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

Abstract

We study the injectivity radius of complete Riemannian surfaces (S,g) with curvature |K(g)| bounded by 1. We show that if S is orientable with nonabelian fundamental group, then there is a point p in S with injectivity radius at least 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. We establish a similar inequality for surfaces with boundary. The proofs rely on a version due to Yau of the Schwarz lemma, and on the work of Bavard. This article is the sequel of a previous one where we studied applications of the Schwarz lemma to hyperbolic surfaces.
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Dates and versions

hal-00980069 , version 1 (17-04-2014)

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Matthieu Gendulphe. Le lemme de Schwarz et la borne supérieure du rayon d'injectivité des surfaces. 2014. ⟨hal-00980069⟩

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