We study the injectivity radius of complete Riemannian surfaces (S, g) with bounded curvature . We show that if S is orientable with nonabelian fundamental group, then there is a point with injectivity radius R arcsinh. 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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