The converse of the Schwarz Lemma is false

Abstract

Abstract. Let h : X → 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 Thurston. We prove that the converse is false in all other cases, strengthening a result of Masumoto.

Conformal surface embeddings and extremal length

“…It is tempting to look for an analogue of Theorem 1 using hyperbolic length instead of extremal length, given that, by the Schwarz lemma, hyperbolic length is decreased under conformal inclusion. However, the results are false for hyperbolic length in almost all cases [Mas00,FB14].…”