Duke Mathematical Journal

No mass drop for mean curvature flow of mean convex hypersurfaces

Jan Metzger and Felix Schulze

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A possible evolution of a compact hypersurface in Rn+1 by mean curvature past singularities is defined via the level set flow. In the case where the initial hypersurface has positive mean curvature, we show that the Brakke flow associated to the level set flow is actually a Brakke flow with equality. As a consequence, we obtain the fact that no mass drop can occur along such a flow. A further application of the techniques used above is to give a new variational formulation for mean curvature flow of mean convex hypersurfaces

Article information

Duke Math. J., Volume 142, Number 2 (2008), 283-312.

First available in Project Euclid: 27 March 2008

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Zentralblatt MATH identifier

Primary: 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.) 49Q20: Variational problems in a geometric measure-theoretic setting


Metzger, Jan; Schulze, Felix. No mass drop for mean curvature flow of mean convex hypersurfaces. Duke Math. J. 142 (2008), no. 2, 283--312. doi:10.1215/00127094-2008-007. https://projecteuclid.org/euclid.dmj/1206642156

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  • K. A. Brakke, The Motion of a Surface by Its Mean Curvature, Math. Notes 20, Princeton Univ. Press, Princeton, 1978.
  • Y. G. Chen, Y. Giga, and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geom. 33 (1991), 749--786.
  • K. Ecker and G. Huisken, Interior estimates for hypersurfaces moving by mean curvature, Invent. Math. 105 (1991), 547--569.
  • L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Stud. Adv. Math., CRC Press, Boca Raton, Fla., 1992.
  • L. C. Evans and J. Spruck, Motion of level sets by mean curvature, I, J. Differential Geom. 33 (1991), 635--681.
  • —, Motion of level sets by mean curvature, IV, J. Geom. Anal. 5 (1995), 77--114.
  • M. E. Gage and R. S. Hamilton, The heat equation shrinking convex plane curves, J. Differential Geom. 23 (1986), 69--96.
  • E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Monogr. Math. 80, Birkhäuser, Basel, 1984.
  • G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom. 20 (1984), 237--266.
  • G. Huisken and T. Ilmanen, The inverse mean curvature flow and the Riemannian Penrose inequality, J. Differential Geom. 59 (2001), 353--437.
  • —, Higher regularity of the inverse mean curvature flow, preprint, 2002.
  • J. E. Hutchinson, Second fundamental form for varifolds and the existence of surfaces minimising curvature, Indiana Univ. Math. J. 35 (1986), 45--71.
  • T. Ilmanen, Elliptic regularization and partial regularity for motion by mean curvature, Mem. Amer. Math. Soc. 108 (1994), no. 520.
  • F. Schulze, Nonlinear evolution by mean curvature and isoperimetric inequalities, preprint,\arxivmath/0606675v1[math.DG]
  • B. White, The size of the singular set in mean curvature flow of mean-convex sets, J. Amer. Math. Soc. 13 (2000), 665--695.