The size of the singular set in mean curvature flow of meanconvex sets
Author:
Brian White
Journal:
J. Amer. Math. Soc. 13 (2000), 665695
MSC (2000):
Primary 53C44; Secondary 49Q20
Published electronically:
April 10, 2000
MathSciNet review:
1758759
Fulltext PDF Free Access
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Abstract: We prove that when a compact meanconvex subset of (or of an dimensional riemannian manifold) moves by meancurvature, the spacetime singular set has parabolic hausdorff dimension at most . Examples show that this is optimal. We also show that, as , the surface converges to a compact stable minimal hypersurface whose singular set has dimension at most . If , the convergence is everywhere smooth and hence after some time , the moving surface has no singularities
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, A local regularity theorem for classical mean curvature flow, preprint.
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, The nature of singularities in mean curvature flow of meanconvex surfaces, preprint (available at http://math.stanford.edu/~white).
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, Subsequent singularities in mean curvature flow of mean convex surfaces, in preparation.
 [B]
 K. Brakke, The motion of a surface by its mean curvature, Princeton Univ. Press, 1978. MR 82c:49035
 [CGG]
 Y. G. Chen, Y. Giga, and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Diff. Geom. 33 (1991), 749786. MR 93a:35093
 [ES]
 L. C. Evans and J. Spruck, Motion of level sets by mean curvature I, J. Diff. Geom. 33 (1991), 635681; II, Trans. Amer. Math. Soc. 330 (1992), 321332; III, J. Geom. Anal. 2 (1992), 121150; IV, J. Geom. Anal. 5 (1995), 77114. MR 92h:35097; MR 92f:58050; MR 93d:58044; MR 96a:35077
 [F]
 H. Federer, The singular sets of area minimizing rectifiable currents with codimension one and of area minimizing flat chains modulo two with arbitrary codimension, Bull. Amer. Math. Soc. 76 (1970), 767771. MR 41:5601
 [GT]
 D. Gilbarg and N. Trudinger, Elliptic partial differential equations of second order, 2nd edition, Springer, New York, 1983. MR 86c:35035
 [H1]
 G. Huisken, Lecture one: mean curvature evolution of closed hypersurfaces, Tsing Hua lectures on geometry and analysis (Hsinchu, 19901991), Internat. Press, Cambridge, MA, 1997, pp. 117123; Lecture two: singularities of the mean curvature flow, pp. 125130. MR 99a:53040
 [H2]
 , Asymptotic behavior for singularities of the mean curvature flow, J. Diff. Geom. 31 (1990), 285299. MR 90m:53016
 [H3]
 , Local and global behavior of hypersurfaces moving by mean curvature, Proc. Sympos. Pure Math. 54 (1993), 175191. MR 94c:58037
 [HS]
 G. Huisken and C. Sinestrari, Convexity estimates for mean curvature flow and singularities of mean convex surfaces, Acta Math. 183 (1999), 4570. CMP 2000:05
 [I1]
 T. Ilmanen, Elliptic regularization and partial regularity for motion by mean curvature, Mem. Amer. Math. Soc. 108 (520) (1994). MR 95d:49060
 [I2]
 , Generalized flow of sets by mean curvature on a manifold, Indiana Univ. Math. J. 41 (1992), 671705. MR 93k:58057
 [I3]
 , The levelset flow on a manifold, Proc. Sympos. Pure Math. 54 (1993), 193204. MR 94d:58040
 [I4]
 , Singularities of mean curvature flow of surfaces, preprint.
 [L]
 G. Lieberman, Second order parabolic differential equations, World Scientific, Singapore, 1996. MR 98k:35003
 [OS]
 S. Osher and J. Sethian, Fronts propagating with curvaturedependent speed: algorithms based on HamiltonJacobi formulations, J. Comput. Phys. 79 (1988), 1249. MR 89h:80012
 [SJ]
 J. Simons, Minimal varieties in riemannian manifolds, Ann. of Math. 88 (1962), 62105. MR 38:1617
 [SL]
 L. Simon, Lectures on geometric measure theory, Proc. Centre for Math. Analysis 3 (1983). MR 87a:49001
 [SS]
 R. Schoen and L. Simon, Regularity of stable minimal hypersurfaces, Comm. Pure Appl. Math. 34 (1981), 741797. MR 82k:49054
 [St]
 A. Stone, A density function and the structure of singularities of the mean curvature flow, Calc. Var. Partial Diff. Equations 2 (1994). MR 97c:58030
 [W1]
 B. White, Partial regularity of meanconvex hypersurfaces flowing by mean curvature, International Math. Research Notices 4 (1994), 185192. MR 95b:58042
 [W2]
 , The topology of hypersurfaces moving by mean curvature, Comm. Analysis and Geom. 3 (1995), 317333. MR 96k:58051
 [W3]
 , Stratification of minimal surfaces, mean curvature flows, and harmonic maps, J. reine angew. Math. 488 (1997), 135. MR 99b:49038
 [W4]
 , A local regularity theorem for classical mean curvature flow, preprint.
 [W5]
 , The nature of singularities in mean curvature flow of meanconvex surfaces, preprint (available at http://math.stanford.edu/~white).
 [W6]
 , Subsequent singularities in mean curvature flow of mean convex surfaces, in preparation.
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Additional Information
Brian White
Affiliation:
Department of Mathematics, Stanford University, Stanford, California 94305
Email:
white@math.stanford.edu
DOI:
http://dx.doi.org/10.1090/S0894034700003386
PII:
S 08940347(00)003386
Keywords:
Mean curvature flow,
mean convex,
singularities
Received by editor(s):
November 16, 1998
Received by editor(s) in revised form:
March 15, 2000
Published electronically:
April 10, 2000
Additional Notes:
The research presented here was partially funded by NSF grant DMS 9803403 and by a Guggenheim Foundation Fellowship.
Article copyright:
© Copyright 2000
American Mathematical Society
