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

  • Matthieu GendulpheEmail author


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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  1. 1.
    Abresch, U., Meyer, W.T.: Injectivity radius estimates and sphere theorems. In: Comparison Geometry (Berkeley, CA, 1993–94), volume 30 of Math. Sci. Res. Inst. Publ., pp. 1–47. Cambridge University Press, Cambridge (1997)Google Scholar
  2. 2.
    Bavard, C.: La borne supérieure du rayon d’injectivité en dimension 2 et 3. Thèse de troisième cycle, université Paris-Sud (1984)Google Scholar
  3. 3.
    Bavard C.: Le rayon d’injectivité des surfaces à courbure majorée. J. Differ. Geom. 20(1), 137–142 (1984)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Bavard C.: La systole des surfaces hyperelliptiques. Prépublication de l’ENS Lyon, Juillet (1992)Google Scholar
  5. 5.
    Bavard C.: Disques extrémaux et surfaces modulaires. Ann. Fac. Sci. Toulouse Math. (6) 5(2), 191–202 (1996)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Bavard C., Pansu P.: Sur le volume minimal de \({\mathbf{R}^2}\). Ann. Sci. École Norm. Sup. (4) 19(4), 479–490 (1986)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Bavard C., Pansu P.: Sur l’espace des surfaces à courbure et aire bornées. Ann. Inst. Fourier (Grenoble) 38(1), 175–203 (1988)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Burago Y.D.: The radius of injectivity on the surfaces whose curvature is bounded above. Ukrain. Geom. Sb. 21, 10–14 (1978)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Buser P.: Geometry and Spectra of Compact Riemann Surfaces, volume 106 of Progress in Mathematics. Birkhäuser, Basel (1992)Google Scholar
  10. 10.
    Cheeger J., Ebin D.G.: Comparison Theorems in Riemannian Geometry. North Holland, NY (1975)zbMATHGoogle Scholar
  11. 11.
    Fricke, R., Klein, F.: Vorlesungen über die Theorie der automorphen Funktionen. Band 1: Die gruppentheoretischen Grundlagen. Teubner (1897)Google Scholar
  12. 12.
    Gendulphe, M.: Trois applications du lemme de Schwarz aux surfaces hyperboliques. (2014). Prépublication disponible à matthieu.gendulphe.comGoogle Scholar
  13. 13.
    Gromov, M.: Systoles and intersystolic inequalities. In: Actes de la Table Ronde de Géométrie Différentielle (Luminy, 1992), volume 1 of Sémin. Congr., pp. 291–362. Soc. Math. France (1996)Google Scholar
  14. 14.
    Jenni F.: Über den ersten Eigenwert des Laplace-Operators auf ausgewählten Beispielen kompakter Riemannscher Flächen. Comment. Math. Helv. 59(2), 193–203 (1984)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Katz M., Sabourau S.: An optimal systolic inequality for CAT(0) metrics in genus two. Pac. J. Math. 227(1), 95–107 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Parlier H.: Hyperbolic polygons and simple closed geodesics. Enseign. Math. (2) 52(3-4), 295–317 (2006)zbMATHMathSciNetGoogle Scholar
  17. 17.
    Suárez-Serrato P., Tapie S.: Conformal entropy rigidity through Yamabe flows. Math. Ann. 353(2), 333–357 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Troyanov M.: The Schwarz lemma for nonpositively curved Riemannian surfaces. Manuscr. Math. 72(3), 251–256 (1991)zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Yamada, A.: On Marden’s universal constant of Fuchsian groups. II.. J. Anal. Math. 41, 234–248 (1982)Google Scholar
  20. 20.
    Yau S.T.: Remarks on conformal transformations. J. Differ. Geom. 8, 369–381 (1973)zbMATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

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

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