The nature of singularities in mean curvature flow of mean-convex sets

White, Brian

doi:10.1090/S0894-0347-02-00406-X

The nature of singularities in mean curvature flow of mean-convex sets
HTML articles powered by AMS MathViewer

by Brian White;

J. Amer. Math. Soc. 16 (2003), 123-138

DOI: https://doi.org/10.1090/S0894-0347-02-00406-X

Published electronically: October 9, 2002

PDF | Request permission

Abstract:

This paper analyzes the singular behavior of the mean curvature flow generated by the boundary of the compact mean-convex region of $\mathbf {R}^{n+1}$ or of an $(n+1)$-dimensional riemannian manifold. If $n<7$, the moving boundary is shown to be very nearly convex in a spacetime neighborhood of any singularity. In particular, the tangent flows at singular points are all shrinking spheres or shrinking cylinders. If $n\ge 7$, the same results are shown up to the first time that singularities occur.

References

Kenneth A. Brakke, The motion of a surface by its mean curvature, Mathematical Notes, vol. 20, Princeton University Press, Princeton, NJ, 1978. MR 485012
S. Gallot, D. Hulin, and J. Lafontaine, Riemannian geometry, Universitext, Springer-Verlag, Berlin, 1987. MR 909697, DOI 10.1007/978-3-642-97026-9
Richard S. Hamilton, Four-manifolds with positive curvature operator, J. Differential Geom. 24 (1986), no. 2, 153–179. MR 862046
Gerhard Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom. 20 (1984), no. 1, 237–266. MR 772132
Gerhard Huisken, Local and global behaviour of hypersurfaces moving by mean curvature, Differential geometry: partial differential equations on manifolds (Los Angeles, CA, 1990) Proc. Sympos. Pure Math., vol. 54, Amer. Math. Soc., Providence, RI, 1993, pp. 175–191. MR 1216584, DOI 10.1090/pspum/054.1/1216584
Gerhard Huisken and Carlo Sinestrari, Mean curvature flow singularities for mean convex surfaces, Calc. Var. Partial Differential Equations 8 (1999), no. 1, 1–14. MR 1666878, DOI 10.1007/s005260050113
Gerhard Huisken and Carlo Sinestrari, Convexity estimates for mean curvature flow and singularities of mean convex surfaces, Acta Math. 183 (1999), no. 1, 45–70. MR 1719551, DOI 10.1007/BF02392946
Tom Ilmanen, Elliptic regularization and partial regularity for motion by mean curvature, Mem. Amer. Math. Soc. 108 (1994), no. 520, x+90. MR 1196160, DOI 10.1090/memo/0520
Tom Ilmanen, The level-set flow on a manifold, Differential geometry: partial differential equations on manifolds (Los Angeles, CA, 1990) Proc. Sympos. Pure Math., vol. 54, Amer. Math. Soc., Providence, RI, 1993, pp. 193–204. MR 1216585, DOI 10.1090/pspum/054.1/1216585

Singularities of mean curvature flow of surfaces

Murray H. Protter and Hans F. Weinberger, Maximum principles in differential equations, Springer-Verlag, New York, 1984. Corrected reprint of the 1967 original. MR 762825, DOI 10.1007/978-1-4612-5282-5
James Simons, Minimal varieties in riemannian manifolds, Ann. of Math. (2) 88 (1968), 62–105. MR 233295, DOI 10.2307/1970556
Richard Schoen and Leon Simon, Regularity of stable minimal hypersurfaces, Comm. Pure Appl. Math. 34 (1981), no. 6, 741–797. MR 634285, DOI 10.1002/cpa.3160340603
Brian White, Partial regularity of mean-convex hypersurfaces flowing by mean curvature, Internat. Math. Res. Notices 4 (1994), 186 ff., approx. 8 pp.}, issn=1073-7928, review= MR 1266114, doi=10.1155/S1073792894000206,
Brian White, The topology of hypersurfaces moving by mean curvature, Comm. Anal. Geom. 3 (1995), no. 1-2, 317–333. MR 1362655, DOI 10.4310/CAG.1995.v3.n2.a5
Brian White, Stratification of minimal surfaces, mean curvature flows, and harmonic maps, J. Reine Angew. Math. 488 (1997), 1–35. MR 1465365, DOI 10.1515/crll.1997.488.1
Brian White, The size of the singular set in mean curvature flow of mean-convex sets, J. Amer. Math. Soc. 13 (2000), no. 3, 665–695. MR 1758759, DOI 10.1090/S0894-0347-00-00338-6

Brakke’s regularity theorem for limits of smooth flows

A strict maximum principle at infinity for mean curvature flow