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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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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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
- Mean curvature flow
- Manifolds with density
- Type I blow-up
Mathematics Subject Classification