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

Matthieu Gendulphe (1)

1 Dipartimento di Matematica Guido Castelnuovo

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.

Mots clés: Injectivity radius; Schwarz lemma

Domaines: Géométrie différentielle [math.DG]

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