Generic mean curvature flow. I: Generic singularities. (English) Zbl 1239.53084

Colding, Tobias H.; Minicozzi, William P. II

Ann. Math. (2) 175, No. 2, 755-833 (2012).

This paper is a first in a series that aims in proving an old conjecture concerning the mean curvature flow. The conjecture says that starting at a generic smooth closed embedded surface in \(\mathbb R^3\), the mean curvature flow remains smooth until it arrives at a singularity in a neighborhood of which the flow looks like concentric spheres or cylinders. In this first approach, it is proved that, in all dimensions, the only singularities that cannot be perturbed away are cylinders and spheres. As application, it is shown that if generic mean curvature flow in \(\mathbb R^3\) disappears in a compact point, it does so in a round point.

Reviewer: Ricardo Miranda Martins (Campinas)

MSC:

53C44	Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
58E10	Variational problems in applications to the theory of geodesics (problems in one independent variable)

Keywords:

curvature flow; mean curvature; singularities

References:

[1]	U. Abresch and J. Langer, ”The normalized curve shortening flow and homothetic solutions,” J. Differential Geom., vol. 23, iss. 2, pp. 175-196, 1986. · Zbl 0592.53002
[2]	W. K. Allard, ”On the first variation of a varifold,” Ann. of Math., vol. 95, pp. 417-491, 1972. · Zbl 0252.49028
[3]	S. Altschuler, S. B. Angenent, and Y. Giga, ”Mean curvature flow through singularities for surfaces of rotation,” J. Geom. Anal., vol. 5, iss. 3, pp. 293-358, 1995. · Zbl 0847.58072
[4]	B. Andrews, ”Classification of limiting shapes for isotropic curve flows,” J. Amer. Math. Soc., vol. 16, iss. 2, pp. 443-459, 2003. · Zbl 1023.53051
[5]	S. B. Angenent, ”Shrinking doughnuts,” in Nonlinear Diffusion Equations and their Equilibrium States, Boston, MA: Birkhäuser, 1992, vol. 7, pp. 21-38. · Zbl 0762.53028
[6]	S. B. Angenent, T. Ilmanen, and D. L. Chopp, ”A computed example of nonuniqueness of mean curvature flow in \(\mathbb R^3\),” Comm. Partial Differential Equations, vol. 20, iss. 11-12, pp. 1937-1958, 1995.
[7]	J. Barta, ”Sur la vibration fundamentale dúne membrane,” C.R. Acad. Sci., vol. 204, pp. 472-473, 1937. · JFM 63.0762.02
[8]	K. A. Brakke, The Motion of a Surface by its Mean Curvature, Princeton, N.J.: Princeton Univ. Press, 1978, vol. 20. · Zbl 0386.53047
[9]	H-D. Cao, R. S. Hamilton, and T. Ilmanen, Gaussian densities and stability for some Ricci solitons, 2004.
[10]	I. Chavel, Eigenvalues in Riemannian Geometry, Orlando, FL: Academic Press, 1984, vol. 115. · Zbl 0551.53001
[11]	Y. G. Chen, Y. Giga, and S. Goto, ”Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations,” J. Differential Geom., vol. 33, iss. 3, pp. 749-786, 1991. · Zbl 0696.35087
[12]	D. L. Chopp, ”Computation of self-similar solutions for mean curvature flow,” Experiment. Math., vol. 3, iss. 1, pp. 1-15, 1994. · Zbl 0811.53011
[13]	T. H. Colding and W. P. Minicozzi II, Generic mean curvature flow II; dynamics of a closed smooth singularity. · Zbl 1341.53098
[14]	T. H. Colding and W. P. Minicozzi II, Smooth compactness of self-shrinkers. · Zbl 1258.53069
[15]	T. H. Colding and W. P. Minicozzi II, Minimal Surfaces, New York: New York University Courant Institute of Mathematical Sciences, 1999, vol. 4. · Zbl 0987.49025
[16]	T. H. Colding and W. P. Minicozzi II, ”Shapes of embedded minimal surfaces,” Proc. Natl. Acad. Sci. USA, vol. 103, iss. 30, pp. 11106-11111, 2006. · Zbl 1175.53008
[17]	T. H. Colding and W. P. Minicozzi II, ”The space of embedded minimal surfaces of fixed genus in a 3-manifold. I. Estimates off the axis for disks,” Ann. of Math., vol. 160, iss. 1, pp. 27-68, 2004. · Zbl 1070.53031
[18]	T. H. Colding and W. P. Minicozzi II, ”The space of embedded minimal surfaces of fixed genus in a 3-manifold. II. Multi-valued graphs in disks,” Ann. of Math., vol. 160, iss. 1, pp. 69-92, 2004. · Zbl 1070.53032
[19]	T. H. Colding and W. P. Minicozzi II, ”The space of embedded minimal surfaces of fixed genus in a 3-manifold. III. Planar domains,” Ann. of Math., vol. 160, iss. 2, pp. 523-572, 2004. · Zbl 1076.53068
[20]	T. H. Colding and W. P. Minicozzi II, ”The space of embedded minimal surfaces of fixed genus in a 3-manifold. IV. Locally simply connected,” Ann. of Math., vol. 160, iss. 2, pp. 573-615, 2004. · Zbl 1076.53069
[21]	T. H. Colding and W. P. Minicozzi II, The space of embedded minimal surfaces of fixed genus in a 3-manifold V; Fixed genus. · Zbl 1322.53059
[22]	T. H. Colding and W. P. Minicozzi II, A course on the heat equation and mean curvature flow. · Zbl 1165.53363
[23]	T. H. Colding and W. P. Minicozzi II, A Course in Minimal Surfaces, Providence, RI: Amer. Math. Soc., 2011, vol. 121. · Zbl 1242.53007
[24]	R. J. Duffin, ”Lower bounds for eigenvalues,” Physical Rev., vol. 71, pp. 827-828, 1947.
[25]	K. Ecker, ”Local monotonicity formulas for some nonlinear diffusion equations,” Calc. Var. Partial Differential Equations, vol. 23, iss. 1, pp. 67-81, 2005. · Zbl 1119.35026
[26]	K. Ecker, Regularity Theory for Mean Curvature Flow, Boston, MA: Birkhäuser, 2004, vol. 57. · Zbl 1058.53054
[27]	K. Ecker, ”A formula relating entropy monotonicity to Harnack inequalities,” Comm. Anal. Geom., vol. 15, iss. 5, pp. 1025-1061, 2007. · Zbl 1167.53055
[28]	K. Ecker, ”Heat equations in geometry and topology,” Jahresber. Deutsch. Math.-Verein., vol. 110, iss. 3, pp. 117-141, 2008. · Zbl 1151.53061
[29]	C. L. Epstein and M. Gage, ”The curve shortening flow,” in Wave Motion: Theory, Modelling, and Computation, New York: Springer-Verlag, 1987, vol. 7, pp. 15-59. · Zbl 0645.53028
[30]	C. L. Epstein and M. I. Weinstein, ”A stable manifold theorem for the curve shortening equation,” Comm. Pure Appl. Math., vol. 40, iss. 1, pp. 119-139, 1987. · Zbl 0602.34026
[31]	L. C. Evans, Partial Differential Equations, Providence, RI: Amer. Math. Soc., 1998, vol. 19. · Zbl 0902.35002
[32]	L. C. Evans and J. Spruck, ”Motion of level sets by mean curvature. I,” J. Differential Geom., vol. 33, iss. 3, pp. 635-681, 1991. · Zbl 0726.53029
[33]	M. Gage and R. S. Hamilton, ”The heat equation shrinking convex plane curves,” J. Differential Geom., vol. 23, iss. 1, pp. 69-96, 1986. · Zbl 0621.53001
[34]	M. A. Grayson, ”The heat equation shrinks embedded plane curves to round points,” J. Differential Geom., vol. 26, iss. 2, pp. 285-314, 1987. · Zbl 0667.53001
[35]	M. A. Grayson, ”A short note on the evolution of a surface by its mean curvature,” Duke Math. J., vol. 58, iss. 3, pp. 555-558, 1989. · Zbl 0677.53059
[36]	M. A. Grayson, ”Shortening embedded curves,” Ann. of Math., vol. 129, iss. 1, pp. 71-111, 1989. · Zbl 0686.53036
[37]	R. S. Hamilton, ”Isoperimetric estimates for the curve shrinking flow in the plane,” in Modern Methods in Complex Analysis, Princeton, NJ: Princeton Univ. Press, 1995, vol. 137, pp. 201-222. · Zbl 0846.51010
[38]	R. S. Hamilton, ”Harnack estimate for the mean curvature flow,” J. Differential Geom., vol. 41, iss. 1, pp. 215-226, 1995. · Zbl 0827.53006
[39]	G. Huisken, ”Flow by mean curvature of convex surfaces into spheres,” J. Differential Geom., vol. 20, iss. 1, pp. 237-266, 1984. · Zbl 0556.53001
[40]	G. Huisken, ”Local and global behaviour of hypersurfaces moving by mean curvature,” Proc. CMA, ANU, vol. 26, 1991. · Zbl 0791.58090
[41]	G. Huisken, ”Asymptotic behavior for singularities of the mean curvature flow,” J. Differential Geom., vol. 31, iss. 1, pp. 285-299, 1990. · Zbl 0694.53005
[42]	G. Huisken, ”Local and global behaviour of hypersurfaces moving by mean curvature,” in Differential Geometry: Partial Differential Equations on Manifolds, Providence, RI: Amer. Math. Soc., 1993, vol. 54, pp. 175-191. · Zbl 0791.58090
[43]	G. Huisken, ”A distance comparison principle for evolving curves,” Asian J. Math., vol. 2, iss. 1, pp. 127-133, 1998. · Zbl 0931.53032
[44]	G. Huisken and A. Polden, ”Geometric evolution equations for hypersurfaces,” in Calculus of Variations and Geometric Evolution Problems, New York: Springer-Verlag, 1999, vol. 1713, pp. 45-84. · Zbl 0942.35047
[45]	G. Huisken and C. Sinestrari, ”Convexity estimates for mean curvature flow and singularities of mean convex surfaces,” Acta Math., vol. 183, iss. 1, pp. 45-70, 1999. · Zbl 0992.53051
[46]	G. Huisken and C. Sinestrari, ”Mean curvature flow singularities for mean convex surfaces,” Calc. Var. Partial Differential Equations, vol. 8, iss. 1, pp. 1-14, 1999. · Zbl 0992.53052
[47]	G. Huisken and C. Sinestrari, ”Mean curvature flow with surgeries of two-convex hypersurfaces,” Invent. Math., vol. 175, iss. 1, pp. 137-221, 2009. · Zbl 1170.53042
[48]	T. Ilmanen, Singularities of Mean Curvature Flow of Surfaces, 1995. · Zbl 0759.53035
[49]	T. Ilmanen, ”Elliptic regularization and partial regularity for motion by mean curvature,” Mem. Amer. Math. Soc., vol. 108, iss. 520, p. x, 1994. · Zbl 0798.35066
[50]	T. Ilmanen, Lectures on Mean Curvature Flow and Related Equations (Trieste Notes), 1995.
[51]	B. H. Lawson Jr., ”Local rigidity theorems for minimal hypersurfaces,” Ann. of Math., vol. 89, pp. 187-197, 1969. · Zbl 0174.24901
[52]	C. Mantegazza and A. Magni, ”Some Remarks on Huisken’s Monotonicity Formula for Mean Curvature Flow,” in Singularities in Nonlinear Evolution Phenomena and Applications, Novaga, M. and Orlandi, G., Eds., , 2009, vol. 9, pp. 157-169. · Zbl 1178.53067
[53]	J. H. Michael and L. M. Simon, ”Sobolev and mean-value inequalities on generalized submanifolds of \(R^n\),” Comm. Pure Appl. Math., vol. 26, pp. 361-379, 1973. · Zbl 0256.53006
[54]	X. H. Nguyen, ”Construction of complete embedded self-similar surfaces under mean curvature flow. I,” Trans. Amer. Math. Soc., vol. 361, iss. 4, pp. 1683-1701, 2009. · Zbl 1166.53046
[55]	X. H. Nguyen, ”Construction of complete embedded self-similar surfaces under mean curvature flow. II,” Adv. Differential Equations, vol. 15, iss. 5-6, pp. 503-530, 2010. · Zbl 1200.53061
[56]	X. H. Nguyen, ”Translating tridents,” Comm. Partial Differential Equations, vol. 34, iss. 1-3, pp. 257-280, 2009. · Zbl 1187.53071
[57]	S. Osher and J. A. Sethian, ”Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations,” J. Comput. Phys., vol. 79, iss. 1, pp. 12-49, 1988. · Zbl 0659.65132
[58]	R. Schoen and L. Simon, ”Regularity of stable minimal hypersurfaces,” Comm. Pure Appl. Math., vol. 34, iss. 6, pp. 741-797, 1981. · Zbl 0497.49034
[59]	R. Schoen, L. Simon, and S. T. Yau, ”Curvature estimates for minimal hypersurfaces,” Acta Math., vol. 134, iss. 3-4, pp. 275-288, 1975. · Zbl 0323.53039
[60]	L. Simon, Lectures on geometric measure theory, Canberra: Australian National University Centre for Mathematical Analysis, 1983. · Zbl 0546.49019
[61]	K. Smoczyk, ”Self-shrinkers of the mean curvature flow in arbitrary codimension,” Int. Math. Res. Not., vol. 2005, iss. 48, pp. 2983-3004, 2005. · Zbl 1085.53059
[62]	A. Stone, ”A density function and the structure of singularities of the mean curvature flow,” Calc. Var. Partial Differential Equations, vol. 2, iss. 4, pp. 443-480, 1994. · Zbl 0833.35062
[63]	J. J. L. Velázquez, ”Curvature blow-up in perturbations of minimal cones evolving by mean curvature flow,” Ann. Scuola Norm. Sup. Pisa Cl. Sci., vol. 21, iss. 4, pp. 595-628, 1994. · Zbl 0926.35023
[64]	B. White, ”Evolution of curves and surfaces by mean curvature,” in Proceedings of the International Congress of Mathematicians, Vol. I, Beijing, 2002, pp. 525-538. · Zbl 1036.53045
[65]	B. White, ”The size of the singular set in mean curvature flow of mean-convex sets,” J. Amer. Math. Soc., vol. 13, iss. 3, pp. 665-695, 2000. · Zbl 0961.53039
[66]	B. White, ”The nature of singularities in mean curvature flow of mean-convex sets,” J. Amer. Math. Soc., vol. 16, iss. 1, pp. 123-138, 2003. · Zbl 1027.53078
[67]	B. White, ”Stratification of minimal surfaces, mean curvature flows, and harmonic maps,” J. Reine Angew. Math., vol. 488, pp. 1-35, 1997. · Zbl 0874.58007
[68]	B. White, ”A local regularity theorem for mean curvature flow,” Ann. of Math., vol. 161, iss. 3, pp. 1487-1519, 2005. · Zbl 1091.53045
[69]	N. Wickramasekera, A general regularity theorem for stable codimension \(1\) integral varifolds, 2009. · Zbl 1307.58005