The size of the singular set in mean curvature flow of mean-convex sets

Author:
Brian White

Journal:
J. Amer. Math. Soc. **13** (2000), 665-695

MSC (2000):
Primary 53C44; Secondary 49Q20

DOI:
https://doi.org/10.1090/S0894-0347-00-00338-6

Published electronically:
April 10, 2000

MathSciNet review:
1758759

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that when a compact mean-convex subset of $\mathbf {R}^{n+1}$ (or of an $(n+1)$-dimensional riemannian manifold) moves by mean-curvature, the spacetime singular set has parabolic hausdorff dimension at most $n-1$. Examples show that this is optimal. We also show that, as $t\to \infty$, the surface converges to a compact stable minimal hypersurface whose singular set has dimension at most $n - 7$. If $n < 7$, the convergence is everywhere smooth and hence after some time $T$, the moving surface has no singularities

- 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**
[ES]ES L. C. Evans and J. Spruck, - Herbert 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), 767–771. MR**260981**, DOI https://doi.org/10.1090/S0002-9904-1970-12542-3 - David Gilbarg and Neil S. Trudinger,
*Elliptic partial differential equations of second order*, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR**737190** - Gerhard Huisken,
*Lecture one: mean curvature evolution of closed hypersurfaces*, Tsing Hua lectures on geometry & analysis (Hsinchu, 1990–1991) Int. Press, Cambridge, MA, 1997, pp. 117–123. MR**1482034** - Gerhard Huisken,
*Asymptotic behavior for singularities of the mean curvature flow*, J. Differential Geom.**31**(1990), no. 1, 285–299. MR**1030675** - Gerhard Huisken,
*Local and global behaviour of hypersurfaces moving by mean curvature*, Differential geometry: partial differential equations on manifolds (Los Angeles, CA, 1990) Proc. Sympos. Pure Math., vol. 54, Amer. Math. Soc., Providence, RI, 1993, pp. 175–191. MR**1216584**, DOI https://doi.org/10.1090/pspum/054.1/1216584
[HS]HS G. Huisken and C. Sinestrari, - 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 - Tom Ilmanen,
*Generalized flow of sets by mean curvature on a manifold*, Indiana Univ. Math. J.**41**(1992), no. 3, 671–705. MR**1189906**, DOI https://doi.org/10.1512/iumj.1992.41.41036 - Tom Ilmanen,
*The level-set flow on a manifold*, Differential geometry: partial differential equations on manifolds (Los Angeles, CA, 1990) Proc. Sympos. Pure Math., vol. 54, Amer. Math. Soc., Providence, RI, 1993, pp. 193–204. MR**1216585**, DOI https://doi.org/10.1090/pspum/054.1/1216585
[I4]I4 ---, - Gary M. Lieberman,
*Second order parabolic differential equations*, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. MR**1465184** - Stanley Osher and James A. Sethian,
*Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations*, J. Comput. Phys.**79**(1988), no. 1, 12–49. MR**965860**, DOI https://doi.org/10.1016/0021-9991%2888%2990002-2 - James Simons,
*Minimal varieties in riemannian manifolds*, Ann. of Math. (2)**88**(1968), 62–105. MR**233295**, DOI https://doi.org/10.2307/1970556 - Leon Simon,
*Lectures on geometric measure theory*, Proceedings of the Centre for Mathematical Analysis, Australian National University, vol. 3, Australian National University, Centre for Mathematical Analysis, Canberra, 1983. MR**756417** - Richard Schoen and Leon Simon,
*Regularity of stable minimal hypersurfaces*, Comm. Pure Appl. Math.**34**(1981), no. 6, 741–797. MR**634285**, DOI https://doi.org/10.1002/cpa.3160340603 - Andrew Stone,
*A density function and the structure of singularities of the mean curvature flow*, Calc. Var. Partial Differential Equations**2**(1994), no. 4, 443–480. MR**1383918**, DOI https://doi.org/10.1007/BF01192093 - Brian White,
*Partial regularity of mean-convex hypersurfaces flowing by mean curvature*, Internat. Math. Res. Notices**4**(1994), 186 ff., approx. 8 pp.}, issn=1073-7928, review=\MR{1266114}, doi=10.1155/S1073792894000206,. - Brian White,
*The topology of hypersurfaces moving by mean curvature*, Comm. Anal. Geom.**3**(1995), no. 1-2, 317–333. MR**1362655**, DOI https://doi.org/10.4310/CAG.1995.v3.n2.a5 - Brian White,
*Stratification of minimal surfaces, mean curvature flows, and harmonic maps*, J. Reine Angew. Math.**488**(1997), 1–35. MR**1465365**, DOI https://doi.org/10.1515/crll.1997.488.1
[W4]W4 ---,

*Motion of level sets by mean curvature I*, J. Diff. Geom.

**33**(1991), 635–681;

*II*, Trans. Amer. Math. Soc.

**330**(1992), 321–332;

*III*, J. Geom. Anal.

**2**(1992), 121–150;

*IV*, J. Geom. Anal.

**5**(1995), 77–114. ; ; ;

*Convexity estimates for mean curvature flow and singularities of mean convex surfaces*, Acta Math.

**183**(1999), 45–70.

*Singularities of mean curvature flow of surfaces*, preprint.

*A local regularity theorem for classical mean curvature flow*, preprint. [W5]W5 ---,

*The nature of singularities in mean curvature flow of mean-convex surfaces*, preprint (available at http://math.stanford.edu/˜white). [W6]W6 ---,

*Subsequent singularities in mean curvature flow of mean convex surfaces*, in preparation.

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

Email:
white@math.stanford.edu

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