Annales Academiæ Scientiarum Fennicæ
Volumen 41, 2016, 235-241
University of Toronto, Department of Mathematics
40 St. George Street, Toronto, Ontario, Canada M5S 2E4; mbourque 'at' math.toronto.edu
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.
2010 Mathematics Subject Classification: Primary 30F45, 30F60, 32G15.
Key words: Schwarz Lemma, hyperbolic surfaces, length of geodesics.
Reference to this article: M. Fortier Bourque: The converse of the Schwarz Lemma is false. Ann. Acad. Sci. Fenn. Math. 41 (2016), 235-241.
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