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Annals of Global Analysis and Geometry

, Volume 53, Issue 1, pp 11–37 | Cite as

Mean curvature flow of area decreasing maps between Riemann surfaces

  • Andreas Savas-HalilajEmail author
  • Knut Smoczyk
Article

Abstract

In this article, we give a complete description of the evolution of an area decreasing map \(f{: }M\rightarrow N\), induced by the mean curvature of their graph, in the situation where M and N are complete Riemann surfaces with bounded geometry, M being compact, for which their sectional curvatures \(\sigma _M\) and \(\sigma _N\) satisfy \(\min \sigma _M\ge \sup \sigma _N\).

Keywords

Mean curvature flow Area decreasing maps Graphical surfaces Riemann surfaces 

Mathematics Subject Classification

53C44 53C42 57R52 35K55 

Notes

Acknowledgements

This work was initiated during the research visit of both authors at the Max-Planck-Institut für Mathematik in den Naturwissenschaften Leipzig in August 2014. The authors would like to express their gratitude to Jürgen Jost and the Institute for the excellent research conditions and the hospitality.

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Authors and Affiliations

  1. 1.Institut für DifferentialgeometrieLeibniz Universität HannoverHannoverGermany
  2. 2.Institut für Differentialgeometrie, Riemann Center for Geometry and PhysicsLeibniz Universität HannoverHannoverGermany

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