|
The nature of singularities in mean curvature flow of mean-convex sets
Author(s):
Brian
White
Journal:
J. Amer. Math. Soc.
16
(2003),
123-138.
MSC (2000):
Primary 53C44;
Secondary 49Q20
Posted:
October 9, 2002
Retrieve article in:
PDF DVI PostScript
Abstract |
References |
Similar articles |
Additional information
Abstract:
This paper analyzes the singular behavior of the mean curvature flow generated by the boundary of the compact mean-convex region of  or of an  -dimensional riemannian manifold. If  , 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  , the same results are shown up to the first time that singularities occur.
References:
-
- [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.
Similar Articles:
Retrieve articles in Journal of the American Mathematical Society
with MSC
(2000):
53C44,
49Q20
Retrieve articles in all Journals with MSC
(2000):
53C44,
49Q20
Additional Information:
Brian
White
Affiliation:
Department of Mathematics, Stanford University, Stanford, California 94305-2060
Email:
white@math.stanford.edu
DOI:
10.1090/S0894-0347-02-00406-X
PII:
S 0894-0347(02)00406-X
Keywords:
Mean curvature flow,
mean convex,
singularity
Received by editor(s):
November 25, 1998
Received by editor(s) in revised form:
September 11, 2002
Posted:
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.
Copyright of article:
Copyright
2002,
American Mathematical Society
|
Comments: webmaster@ams.org