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Minimal surfaces and harmonic diffeomorphisms from the complex plane onto certain Hadamard surfaces. (English) Zbl 1229.53064
The authors prove the following result: Suppose that $M$ is a Hadamard surface whose sectional curvature satisfies the inequality $K_M\le a$, for some $a< 0$. Then there exists a harmonic diffeomorphism $\bbfC\to M$. This answers a question posed by R. Schoen. The authors proceed by showing the existence of entire minimal graphs in $\bbfM\times\bbfR$, where $\bbfM$ is a complete simply connected Riemannian surface of sectional curvature $K_{\bbfM}\le a< 0$. The proof proceeds in a similiar way as in [{\it P. Collin} and {\it H. Rosenberg}, Ann. Math. (2) 172, No. 3, 1879--1906 (2010; Zbl 1209.53010)]. The article contains other interesting results, in particular on the solution of the Dirichlet problem for minimal surfaces given by ideal polygons in $\bbfM$ (Theorem 3.1).

53C42Immersions (differential geometry)
53C43Differential geometric aspects of harmonic maps
58E20Harmonic maps between infinite-dimensional spaces
53A10Minimal surfaces, surfaces with prescribed mean curvature
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