The nature of singularities in mean curvature flow of meanconvex sets
Author:
Brian White
Journal:
J. Amer. Math. Soc. 16 (2003), 123138
MSC (2000):
Primary 53C44; Secondary 49Q20
Published electronically:
October 9, 2002
MathSciNet review:
1937202
Fulltext PDF Free Access
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Abstract: This paper analyzes the singular behavior of the mean curvature flow generated by the boundary of the compact meanconvex 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.
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 [Ha]
 R. S. Hamilton, Fourmanifolds with positive curvature operator, J. Differential Geom. 24 (1986), 153179. MR 87m:53055
 [H1]
 G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom. 20 (1984), 237266. MR 86j:53097
 [H2]
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 [HS1]
 G. Huisken and C. Sinestrari, Mean curvature flow singularities for mean convex surfaces, Calc. Var. Partial Differential Equations 8 (1999), 114. MR 99m:58057
 [HS2]
 , Convexity estimates for mean curvature flow and singularities of mean convex surfaces, Acta Math. 183 (1999), 4570. 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 levelset flow on a manifold, Proc. Symp. Pure Math. 54 (1993), 193204. MR 94d:58040
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 , Singularities of mean curvature flow of surfaces, preprint.
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 M. H. Protter and H. F. Weinberger, Maximum principles in differential equations, SpringerVerlag, 1984. MR 86f:35034
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 R. Schoen and L. Simon, Regularity of stable minimal hypersurfaces, Comm. Pure Appl. Math. 34 (1981), 741797. MR 82k:49054
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 [W3]
 , Stratification of minimal surfaces, mean curvature flows, and harmonic maps, J. reine angew. Math. 488 (1997), 135. MR 99b:49038
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Additional Information
Brian White
Affiliation:
Department of Mathematics, Stanford University, Stanford, California 943052060
Email:
white@math.stanford.edu
DOI:
http://dx.doi.org/10.1090/S089403470200406X
PII:
S 08940347(02)00406X
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
