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 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.

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Correspondence to Matthieu Gendulphe.

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This work has been fully supported by FIRB 2010 (RBFR10GHHH003).

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Gendulphe, M. Le lemme de Schwarz et la borne supérieure du rayon d’injectivité des surfaces. manuscripta math. 148, 399–413 (2015). https://doi.org/10.1007/s00229-015-0751-9

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Mathematics Subject Classification

  • 53C20 (primary)
  • 30F45 (secondary)