# A Schwarz-Pick lemma for minimal maps

## TL;DR

This paper establishes a Schwarz-Pick type lemma for minimal maps between negatively curved Riemann surfaces, showing such maps are area decreasing under certain curvature conditions.

## Contribution

It introduces a novel Schwarz-Pick lemma for minimal maps between negatively curved surfaces, extending classical results to this geometric setting.

## Key findings

- Minimal maps with bounded Jacobian are area decreasing.
- The lemma applies when curvature bounds satisfy infσ_M ≥ supσ_N.
- The result generalizes classical Schwarz-Pick lemmas to minimal maps.

## Abstract

In this note, we prove a Schwarz-Pick type lemma for minimal maps between negatively curved Riemannian surfaces. More precisely, we prove that if $f:M \to N$ is a minimal map with bounded Jacobian between two complete negatively curved Riemann surfaces M and N whose sectional curvatures $\sigma_M$ and $\sigma_N$ satisfy $inf\sigma_M \ge sup\sigma_N$, then f is area decreasing.