Gaussian curvature of minimal graphs in $M\times \mathbb{R}$

TL;DR

This paper investigates the Gaussian curvature of minimal graphs in three-dimensional Riemannian manifolds, providing estimates and new results using harmonic mappings and classical complex analysis techniques.

Contribution

It introduces new curvature estimates for minimal graphs in product manifolds and applies harmonic mapping theory to derive Schwarz and Heinz type results.

Findings

Gaussian curvature estimates for minimal graphs in $M\times \mathbb{R}$

Schwarz lemma type results for harmonic mappings

Heinz type results for harmonic mappings between geodesic disks

Abstract

In this paper, we consider minimal graphs in the three-dimensional Riemannian manifold $M\times\mathbb{R}$. We mainly estimate the Gaussian curvature of such surfaces. We consider the minimal disks and minimal graphs bounded by two Jordan curves in parallel planes. The key to the proofs is the Weierstrass representation of those surfaces via $\wp-$harmonic mappings. We also prove some Schwarz lemma type results and some Heinz type results for harmonic mappings between geodesic disks in Riemannian surfaces.