# On Heinz type inequality for the half-plane and Gaussian curvature of Minimal surfaces

## TL;DR

This paper establishes a Heinz type inequality for harmonic diffeomorphisms of the half-plane and applies it to derive sharp bounds on the Gaussian curvature of certain minimal surfaces.

## Contribution

It introduces a Heinz type inequality for harmonic maps of the half-plane and uses it to obtain curvature bounds for minimal surfaces lying above the half-plane.

## Key findings

- Proved a Heinz type inequality for harmonic diffeomorphisms of the half-plane.
- Derived sharp bounds for the Gaussian curvature of minimal surfaces.
- Applied the inequality to minimal surfaces above the half-plane in bc^3.

## Abstract

We prove a Heinz type inequality for harmonic diffeomorphisms of of the half-plane onto itself. We then apply this result to prove some sharp bound of the Gaussian curvature of a minimal surface, provided that it lies above the whole half-plane in $\mathbf{R}^3$.