Minimal graphs over Riemannian surfaces and harmonic diffeomorphisms

TL;DR

This paper constructs a special minimal graph over a hyperbolic surface, producing a harmonic diffeomorphism, and extends a theorem relating surface metrics and harmonic maps, with implications for geometric analysis.

Contribution

It introduces a method to construct parabolic minimal graphs over hyperbolic surfaces and generalizes a theorem on harmonic diffeomorphisms between surfaces with specific curvature conditions.

Findings

Constructed a parabolic minimal graph over a hyperbolic surface.

Established a harmonic diffeomorphism from the graph to the surface.

Extended Schoen's theorem relating surface metrics and harmonic maps.

Abstract

We construct a parabolic entire minimal graph $S$ over a finite topology complete Riemannian surface $\Sigma$ 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 $\Sigma$. The proof uses the theory of divergence lines to construct minimal graphs. We also generalize a theorem of R. Schoen. Let $g_1$ and $g_2$ be two complete metrics on a orientable surface $S$ with compact boundary and suppose $$\int_{S_r^2}K_{g_2}^-d\sigma_{g_2}\le C\ln(2+r)$$ for some $C>0$ and all $r>0$. If there is a harmonic diffeomorphism from $(S,g_1)$ to $(S,g_2)$, then $(S,g_1)$ is parabolic.