Advertisement

Gromov’s Schwarz lemma as an estimate of the gradient for holomorphic curves

  • Marie-Paule Muller
Chapter
Part of the Progress in Mathematics book series (PM, volume 117)

Abstract

In a symplectic manifold (M 2n , ω) with a tamed almost complex structure J, the 2-form ω induces an area form on (immersed) J-curves f: (Σ, i) → (M, J), where (Σ, i) is a Riemann surface which will be closed, open or compact with boundary, according to the context. The image f(Σ) will be denoted by S. Throughout this chapter, we shall assume that S is contained in a fixed compact subset of M.

Keywords

Sectional Curvature Isoperimetric Inequality Compact Riemann Surface Holomorphic Curve Holomorphic Curf 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. [1]
    L.V. Ahlfors, An extension of Schwarz’s lemma, Trans. Amer. Math. Soc. 43 (1938), 359–364.MathSciNetGoogle Scholar
  2. [2]
    F. Almgren, Optimal isoperimetric inequalities, Bull. Amer. Math. Soc. 13 (1985), 123–126.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    E. Beckenbach, T. Rado, Subharmonic functions and surfaces of negative curvature, Trans. Amer. Math. Soc. 35 (1933), 662–674.MathSciNetCrossRefGoogle Scholar
  4. [4]
    G. Bol, Isoperimetrische Ungleichungen für Bereiche auf Flächen, Jahresbericht der Deutchen Mathem. Vereinigung 3 (1951), 219–257.Google Scholar
  5. [5]
    A. Douglis, L. Nirenberg, Interior estimates for elliptic systems of partial differential equations, Commun. in Pure Appl. Math. 8 (1955), 503–538.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    B. Dubrovin, S. Novikov, A. Fomenko, Géométrie contemporaine, MIR, Moscou, 1987.Google Scholar
  7. [7]
    H. Grauert, H. Reckziegel, Hermitesche Metriken und normale Familien holomorpher Abbildungen, Math. Z. 89 (1965), 108–125.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    P. Lelong, FrPropriétés métriques des variétés analytiques complexes définies par une équation, Ann. Ec. Norm. Sup. 67 (1950), 393–419.MathSciNetzbMATHGoogle Scholar
  9. [9]
    A. Nijenhuis, W.B. Woolf, Some integration problems in almost complex and complex manifolds, Ann. of Math. 77 (1963), 424–489.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Y. G. Oh, Removal of boundary singularities of pseudo-holomorphic curves with Lagrangian boundary conditions, Comm. Pure Appl. Math. 45 (1992), 121–139.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    A. Weil, Sur les surfaces à courbure négative, C. R. Acad. Sci. Paris 182 (1926), 1069–1071.zbMATHGoogle Scholar
  12. [12]
    Hung-Hsi Wu, Contemporary Geometry, Zhong Memorial volume, Plenum Press, 1991.Google Scholar

Copyright information

© Springer Basel AG 1994

Authors and Affiliations

  • Marie-Paule Muller

There are no affiliations available

Personalised recommendations