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

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• 作者：Matthieu Gendulphe
• 关键词：53C20 (primary) ; 30F45 (secondary)
• 刊名：manuscripta mathematica
• 出版年：2015
• 出版时间：November 2015
• 年：2015
• 卷：148
• 期：3-4
• 页码：399-413
• 全文大小：428 KB
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  2.Bavard, C.: La borne supérieure du rayon d’injectivité en dimension 2 et 3. Thèse de troisième cycle, université Paris-Sud (1984)
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  4.Bavard C.: La systole des surfaces hyperelliptiques. Prépublication de l’ENS Lyon, Juillet (1992)
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  11.Fricke, R., Klein, F.: Vorlesungen über die Theorie der automorphen Funktionen. Band 1: Die gruppentheoretischen Grundlagen. Teubner (1897)
  12.Gendulphe, M.: Trois applications du lemme de Schwarz aux surfaces hyperboliques. (2014). Prépublication disponible à matthieu.gendulphe.com
  13.Gromov, M.: Systoles and intersystolic inequalities. In: Actes de la Table Ronde de Géométrie Différentielle (Luminy, 1992), volume 1 of Sémin. Congr., pp. 291-62. Soc. Math. France (1996)
  14.Jenni F.: über den ersten Eigenwert des Laplace-Operators auf ausgew?hlten Beispielen kompakter Riemannscher Fl?chen. Comment. Math. Helv. 59(2), 193-03 (1984)MATH MathSciNet CrossRef
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  19.Yamada, A.: On Marden’s universal constant of Fuchsian groups. II.. J. Anal. Math. 41, 234-48 (1982)
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• 作者单位：Matthieu Gendulphe (1)
  
  1. Dipartimento di Matematica Guido Castelnuovo, Sapienza università di Roma, Piazzale Aldo Moro, 00185, Rome, Italy
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  Algebraic Geometry
  Topological Groups and Lie Groups
  Geometry
  Number Theory
  Calculus of Variations and Optimal Control
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-1785

文摘

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 8:369-81, 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)