Hostname: page-component-848d4c4894-2pzkn Total loading time: 0 Render date: 2024-05-22T22:41:20.940Z Has data issue: false hasContentIssue false
  • English
  • Français

A pinching estimate for convex hypersurfaces evolving under a non-homogeneous variant of mean curvature flow

Part of: Partial differential equations

Published online by Cambridge University Press:  18 April 2022

Tim Espin
Tim Espin*
Affiliation:
School of Mathematics, Maxwell Institute for Mathematical Sciences, University of Edinburgh, Edinburgh, EH9 3FD, UK (Tim.Espin@ed.ac.uk)
Get access Rights & Permissions [Opens in a new window]

Abstract

We study a variant of the mean curvature flow for closed, convex hypersurfaces where the normal velocity is a non-homogeneous function of the principal curvatures. We show that if the initial hypersurface satisfies a certain pinching condition, then this is preserved and the flow converges to a sphere under rescaling.

Keywords

convex hypersurfacesnon-homogeneous curvature flow

MSC classification

Secondary: 35B40: Asymptotic behavior of solutions
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 65 , Issue 2 , May 2022 , pp. 376 - 391
Check for updates
Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alessandroni, R. and Sinestrari, C., Convexity estimates for a nonhomogeneous mean curvature flow, Math. Z. 266 (2010), 65–82.CrossRefGoogle Scholar
Andrews, B., Contraction of convex hypersurfaces in Euclidean space, Calc. Var. 2 (1994), 151–171.CrossRefGoogle Scholar
Andrews, B., Langford, M. and McCoy, J., Convexity estimates for surfaces moving by curvature functions, J. Diff. Geo. 99 (2015), 47–75.Google Scholar
Chou, K.-S. and Wang, X.-J., A logarithmic Gauss curvature flow and the Minkowski problem, Ann. Inst. H. Poincaré Anal. Non Linéaire 17(6) (2000), 733–751.CrossRefGoogle Scholar
Chow, B., Deforming convex hypersurfaces by the nth root of the Gaussian curvature, J. Diff. Geo. 22 (1985), 117–138.Google Scholar
Ecker, K., Regularity theory for mean curvature flow (Birkhäuser, Boston, 2004).CrossRefGoogle Scholar
Gerhardt, C., Curvature problems. IP Series in Geometry and Topology, Vol. 39 (2006).Google Scholar
Huisken, G., Flow by mean curvature of convex surfaces into spheres, J. Diff. Geo. 20 (1984), 237–266.Google Scholar
Huisken, G. and Sinestrari, C., Convexity estimates for mean curvature flow and singularities of mean convex surfaces, Acta Math. 183 (1999), 45–70.CrossRefGoogle Scholar
Huisken, G. and Sinestrari, C., Mean curvature flow singularities for mean convex surfaces, Calc. Var. Partial Diff. Equ. 8 (1999), 1–14.CrossRefGoogle Scholar
Krylov, N. V., Nonlinear elliptic and parabolic equations of second order. Mathematics and Its Applications (Soviet Series), (D. Reidel Publishing Company, 1978).Google Scholar
Schulze, F., Evolution of convex hypersurfaces by powers of the mean curvature, Math. Z. 251(4) (2005), 721–733.CrossRefGoogle Scholar
Schulze, F., Convexity estimates for flows by powers of the mean curvature, Annali della Scuola Normale Superiore di Pisa – Classe di Scienze, Serie 5 5(2) (2006), 261–277.Google Scholar
Zhu, X., Lectures on mean curvature flows. AMS/IP Studies in Advanced Mathematics, Volume 32 (2002).CrossRefGoogle Scholar