Minimal graphs over Riemannian surfaces and harmonic diffeomorphisms

2016/07/18 by Mazet, Laurent, Rodriguez, Magdalena, Rosenberg, Harold
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1607.05061

Abstract

We construct a parabolic entire minimal graph S over a finite topology complete Riemannian surface Σ of curvature -1 and infinite area (thus of non-parabolic conformal type). The vertical projection of this graph yields a harmonic diffeomorphism from S onto Σ. The proof uses the theory of divergence lines to construct minimal graphs. We also generalize a theorem of R. Schoen. Let g₁ and g₂ be two complete metrics on a orientable surface S with compact boundary and suppose ∫_Sr²K_g2^-dσ_g2≤ Cln(2+r) for some C>0 and all r>0. If there is a harmonic diffeomorphism from (S,g₁) to (S,g₂), then (S,g₁) is parabolic.

Related

• Minimal surfaces and harmonic diffeomorphisms from the complex plane onto a Hadamard surface
• Minimal Surfaces in \widetildePSL₂(ℝ)
• Capacity for minimal graphs over manifolds and the half-space property
• Uniformizing surfaces via discrete harmonic maps
• Minimal Lagrangian diffeomorphisms between domains in the hyperbolic plane