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The Journal of Geometric Analysis

, Volume 28, Issue 4, pp 3928–3949 | Cite as

Evolution of Area-Decreasing Maps Between Two-Dimensional Euclidean Spaces

  • Felix Lubbe
Article

Abstract

We consider the mean curvature flow of the graph of a smooth map \(f:{\mathbb {R}}^2\rightarrow {\mathbb {R}}^2\) between two-dimensional Euclidean spaces. If f satisfies an area-decreasing property, the solution exists for all times and the evolving submanifold stays the graph of an area-decreasing map \(f_t\). Further, we prove uniform decay estimates for the mean curvature vector of the graph and all higher-order derivatives of the corresponding map \(f_t\).

Keywords

Mean curvature flow Area-decreasing maps Euclidean space 

Mathematics Subject Classification

Primary 53C44 53C42 53A07 

Notes

Acknowledgements

The author would like to thank Andreas Savas-Halilaj for stimulating discussions.

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Copyright information

© Mathematica Josephina, Inc. 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of HamburgHamburgGermany

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