Construction of harmonic diffeomorphisms and minimal graphs

Pascal Collin (1) , Harold Rosenberg (2)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 IMJ - Institut de Mathématiques de Jussieu

We study complete minimal graphs in HxR, 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 "lenghts" 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 HxR 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.

Mots clés: minimal surfaces; harmonic diffeomorphisms

Domaines: Géométrie différentielle [math.DG]

Pascal Collin, Harold Rosenberg. Construction of harmonic diffeomorphisms and minimal graphs. 2007. ⟨hal-00125426⟩