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

Author: Brian White
Journal: J. Amer. Math. Soc. 16 (2003), 123-138
MSC (2000): Primary 53C44; Secondary 49Q20
Published electronically: October 9, 2002
MathSciNet review: 1937202
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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.

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  • [B] K. Brakke, The motion of a surface by its mean curvature, Princeton U. Press, 1978. MR 82c:49035
  • [GHL] S. Gallot, D. Hulin, and J. Lafontaine, Riemannian geometry, Springer-Verlag, 1987. MR 88k:53001
  • [Ha] R. S. Hamilton, Four-manifolds with positive curvature operator, J. Differential Geom. 24 (1986), 153-179. MR 87m:53055
  • [H1] G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom. 20 (1984), 237-266. MR 86j:53097
  • [H2] -, Local and global behavior of hypersurfaces moving by mean curvature, Proc. Symp. Pure Math. 54 (1993), 175-191. MR 94c:58037
  • [HS1] G. Huisken and C. Sinestrari, Mean curvature flow singularities for mean convex surfaces, Calc. Var. Partial Differential Equations 8 (1999), 1-14. MR 99m:58057
  • [HS2] -, Convexity estimates for mean curvature flow and singularities of mean convex surfaces, Acta Math. 183 (1999), 45-70. MR 2001c:53094
  • [I1] T. Ilmanen, Elliptic regularization and partial regularity for motion by mean curvature, Mem. Amer. Math. Soc. 108 (520) (1994). MR 95d:49060
  • [I2] -, The level-set flow on a manifold, Proc. Symp. Pure Math. 54 (1993), 193-204. MR 94d:58040
  • [I3] -, Singularities of mean curvature flow of surfaces, preprint.
  • [PW] M. H. Protter and H. F. Weinberger, Maximum principles in differential equations, Springer-Verlag, 1984. MR 86f:35034
  • [SJ] J. Simons, Minimal varieties in riemannian manifolds, Ann. of Math. 88 (1962), 62-105. MR 38:1617
  • [SS] R. Schoen and L. Simon, Regularity of stable minimal hypersurfaces, Comm. Pure Appl. Math. 34 (1981), 741-797. MR 82k:49054
  • [W1] Brian White, Partial regularity of mean-convex hypersurfaces flowing by mean curvature, International Math. Research Notices 4 (1994), 185-192. MR 95b:58042
  • [W2] -, The topology of hypersurfaces moving by mean curvature, Comm. Analysis and Geom. 3 (1995), 317-333. MR 96k:58051
  • [W3] -, Stratification of minimal surfaces, mean curvature flows, and harmonic maps, J. reine angew. Math. 488 (1997), 1-35. MR 99b:49038
  • [W4] -, The size of the singular set in mean curvature flow of mean-convex surfaces, J. Amer. Math. Soc. 13 (2000), 665-695. MR 2001j:53098
  • [W5] -, Brakke's regularity theorem for limits of smooth flows, in preparation.
  • [W6] -, A strict maximum principle at infinity for mean curvature flow, in preparation.

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Additional Information

Brian White
Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305-2060

Keywords: Mean curvature flow, mean convex, singularity
Received by editor(s): November 25, 1998
Received by editor(s) in revised form: September 11, 2002
Published electronically: October 9, 2002
Additional Notes: The research presented here was partially funded by NSF grants DMS 9803403, DMS 0104049, and by a Guggenheim Foundation Fellowship.
Article copyright: © Copyright 2002 American Mathematical Society

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