The converse of the Schwarz Lemma is false

TL;DR

This paper demonstrates that the converse of the Schwarz Lemma does not hold universally for hyperbolic surfaces, providing counterexamples beyond disks, annuli, and closed surfaces, thus refining the understanding of geometric mappings.

Contribution

It proves that the converse of the Schwarz Lemma fails for most hyperbolic surfaces, extending previous results and clarifying the limitations of length comparisons under homotopic holomorphic maps.

Findings

Counterexamples to the converse of Schwarz Lemma for hyperbolic surfaces

The converse holds only for disks, annuli, and closed surfaces

The result strengthens Masumoto's previous work

Abstract

Let $h:X \to Y$ be a homeomorphism between hyperbolic surfaces with finite topology. If $h$ is homotopic to a holomorphic map, then every closed geodesic in $X$ is at least as long as the corresponding geodesic in $Y$, by the Schwarz Lemma. The converse holds trivially when $X$ and $Y$ are disks or annuli, and it holds when $X$ and $Y$ are closed surfaces by a theorem of W. Thurston. We prove that the converse is false in all other cases, strengthening a result of Masumoto.