discussions concerning ancient solutions of the mean curvature flow. Finally, I wish to acknowledge the generous financial support of the Alexander von Humboldt Foundation.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by N. Trudinger.
Rights and permissions
About this article
Cite this article
Langford, M. A general pinching principle for mean curvature flow and applications. Calc. Var. 56, 107 (2017). https://doi.org/10.1007/s00526-017-1193-x
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00526-017-1193-x