The Poincaré-Lelong equation on complete Kähler manifolds

Ngaiming Mok; Yum-Tong Siu; Shing-Tung Yau

Compositio Mathematica (1981)

  • Volume: 44, Issue: 1-3, page 183-218
  • ISSN: 0010-437X

How to cite


Mok, Ngaiming, Siu, Yum-Tong, and Yau, Shing-Tung. "The Poincaré-Lelong equation on complete Kähler manifolds." Compositio Mathematica 44.1-3 (1981): 183-218. <>.

author = {Mok, Ngaiming, Siu, Yum-Tong, Yau, Shing-Tung},
journal = {Compositio Mathematica},
keywords = {Poincare-Lelong equation; complete Kähler manifold; growth condition on current; Harnack inequality; isometry},
language = {eng},
number = {1-3},
pages = {183-218},
publisher = {Sijthoff et Noordhoff International Publishers},
title = {The Poincaré-Lelong equation on complete Kähler manifolds},
url = {},
volume = {44},
year = {1981},

AU - Mok, Ngaiming
AU - Siu, Yum-Tong
AU - Yau, Shing-Tung
TI - The Poincaré-Lelong equation on complete Kähler manifolds
JO - Compositio Mathematica
PY - 1981
PB - Sijthoff et Noordhoff International Publishers
VL - 44
IS - 1-3
SP - 183
EP - 218
LA - eng
KW - Poincare-Lelong equation; complete Kähler manifold; growth condition on current; Harnack inequality; isometry
UR -
ER -


  1. [1] A. Andreotti and E. Vesentini, Carleman estimates for the Laplace-Beltrami operator on complex manifolds, Publ. Math. Inst. Hautes Etudes Sci.25 (1965), 81-130. Zbl0138.06604MR175148
  2. [2] E. Bombieri and E. Giusti, Harnack's inequality for elliptic differential equations on minimal surfaces, Inv. math.15 (1972), 24-46. Zbl0227.35021MR308945
  3. [3] J. Cheeger and D. Gromoll, The structure of complete manifolds of non-negative curvature, Bull. Am. Math. Soc.74(6), 413-433. Zbl0169.24101MR232310
  4. [4] S.Y. Cheng and S.-T. Yau, Differential equations on Riemannian manifolds and their geometric applications, Comm. Pure Appl. Math. Vol.28 (1975), 333-354. Zbl0312.53031MR385749
  5. [5] S.S. Chern, H. Levine, and L. Nirenberg, Intrinsic norms on a complex manifold, Global analysis, (Papers in honor of K. Kodaira), Tokyo- University of Tokyo Press1969, 119-139. Zbl0202.11603MR254877
  6. [6] C. Croke, Some isoperimetric inequalities and consequences, to appear in Ann. Sci. Ec. Norm. Sup. Pisa. 
  7. [7] R.E. Greene and H. Wu, Function Theory on Manifolds which Possess a Pole, Vol. 669. Springer-Verlag, Berlin- Heidelberg-New York, 1979. Zbl0414.53043MR521983
  8. [8] G.M. Henkin, The Lewy equation and analysis on pseudoconvex manifolds, Russian Math. Surveys32, 3 (1977), 59-130. Zbl0382.35038MR454067
  9. [9] L. Hörmander, L2-estimates and existence theorems for the ∂-operator, Acta Math.113 (1965), 89-152. Zbl0158.11002
  10. [10] A. Huber, On subharmonic functions and differential geometry in the large, Comm. Math. Helv.32 (1957), 13-72. Zbl0080.15001MR94452
  11. [11] P. Lelong, Fonctions entières (n variables) et fonctions plurisousharmoniques d'ordre fini dans Cn, J. Anal. Math.12 (1964), 365-407. Zbl0126.29602MR166391
  12. [12] J. Moser, On Harnack's theorem for elliptic differential equations, Comm. Pure and Appl. Math14 (1961), 577-591. Zbl0111.09302MR159138
  13. [13] Y.T. Siu and S.-T. Yau, Complete Kähler manifolds with non-positive curvature of faster than quadratic decay, Ann. Math.105 (1977), 225-264. Zbl0358.32006MR437797
  14. [14] H. Skoda, Valeurs au bord pour les solutions de l'operateur d'' et caracterisation des zeros des fonctions de la classe de Navanlinna, Bull. Soc. Math. France104 (1976), 225-299. Zbl0351.31007MR450620
  15. [15] G. Stampacchia, Equations elliptiques du second ordre à coefficients discontinués, 1966 (Séminaire de Mathématiques Supérieures 16). Zbl0151.15501MR251373
  16. [16] S.-T. Yau, Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math.28 (1975), 201-228. Zbl0291.31002MR431040
  17. [17] R.L. Bishop and S.I. Goldberg, On the second cohomology group of a Kähler manifold of positive curvature, Proc. Amer. Math. Soc.16 (1965), 119-122. Zbl0125.39403MR172221
  18. [18] S.I. Goldberg and S. Kobayashi, Holomorphic bisectional curvature, J. Diff. Geom.1 (1967), 225-233. Zbl0169.53202MR227901
  19. [19] R. Greene and H.-H. Wu, On a new gap phenomenon in Riemannian geometry, preprint. Zbl0487.53034MR648065

NotesEmbed ?


You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.


Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.