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Singularities of the area preserving curve shortening flow with a free boundary condition

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Abstract

We consider the area preserving curve shortening flow with Neumann free boundary conditions outside of a convex domain or at a straight line. We give a criterion on initial curves that guarantees the appearance of a singularity in finite time. We prove that the singularity is of type II. Furthermore, if these initial curves are convex, then an appropriate rescaling at the finite maximal time of existence yields a grim reaper or half a grim reaper as limit flow. We construct examples of initial curves satisfying the mentioned criterion.

Notes

Acknowledgements

The author would like to thank Jonas Hirsch for very useful discussions. Furthermore, the author would like to express her gratitude to the referee for all the useful comments and suggestions. The author is funded by the Deutsche Forschungsgemeinschaft (DFG), LA 3444/1-1 and MA 7559/1-1.

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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Karlsruhe Institute of Technology, Institute for AnalysisKalrsruheGermany

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