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Type I Singularities in the Curve Shortening Flow Associated to a Density


We define Type I singularities for the mean curvature flow associated to a density \(\psi \) (\(\psi \)MCF ) and describe the blow-up at any singular time of these singularities. Special attention is paid to the case where the singularity comes from the part of the \(\psi \)-curvature due to the density. We describe a family of curves whose evolution under \(\psi \)MCF (in a Riemannian surface of non-negative curvature with a density that is singular at a geodesic of the surface) produces only Type I singularities and study the limits of their rescalings.

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  1. Altschuler, S., Angenent, S.B., Giga, Yoshikazu: Mean curvature flow through singularities for surfaces of rotation. J. Geom. Anal. 5(3), 293–358 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  2. Angenent, Sigur: The zero set of a solution of a parabolic equation. J. Reine Angew. Math. 390, 79–96 (1988)

    MathSciNet  MATH  Google Scholar 

  3. Angenent, S.: Shrinking Doughnuts. In: Lloyd, N.G., Ni, W.M., Peletier, L.A., Serrin, J. (eds.) Nonlinear Diffusion Equations and Their Equilibrium States, 3. Progress in Nonlinear Differential Equations and Their Applications, vol. 7, pp. 21–38. Birkhäuser, Boston, MA (1992)

  4. Bayle, V.: Propriétés de concavité du profil isopérimétrique et applications. Graduate Thesis, Institut Fourier, Université Joseph-Fourier - Grenoble I, (2004)

  5. Cabezas-Rivas, E., Miquel, V.: Volume-preserving mean curvature flow of revolution hypersurfaces in a rotationally symmetric space. Math. Z. 261(3), 489–510 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cabezas-Rivas, E., Miquel, V.: Volume preserving mean curvature flow of revolution hypersurfaces between two equidistants. Calc. Var. PDE 43, 185–210 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chow, B., Chu, S.C., Glickenstein, D., Guenter, C., Isenberg, J., Ivey, T., Knopf, D., Lu, P., Luo, F., Ni, L.: The Ricci Flow: Techniques and Applications. Part II: Analytic Aspects, A. M. S., SURV 144, Providence (2008)

  8. Chen, B.Y., Vanhecke, L.: Differential geometry of geodesic spheres. J. Reine Angew. Math. 325, 28–67 (1982)

    MathSciNet  MATH  Google Scholar 

  9. Gromov, M.: Isoperimetry of waists and concentration of maps. Geom. Funct. Anal. 13, 178–215 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. Huisken, G.: Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature. Invent. Math. 84, 463–480 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  11. Huisken, G.: Asymptotic behavior for singularities of the mean curvature flow. J. Differ. Geom. 31, 285–299 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  12. Mantegazza, C.: Lecture Notes on Mean Curvature Flow Progress in Mathematics, 290. Birkhäuser/Springer Basel AG, Basel (2011)

    Book  MATH  Google Scholar 

  13. Miquel, V., Viñado-Lereu, F.: The curve shortening problem associated to a density. Calc. Variat. PDE 55, 61 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  14. Morgan, F.: Manifolds with density. Notices Am. Math. Soc. 52, 853–858 (2005)

    MathSciNet  MATH  Google Scholar 

  15. O’Neill, B.: Semi-Riemannian Geometry. Academic Press, Cambridge (1983)

    MATH  Google Scholar 

  16. Rosales, C., Cañete, A., Bayle, V., Morgan, F.: On the isoperimetric problem in Euclidean space with density. Calc. Var. Differ. Equ. 31, 27–46 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. Smoczyk, K.: Symmetric hypersurfaces in Riemannian manifolds contracting to Lie-groups by their mean curvature. Calc. Var. Partial Differ. Equ. 4, 155–170 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  18. Smoczyk, K.: A relation between Mean curvature flow solitons and minimal submanifolds. Math. Nachr. 229, 175–186 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  19. Zhu, X.P.: Lectures on Mean Curvature Flows. AMS/IP, Providence (2002)

    Book  MATH  Google Scholar 


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Research partially supported by the MINECO (Spain) and FEDER project MTM2016-77093-P, and the Generalitat Valenciana Project PROMETEOII/2014/064. The second author has been partially supported by a Grant of the Programa Nacional de Formación de Personal Investigador 2011 Subprograma FPI-MICINN ref: BES-2011-045388 and partially by a contract CPI-15-209 associated with Project PROMETEOII/2014/064.

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Correspondence to Francisco Viñado-Lereu.

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Miquel, V., Viñado-Lereu, F. Type I Singularities in the Curve Shortening Flow Associated to a Density. J Geom Anal 28, 2361–2394 (2018).

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  • Mean curvature flow
  • Manifolds with density
  • Type I blow-up

Mathematics Subject Classification

  • 53C44
  • 35R01