A Computed example of nonuniqueness of mean curvature flow in R³

References

• S. Altschuler, S. Angenent, and Y. Giga, "Mean Curvature Flow Through Singularities for Surfaces of Rotation," J. Geom. Analysis, to appear.
   Google Scholar
• L. Ambrosio, "Minimizing Movements," lecture notes for a summer school course in Padova, Italy, June 20-24, 1994.
   Google Scholar
• Angenent , S. 1989 . Shrinking Donuts . Proc. of the Conf. on Elliptic and Parabolic Equations held at Gregynog . 08 1989 , Wales.
   Google Scholar
• S. Angenent, T. Ilmanen, and J.J.L. Velazquez, "Nonuniqueness in Geometric Heat Flows," in preparation.
   Google Scholar
• Barles , G. , Soner , H.M. and Sounganidis , P.E. 1993 . Front Propagation and Phase Field Theory . SIAM J. Cont. Optim. , 31 08 : 439 – 469 .
   Web of Science ®Google Scholar
• Bellettini , G. and Paolini , M. 1994 . Two Examples of Fattening for the Mean Curvature Flow with a Driving Force . Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. (9) Mat. Appl. , 5 08 : 229 – 236 .
   Google Scholar
• Brakke , K. 1978 . The Motion of a Surface by its Mean Curvature , Princeton Univ. Press .
   Google Scholar
• Chen , Y.G. , Giga , Y. and Goto , S. 1991 . Uniqueness and Existence of Viscosity Solutions of Generalized Mean Curvature Flow Equations . J. Diff. Geom. , 33 : 749 – 786 .
   Web of Science ®Google Scholar
• Chopp , D.L. 1994 . Numerical Computation of Self-Similar Solutions for Mean Curvature Flow . Exper. Math. , 3 : 1 – 16 .
   Google Scholar
• E. De Giorgi, "Some Conjectures on Flow by Mean Curvature," white paper, 1990.
   Google Scholar
• De Giorgi , E. 1992 . “ New Problems on Minimizing Movements ” . In Boundary Value Problems for PDE and Applications , Edited by: Baiocchi , C. and Lions , J.L. 81 – 88 . Masson .
   Google Scholar
• Ecker , K. and Huisken , G. 1991 . Interior Estimates for Hypersurfaces moving by Mean Curvature . Invent. Math. , 105 : 635 – 681 .
   Web of Science ®Google Scholar
• Ecker , K. and Huisken , G. 1989 . Mean Curvature Evolution of Entire Graphs . Ann. of Math. , 130 : 453 – 471 .
   Web of Science ®Google Scholar
• Evans , L.C. and Spruck , J. 1991 . Motion of Level Sets by Mean Curvature I . J. Diff. Geom. , 33 : 635 – 681 .
   Web of Science ®Google Scholar
• Evans , L.C. and Spruck , J. 1992 . Motion of Level Sets by Mean Curvature III . J. Geom. Anal. , 2 : 121 – 150 .
   Google Scholar
• Huisken , G. 1990 . Asymptotic Behavior for Singularities of the Mean Curvature Flow . J. Diff. Geom. , 31 : 285 – 299 .
   Web of Science ®Google Scholar
• Ilmanen , T. 1992 . Generalized Flow of Sets by Mean Curvature on a Manifold . Indiana J. Math. , 41 : 671 – 705 .
   Web of Science ®Google Scholar
• Ilmanen , T. 1993 . The Level-Set Flow on a Manifold . Proc. of Symp. in Pure Math. , 54 : 193 – 204 . Part 1
   Google Scholar
• Ilmanen , T. 1993 . Elliptic Regularization and Partial Regularity for Motion by Mean Curvature . Mem. Amer. Math. Soc. , 520.
   Google Scholar
• T. Illmanen, "Dynamics of Stationary Cones," in preparation.
   Google Scholar
• T.Ilmanen, unpublished notes.
   Google Scholar
• T. Ilmanen, "Singularities of Mean Curvature Flow of Surfaces," preprint.
   Google Scholar
• Osher , S. and Sethian , J.A. 1988 . Fronts Progagating with Curvature-Dependent Speed: Algorithms Based on Hamilton-Jacobi Formulations . J. Comp. Phys. , 79 : 12 – 49 .
   Web of Science ®Google Scholar
• Simon , L. 1983 . Lectures on Geometric Measure Theory . Proc. Centre Math. Analysis , 3 Austr. Nat. Univ.
   Google Scholar
• Soner , H.M. and Souganidis , P.E. 1993 . Singularities and Uniqueness of Cyclindrically Symmetric Surfaces Moving by Mean Curvature . Comm. Partial Diff. Eq. , 18 : 859 – 894 .
   Web of Science ®Google Scholar