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

The Journal of Geometric Analysis volume 28, pages 2361–2394 (2018)Cite this article

Abstract

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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Acknowledgements

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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Authors and Affiliations

  1. Department of Mathematics, University of Valencia, 46100, Burjassot, Valencia, Spain

    Vicente Miquel & Francisco Viñado-Lereu

Authors
  1. Vicente Miquel
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  2. Francisco Viñado-Lereu
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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). https://doi.org/10.1007/s12220-017-9907-z

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  • DOI: https://doi.org/10.1007/s12220-017-9907-z

Keywords

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

Mathematics Subject Classification

  • 53C44
  • 35R01