manuscripta mathematica

, Volume 72, Issue 1, pp 251–256 | Cite as

The Schwarz lemma for nonpositively curved Riemmanian surfaces

  • Marc Troyanov


In this paper, we prove that if f is a conformal map between two Riemannian surfaces, and if the curvature of the target is nonpositive and less than or equal to the curvature of the source, then the map is contracting.


Riemannian Surface Conical Singularity Schwarz Lemma Open Riemann Surface Generalize Maximum Principle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Ahlfors L.V.,An extension of Schwarz’s Lemma, Trans. Amer. Math. Soc.43 (1938), 359–364zbMATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    Hulin, D. et Troyanov, M.,Sur la courbure des surfaces ouvertes, C. R. Acad. Sci. Paris Série 1310 (1990), 203–206zbMATHMathSciNetGoogle Scholar
  3. [3]
    Hulin, D. et Troyanov, M.,Prescribing curvature on open surfaces, Préprint Ecole Polytechnique, Palaiseau (1990)Google Scholar
  4. [4]
    Mok N. & Yau S.T.,Completeness of the Kähler-Einstein Metric on Bounded Domain and the Characterization of Domains of Holomorphy by Curvature Conditions, The Mathematical Heritage of Henri Poincaré, part 1 Proc. Symposia Pure Math.39 (1983), pp. 41–60MathSciNetGoogle Scholar
  5. [5]
    Osserman R.,On the inequality Δu≥f(u), Pacific J. MathVII (1957), 1641–1647MathSciNetGoogle Scholar
  6. [6]
    Yau S.T.,Harmonic functions on complete Riemannian Manifolds, Comm. Pure Appl. Math.28 (1975), 201–228zbMATHMathSciNetGoogle Scholar
  7. [7]
    Yau S.T.,A general Schwarz Lemma for Kahler Manifolds., Amer. Jour. Math. 100 (1978), 197–203zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • Marc Troyanov
    • 1
  1. 1.Département de MathématiquesUniversité du Québec à MontrealMontréalCanada

Personalised recommendations