Manuscripta Mathematica

, Volume 148, Issue 3–4, pp 399–413 | Cite as

Le lemme de Schwarz et la borne supérieure du rayon d’injectivité des surfaces



We study the injectivity radius of complete Riemannian surfaces (S, g) with bounded curvature \({|K(g)|\leq 1}\). We show that if S is orientable with nonabelian fundamental group, then there is a point \({p\in S}\) with injectivity radius R\({_p(g)\geq}\) arcsinh\({(2/\sqrt{3})}\). This lower bound is sharp independently of the topology of S. This result was conjectured by Bavard who has already proved the genus zero cases (Bavard 1984). We establish a similar inequality for surfaces with boundary. The proofs rely on a version due to Yau (J Differ Geom 8:369–381, 1973) of the Schwarz lemma, and on the work of Bavard (1984). This article is the sequel of Gendulphe (2014) where we studied applications of the Schwarz lemma to hyperbolic surfaces.

Mathematics Subject Classification

53C20 (primary) 30F45 (secondary) 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Dipartimento di Matematica Guido CastelnuovoSapienza università di RomaRomeItaly

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