Construction of harmonic diffeomorphisms and minimal graphs

By P. Collin, H. Rosenberg. 2007

134 citations recorded by the source corpus.

We study complete minimal graphs in ℍ x ℝ, which take asymptotic boundary values plus and minus infinity on alternating sides of an ideal inscribed polygon Γ in ℍ. We give necessary and sufficient conditions on the "lengths" of the sides of the polygon (and all inscribed polygons in Γ) that ensure the existence of such a graph. We then apply this to construct entire minimal graphs in ℍ x ℝ that are conformally the complex plane C. The vertical projection of such a graph yields a harmonic diffeomorphism from C onto ℍ, disproving a conjecture of Rick Schoen and S.-T. Yau.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f3f7cebeee178c4f26a5e8afb5e7607a0bc5b934

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Holomorphic quadratic differentials and the Bernstein problem in Heisenberg space (2007, background): “Remark 18 The existence of harmonic diffeomorphisms from C onto H(2) has been recently proved by Collin and Rosenberg in [CoRo].”
PARABOLIC AND HYPERBOLIC SCREW MOTION SURFACES IN ℍ2×ℝ (2008, methodology): “This is a result of Nelli and Rosenberg [15].”
Minimal surfaces and harmonic diffeomorphisms from the complex plane onto certain Hadamard surfaces (2010, methodology): “In addition, as it was explained in [ CR ], the previous maximum principle also applies when the solutions u,v to the minimal surface equation have infinite boundary value s along some geodesic arcs of @D. When@D is made of complete geodesics andu, v take the same infinite value on the whole boundar”
Complete constant mean curvature surfaces and Bernstein type theorems in $M^2 x R$ (2008, background): “There are many (interesting) complete minimal and constant mean curvature graphs in M 2 × R (e.g., in H 2 × R, [CoR]).”
The Dirichlet problem for the minimal surface equation, with possible infinite boundary data, over domains in a Riemannian surface (2008, methodology): “P. Collin and H. Rosenberg, ‘Construction of harmonic diffeomorphisms and minimal graphs’, Ann. of Math.”