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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 a, for some a<0. Then there exists a harmonic diffeomorphism M. This answers a question posed by R. Schoen. The authors proceed by showing the existence of entire minimal graphs in 𝕄×, where 𝕄 is a complete simply connected Riemannian surface of sectional curvature K 𝕄 a<0. The proof proceeds in a similiar way as in [P. Collin and 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 𝕄 (Theorem 3.1).

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