Singularity structure in mean curvature flow of mean-convex sets
Authors:
Tobias H. Colding and Bruce Kleiner
Journal:
Electron. Res. Announc. Amer. Math. Soc. 9 (2003), 121-124
MSC (2000):
Primary 53C44
DOI:
https://doi.org/10.1090/S1079-6762-03-00119-7
Published electronically:
November 26, 2003
MathSciNet review:
2029473
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Abstract: In this note we announce results on the mean curvature flow of mean-convex sets in three dimensions. Loosely speaking, our results justify the naive picture of mean curvature flow where the only singularities are neck pinches, and components which collapse to asymptotically round spheres.
- Kenneth A. Brakke, The motion of a surface by its mean curvature, Mathematical Notes, vol. 20, Princeton University Press, Princeton, N.J., 1978. MR 485012
- Yun Gang Chen, Yoshikazu Giga, and Shun’ichi Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geom. 33 (1991), no. 3, 749–786. MR 1100211
- L. C. Evans and J. Spruck, Motion of level sets by mean curvature. I, J. Differential Geom. 33 (1991), no. 3, 635–681. MR 1100206
[HS]huiskensinestrari G. Huisken and C. Sinestrari, in preparation.
- 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 https://doi.org/10.1007/BF02392946
- 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 https://doi.org/10.1007/s005260050113
- 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 https://doi.org/10.1090/memo/0520
[Per02]perelman1 G. Perelman, The entropy formula for the Ricci flow and its geometric applications, math.DG/0211159.
[Whi95]white3 B. White, The topology of hypersurfaces moving by mean curvature, Comm. Anal. Geom. 3 (1995), no. 1-2, 317–333.
- 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 https://doi.org/10.1090/S0894-0347-00-00338-6
- Brian White, The nature of singularities in mean curvature flow of mean-convex sets, J. Amer. Math. Soc. 16 (2003), no. 1, 123–138. MR 1937202, DOI https://doi.org/10.1090/S0894-0347-02-00406-X
[Bra78]brakke K. Brakke, The motion of a surface by its mean curvature, Mathematical Notes, vol. 20, Princeton University Press, Princeton, N.J., 1978.
[CGG91]chengigagoto Y. G. Chen, Y. Giga, and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geom. 33 (1991), no. 3, 749–786.
[ES91]evansspruck L. C. Evans and J. Spruck, Motion of level sets by mean curvature. I, J. Differential Geom. 33 (1991), no. 3, 635–681.
[HS]huiskensinestrari G. Huisken and C. Sinestrari, in preparation.
[HS99a]huiskensinestrari0 ---, Convexity estimates for mean curvature flow and singularities of mean-convex surfaces, Acta Math. 183 (1999), no. 1, 45–70.
[HS99b]huiskensinestrari1 ---, Mean curvature flow singularities for mean-convex surfaces, Calc. Var. Partial Differential Equations 8 (1999), no. 1, 1–14.
[Ilm94]ilmanen T. Ilmanen, Elliptic regularization and partial regularity for motion by mean curvature, Mem. Amer. Math. Soc. 108 (1994), no. 520, x+90.
[Per02]perelman1 G. Perelman, The entropy formula for the Ricci flow and its geometric applications, math.DG/0211159.
[Whi95]white3 B. White, The topology of hypersurfaces moving by mean curvature, Comm. Anal. Geom. 3 (1995), no. 1-2, 317–333.
[Whi00]white1 ---, The size of the singular set in mean curvature flow of mean-convex sets, J. Amer. Math. Soc. 13 (2000), no. 3, 665–695.
[Whi03]white2 ---, The nature of singularities in mean curvature flow of mean-convex sets, J. Amer. Math. Soc. 16 (2003), no. 1, 123–138.
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Additional Information
Tobias H. Colding
Affiliation:
Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, New York 10012
MR Author ID:
335440
Email:
colding@cims.nyu.edu
Bruce Kleiner
Affiliation:
Department of Mathematics, University of Michigan, 2072 East Hall, 525 E University Avenue, Ann Arbor, Michigan 48109-1109
Email:
bkleiner@umich.edu
Keywords:
Mean-convex,
mean curvature,
singularities
Received by editor(s):
September 24, 2003
Published electronically:
November 26, 2003
Additional Notes:
The first author was supported by NSF grant DMS-0104453
The second author was supported by NSF grant DMS-0204506
Communicated by:
Svetlana Katok
Article copyright:
© Copyright 2003
American Mathematical Society
