Neckpinch singularities in fractional mean curvature flows

Cinti, Eleonora; Sinestrari, Carlo; Valdinoci, Enrico

doi:10.1090/proc/14002

Neckpinch singularities in fractional mean curvature flows
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by Eleonora Cinti, Carlo Sinestrari and Enrico Valdinoci PDF

Proc. Amer. Math. Soc. 146 (2018), 2637-2646 Request permission

Abstract:

In this paper we consider the evolution of boundaries of sets by a fractional mean curvature flow. We show that for any dimension $n\geqslant 2$, there exist embedded hypersurfaces in $\mathbb {R}^n$ which develop a singularity without shrinking to a point. Such examples are well known for the classical mean curvature flow for $n \geqslant 3$. Interestingly, when $n=2$, our result provides instead a counterexample in the nonlocal framework to the well-known Grayson’s Theorem, which states that any smooth embedded curve in the plane evolving by (classical) MCF shrinks to a point. The essential step in our construction is an estimate which ensures that a suitably small perturbation of a thin strip has positive fractional curvature at every boundary point.

References

Roberta Alessandroni and Carlo Sinestrari, Evolution of hypersurfaces by powers of the scalar curvature, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 9 (2010), no. 3, 541–571. MR 2722655
Sigurd B. Angenent, Shrinking doughnuts, Nonlinear diffusion equations and their equilibrium states, 3 (Gregynog, 1989) Progr. Nonlinear Differential Equations Appl., vol. 7, Birkhäuser Boston, Boston, MA, 1992, pp. 21–38. MR 1167827
Begoña Barrios, Alessio Figalli, and Enrico Valdinoci, Bootstrap regularity for integro-differential operators and its application to nonlocal minimal surfaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 13 (2014), no. 3, 609–639. MR 3331523
X. Cabré, M. M. Fall, J. Solá-Morales and T. Weth, Curves and surfaces with constant nonlocal mean curvature: meeting Alexandrov and Delaunay, to appear in J. Reine Angew. Math.
L. Caffarelli, J.-M. Roquejoffre, and O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math. 63 (2010), no. 9, 1111–1144. MR 2675483, DOI 10.1002/cpa.20331
Luis A. Caffarelli and Panagiotis E. Souganidis, Convergence of nonlocal threshold dynamics approximations to front propagation, Arch. Ration. Mech. Anal. 195 (2010), no. 1, 1–23. MR 2564467, DOI 10.1007/s00205-008-0181-x
Luis Caffarelli and Enrico Valdinoci, Regularity properties of nonlocal minimal surfaces via limiting arguments, Adv. Math. 248 (2013), 843–871. MR 3107529, DOI 10.1016/j.aim.2013.08.007
Antonin Chambolle, Massimiliano Morini, and Marcello Ponsiglione, A nonlocal mean curvature flow and its semi-implicit time-discrete approximation, SIAM J. Math. Anal. 44 (2012), no. 6, 4048–4077. MR 3023439, DOI 10.1137/120863587
Antonin Chambolle, Massimiliano Morini, and Marcello Ponsiglione, Nonlocal curvature flows, Arch. Ration. Mech. Anal. 218 (2015), no. 3, 1263–1329. MR 3401008, DOI 10.1007/s00205-015-0880-z
Antonin Chambolle, Matteo Novaga, and Berardo Ruffini, Some results on anisotropic fractional mean curvature flows, Interfaces Free Bound. 19 (2017), no. 3, 393–415. MR 3713894, DOI 10.4171/IFB/387
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
E. Cinti, J. Serra, and E. Valdinoci, Quantitative flatness results and $BV$-estimates for stable nonlocal minimal surfaces, to appear in J. Differential Geom.
Matteo Cozzi, On the variation of the fractional mean curvature under the effect of $C^{1,\alpha }$ perturbations, Discrete Contin. Dyn. Syst. 35 (2015), no. 12, 5769–5786. MR 3393254, DOI 10.3934/dcds.2015.35.5769
Klaus Ecker, Regularity theory for mean curvature flow, Progress in Nonlinear Differential Equations and their Applications, vol. 57, Birkhäuser Boston, Inc., Boston, MA, 2004. MR 2024995, DOI 10.1007/978-0-8176-8210-1
L. C. Evans and J. Spruck, Motion of level sets by mean curvature. I, J. Differential Geom. 33 (1991), no. 3, 635–681. MR 1100206
M. Gage and R. S. Hamilton, The heat equation shrinking convex plane curves, J. Differential Geom. 23 (1986), no. 1, 69–96. MR 840401
Matthew A. Grayson, A short note on the evolution of a surface by its mean curvature, Duke Math. J. 58 (1989), no. 3, 555–558. MR 1016434, DOI 10.1215/S0012-7094-89-05825-0
Matthew A. Grayson, The heat equation shrinks embedded plane curves to round points, J. Differential Geom. 26 (1987), no. 2, 285–314. MR 906392
Gerhard Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom. 20 (1984), no. 1, 237–266. MR 772132
Gerhard Huisken, A distance comparison principle for evolving curves, Asian J. Math. 2 (1998), no. 1, 127–133. MR 1656553, DOI 10.4310/AJM.1998.v2.n1.a2
Cyril Imbert, Level set approach for fractional mean curvature flows, Interfaces Free Bound. 11 (2009), no. 1, 153–176. MR 2487027, DOI 10.4171/IFB/207
Carlo Mantegazza, Lecture notes on mean curvature flow, Progress in Mathematics, vol. 290, Birkhäuser/Springer Basel AG, Basel, 2011. MR 2815949, DOI 10.1007/978-3-0348-0145-4
M. Sáez and E. Valdinoci, On the evolution by fractional mean curvature, to appear in Comm. Anal. Geom. (Available at http://arxiv.org/pdf/1511.06944.pdf.)
Ovidiu Savin and Enrico Valdinoci, Regularity of nonlocal minimal cones in dimension 2, Calc. Var. Partial Differential Equations 48 (2013), no. 1-2, 33–39. MR 3090533, DOI 10.1007/s00526-012-0539-7