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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 $$\min \sigma _M\ge \sup \sigma _N$$ . [ABSTRACT FROM AUTHOR]