Uniqueness of blowups and Łojasiewicz inequalities. (English) Zbl 1337.53082

In this paper, the authors study the singularities of a mean curvature flow and they prove that at each generic singularity the blow-up is unique, i.e., it does not depend on the sequence of rescalings. This is the first general uniqueness theorem of blowups for a geometric PDE at a noncompact singularity and implies the regularity of the singular set of the flow. The proof of the uniqueness result is based on two completely new infinite-dimensional Łojasiewicz-type inequalities. In real algebraic geometry, Łojasiewicz proved in the sixties some inequalities for real analytic functions defined on an open set \(U\) of \({\mathbb R}^n\). Infinite-dimensional versions of these inequalities were proven by Leon Simon for the area and related functionals and applied to the study of tangent cones with smooth cross section of minimal surfaces. The proofs of Simon are based on a reduction of the infinite-dimensional case to the classical Łojasiewicz inequality by a Ljapunov-Schmidt reduction argument.
In the paper under review, the infinite-dimensional Łojasiewicz inequalities are proved for the functional \(F\), defined on the space of hypersurfaces, obtained by integrating the Gaussian over a hypersurface \(\Sigma \subset {\mathbb R}^{n+1}\), namely \[ F(\Sigma) = (4\pi)^{-n/2}\int_{\Sigma}\mathrm{e}^{-{{|x|^2}\over 4}}\mathrm{d}\mu. \] The inequalities are directly proved on the functional \(F\), without any finite-dimensional reduction, and play a basic role in the proof of the uniqueness of blowups.


53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
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[1] W. K. Allard and F. J. Almgren Jr., ”On the radial behavior of minimal surfaces and the uniqueness of their tangent cones,” Ann. of Math., vol. 113, iss. 2, pp. 215-265, 1981. · Zbl 0437.53045
[2] 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
[3] B. Andrews, ”Noncollapsing in mean-convex mean curvature flow,” Geom. Topol., vol. 16, iss. 3, pp. 1413-1418, 2012. · Zbl 1250.53063
[4] P. A. Beck, Metal Interfaces, Cleveland, OH: Amer. Soc. for Testing Materials, 1952.
[5] K. A. Brakke, The Motion of a Surface by its Mean Curvature, Princeton, N.J.: Princeton Univ. Press, 1978, vol. 20. · Zbl 0386.53047
[6] D. Bakry and M. Émery, ”Diffusions hypercontractives,” in Séminaire de Probabilités, XIX, 1983/84, New York: Springer-Verlag, 1985, vol. 1123, pp. 177-206. · Zbl 0561.60080
[7] H. Brezis, J. Coron, and E. H. Lieb, ”Harmonic maps with defects,” Comm. Math. Phys., vol. 107, iss. 4, pp. 649-705, 1986. · Zbl 0608.58016
[8] J. Burke, ”Some factors affecting the rate of grain growth in metals,” AIME Trans., vol. 180, pp. 73-91, 1949.
[9] T. H. Colding, T. Ilmanen, and W. P. Minicozzi II, Rigidity of generic singularities of mean curvature flow, 2013. · Zbl 1331.53098
[10] T. H. Colding, T. Ilmanen, W. P. Minicozzi II, and B. White, ”The round sphere minimizes entropy among closed self-shrinkers,” J. Differential Geom., vol. 95, iss. 1, pp. 53-69, 2013. · Zbl 1278.53069
[11] T. H. Colding and W. P. Minicozzi II, ”Generic mean curvature flow I: generic singularities,” Ann. of Math., vol. 175, iss. 2, pp. 755-833, 2012. · Zbl 1239.53084
[12] T. H. Colding and W. P. Minicozzi II, A Course in Minimal Surfaces, Providence, RI: Amer. Math. Soc., 2011, vol. 121. · Zbl 1242.53007
[13] T. H. Colding and W. P. Minicozzi II, ”On uniqueness of tangent cones for Einstein manifolds,” Invent. Math., vol. 196, iss. 3, pp. 515-588, 2014. · Zbl 1302.53048
[14] T. H. Colding and W. P. Minicozzi II, The singular set of mean curvature flow with generic singularities, 2014. · Zbl 1341.53098
[15] T. H. Colding and W. P. Minicozzi II, Łojasiewicz inequalities and applications, 2014. · Zbl 1329.53092
[16] T. H. Colding and W. P. Minicozzi II, Differentiability of the arrival time, 2015. · Zbl 1353.53068
[17] T. H. Colding, W. P. Minicozzi II, and E. K. Pedersen, ”Mean curvature flow,” Bull. Amer. Math. Soc., vol. 52, iss. 2, pp. 297-333, 2015. · Zbl 1442.53064
[18] K. Ecker and G. Huisken, ”Interior estimates for hypersurfaces moving by mean curvature,” Invent. Math., vol. 105, iss. 3, pp. 547-569, 1991. · Zbl 0707.53008
[19] H. Federer and W. H. Fleming, ”Normal and integral currents,” Ann. of Math., vol. 72, pp. 458-520, 1960. · Zbl 0187.31301
[20] Z. Gang and D. Knopf, Universality in mean curvature flow neckpinches, 2013. · Zbl 1329.53093
[21] Z. Gang, D. Knopf, and I. M. Sigal, Neckpinch dynamics for asymmetric surfaces evolving by mean curvature flow, 2011.
[22] Z. Gang and I. M. Sigal, ”Neck pinching dynamics under mean curvature flow,” J. Geom. Anal., vol. 19, iss. 1, pp. 36-80, 2009. · Zbl 1179.53065
[23] M. Giga, Y. Giga, and J. Saal, Nonlinear Partial Differential Equations. Asymptotic Behavior of Solutions and Self-Similar Solutions, Boston: Birkhäuser, 2010, vol. 79. · Zbl 1215.35001
[24] A. Grigor’yan, Heat Kernel and Analysis on Manifolds, Providence, RI: Amer. Math. Soc., 2009, vol. 47. · Zbl 1206.58008
[25] R. M. Hardt, ”Singularities of harmonic maps,” Bull. Amer. Math. Soc., vol. 34, iss. 1, pp. 15-34, 1997. · Zbl 0673.58016
[26] R. M. Hardt and F. Lin, ”Stability of singularities of minimizing harmonic maps,” J. Differential Geom., vol. 29, iss. 1, pp. 113-123, 1989. · Zbl 0673.58016
[27] D. Harker and E. Parker, ”Grain shape and grain growth,” Trans. Amer. Soc. Met., vol. 34, pp. 156-201, 1945.
[28] R. Haslhofer and B. Kleiner, Mean curvature flow of mean convex hypersurfaces, 2013. · Zbl 1360.53069
[29] 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
[30] 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
[31] 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
[32] 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
[33] 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
[34] 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
[35] T. Ilmanen, Singularities of Mean Curvature Flow of Surfaces, 1995. · Zbl 0759.53035
[36] O. D. Kellogg, ”On bounded polynomials in several variables,” Math. Z., vol. 27, iss. 1, pp. 55-64, 1928. · JFM 53.0082.03
[37] S. Łojasiewicz, Ensembles semi-analytiques, 1965.
[38] W. W. Mullins, ”Two-dimensional motion of idealized grain boundaries,” J. Appl. Phys., vol. 27, pp. 900-904, 1956. · Zbl 0112.23801
[39] J. von Neumann, ,” in Metal Interfaces, Herring, C., Ed., Cleveland, OH: Amer. Soc. for Metals, 1952, pp. 108-110.
[40] F. Schulze, ”Uniqueness of compact tangent flows in mean curvature flow,” J. Reine Angew. Math., vol. 690, pp. 163-172, 2014. · Zbl 1290.53066
[41] N. Sesum, ”Rate of convergence of the mean curvature flow,” Comm. Pure Appl. Math., vol. 61, iss. 4, pp. 464-485, 2008. · Zbl 1143.53066
[42] L. Simon, ”Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems,” Ann. of Math., vol. 118, iss. 3, pp. 525-571, 1983. · Zbl 0549.35071
[43] L. Simon, ”A general asymptotic decay lemma for elliptic problems,” in Handbook of Geometric Analysis. No. 1, Somerville, MA: Int. Press, 2008, vol. 7, pp. 381-411. · Zbl 1159.53004
[44] L. Simon, ”Rectifiability of the singular set of energy minimizing maps,” Calc. Var. Partial Differential Equations, vol. 3, iss. 1, pp. 1-65, 1995. · Zbl 0818.49023
[45] L. Simon, ,” in Theorems on Regularity and Singularity of Energy Minimizing Maps, Basel: Birkhäuser Verlag, 1996, p. viii. · Zbl 0864.58015
[46] L. Simon, ”Rectifiability of the singular sets of multiplicity \(1\) minimal surfaces and energy minimizing maps,” in Surveys in Differential Geometry, Vol. II, Cambridge, MA: Int. Press, 1995, pp. 246-305. · Zbl 0874.49033
[47] H. M. Soner and P. E. Souganidis, ”Singularities and uniqueness of cylindrically symmetric surfaces moving by mean curvature,” Comm. Partial Differential Equations, vol. 18, iss. 5-6, pp. 859-894, 1993. · Zbl 0804.53006
[48] T. Sutoki, ”On the mechanism of crystal growth by annealing,” Scientific Reports of Tohoku, Imperial University, vol. 17, pp. 857-876, 1928.
[49] 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
[50] 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
[51] B. White, ”A local regularity theorem for mean curvature flow,” Ann. of Math., vol. 161, iss. 3, pp. 1487-1519, 2005. · Zbl 1091.53045
[52] S. Brendle, An inscribed radius estimate for mean curvature flow in Riemannian manifolds. · Zbl 1365.53059
[53] J. E. Taylor, ”Regularity of the singular sets of two-dimensional area-minimizing flat chains modulo \(3\) in \(R^3\),” Invent. Math., vol. 22, pp. 119-159, 1973. · Zbl 0278.49046
[54] B. White, ”The mathematics of F. J. Almgren, Jr.,” J. Geom. Anal., vol. 8, iss. 5, pp. 681-702, 1998. · Zbl 0955.01020
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