Neckpinch singularities in fractional mean curvature flows

In this paper we consider the evolution of boundaries of sets by a fractional mean curvature flow. We show that for any dimension n >= 2, there exist embedded hypersurfaces in Rn which develop a singularity without shrinking to a point. Such examples are well known for the classical mean curvature flow for n >= 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.

[1]  A. Figalli,et al.  Bootstrap regularity for integro-differential operators and its application to nonlocal minimal surfaces , 2012, 1202.4606.

[2]  M. Grayson.  The heat equation shrinks embedded plane curves to round points , 1987 .

[3]  Yun-Gang Chen,et al.  Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations , 1989 .

[4]  O. Savin,et al.  Nonlocal minimal surfaces , 2009, 0905.1183.

[5]  Cyril Imbert,et al.  Level set approach for fractional mean curvature flows , 2008, 0807.2627.

[6]  G. Huisken.  Flow by mean curvature of convex surfaces into spheres , 1984 .

[7]  P. Souganidis,et al.  Convergence of Nonlocal Threshold Dynamics Approximations to Front Propagation , 2008, 0805.2618.

[8]  Klaus Ecker,et al.  Regularity Theory for Mean Curvature Flow , 2003 .

[9]  Antonin Chambolle,et al.  A Nonlocal Mean Curvature Flow and Its Semi-implicit Time-Discrete Approximation , 2012, SIAM J. Math. Anal..

[10]  Enrico Valdinoci,et al.  On the evolution by fractional mean curvature , 2015, Communications in Analysis and Geometry.

[11]  L. Evans,et al.  Motion of level sets by mean curvature IV , 1995 .

[12]  ON THE VARIATION OF THE FRACTIONAL MEAN CURVATURE UNDER THE EFFECT OF C 1 , α PERTURBATIONS , 2015 .

[13]  On the variation of the fractional mean curvature under the effect of $C^{1, \alpha}$ perturbations , 2015 .

[14]  G. Huisken,et al.  Convexity estimates for mean curvature flow and singularities of mean convex surfaces , 1999 .

[15]  Antonin Chambolle,et al.  Nonlocal Curvature Flows , 2014, Archive for Rational Mechanics and Analysis.

[16]  Gerhard Huisken,et al.  Mean curvature flow singularities for mean convex surfaces , 1999 .

[17]  L. Caffarelli,et al.  Uniform estimates and limiting arguments for nonlocal minimal surfaces , 2011 .

[18]  M. Fall,et al.  Curves and surfaces with constant nonlocal mean curvature: Meeting Alexandrov and Delaunay , 2015, Journal für die reine und angewandte Mathematik (Crelles Journal).

[19]  L. Ambrosio,et al.  Gamma-convergence of nonlocal perimeter functionals , 2010, 1007.3770.

[20]  E. Valdinoci,et al.  Regularity of nonlocal minimal cones in dimension 2 , 2012, 1202.0973.

[21]  Antonin Chambolle,et al.  Convergence of an Algorithm for the Anisotropic and Crystalline Mean Curvature Flow , 2006, SIAM J. Math. Anal..

[22]  Gerhard Huisken,et al.  Mean curvature flow with surgeries of two–convex hypersurfaces , 2009 .

[23]  L. Caffarelli,et al.  Regularity properties of nonlocal minimal surfaces via limiting arguments , 2011, 1105.1158.

[24]  Matthew A. Grayson,et al.  A short note on the evolution of a surface by its mean curvature , 1989 .

[25]  E. Valdinoci,et al.  Quantitative flatness results and $BV$-estimates for stable nonlocal minimal surfaces , 2016, Journal of Differential Geometry.

[26]  Carlo Mantegazza,et al.  Lecture Notes on Mean Curvature Flow , 2011 .

[27]  Gerhard Huisken,et al.  A distance comparison principle for evolving curves , 1998 .

[28]  M. Gage,et al.  The heat equation shrinking convex plane curves , 1986 .

[29]  Antonin Chambolle,et al.  Some results on anisotropic fractional mean curvature flows , 2016, 1603.07239.