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Minimal surfaces and harmonic diffeomorphisms from the complex plane onto certain Hadamard surfaces

From: American Journal of Mathematics
Volume 132, Number 5, October 2010
pp. 1249-1273 | 10.1353/ajm.2010.0001



We construct harmonic diffeomorphisms from the complex plane ${\bf C}$ onto any Hadamard surface $\mathbb{M}$ whose curvature is bounded above by a negative constant. For that, we prove a Jenkins-Serrin type theorem for minimal graphs in $\mathbb{M} \times \mathbb{R}$ over domains of $\mathbb{M}$ bounded by ideal geodesic polygons and show the existence of a sequence of minimal graphs over polygonal domains converging to an entire minimal graph in $\mathbb{M} \times \mathbb{R}$ with the conformal structure of ${\bf C}$.