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The Poincaré-Lelong equation on complete Kähler manifolds. (English) Zbl 0531.32007
P. Lelong [J. Anal. Math. 12, 365-407 (1964; Zbl 0126.296)] studied the Poincaré-Lelong equation $$i\partial {\bar \partial}u=\rho$$ for a positive d-closed (1,1) current $$\rho$$ defined on $${\mathbb{C}}^ n$$ and reduced it to $$\frac{1}{2}\Delta u=tr(\rho)$$ under suitable growth condition on $$\rho$$. The authors investigate the P-L equation on a complete Kähler manifold by means of two different techniques. When the complete Kähler manifold M has a nonnegative holomorphic bisectional curvature, $$\| i\partial {\bar \partial}-\rho \|^ 2$$ is nonnegative. When $$\rho$$ grows suitably, the mean value inequality implies that $$i\partial {\bar \partial}u=\rho$$. The authors use this technique to prove an isometry theorem on M. Secondly the $$L^ 2$$- estimate of $${\bar \partial}$$ of A. Andreotti and E. Vesentini [Publ. Math., Inst. Hautes Étud. Sci. 25, 81-130 (1965; Zbl 0138.066)] assures a solution of the P-L equation. Making use of the Harnack inequality of J. Moser [Commun. Pure Appl. Math. 14, 577- 591 (1961; Zbl 0111.093)] and results of Y.-T. Siu and S.-T. Yau [Ann. Math., II. Ser. 105, 225-264 (1977; Zbl 0358.32006)], they prove that M is isometrically biholomorphic to $${\mathbb{C}}^ n$$.
Reviewer: J.Kajiwara

##### MSC:
 32C30 Integration on analytic sets and spaces, currents 53C55 Global differential geometry of Hermitian and Kählerian manifolds 58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
##### Citations:
Zbl 0126.296; Zbl 0138.066; Zbl 0111.093; Zbl 0358.32006
Full Text:
##### References:
 [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. · Zbl 0138.06604 [2] E. Bombieri and E. Giusti , Harnack’s inequality for elliptic differential equations on minimal surfaces , Inv. math. 15 (1972), 24-46. · Zbl 0227.35021 [3] J. Cheeger and D. Gromoll , The structure of complete manifolds of non-negative curvature , Bull. Am. Math. Soc. 74(6), 413-433. · Zbl 0169.24101 [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. · Zbl 0312.53031 [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 Press 1969, 119-139. · Zbl 0202.11603 [6] C. Croke , Some isoperimetric inequalities and consequences , to appear in Ann. Sci. Ec. Norm. Sup. Pisa. [7] R.E. Greene and H. Wu , Function Theory on Manifolds which Possess a Pole , Vol. 669. Springer-Verlag, Berlin- Heidelberg-New York, 1979. · Zbl 0414.53043 [8] G.M. Henkin , The Lewy equation and analysis on pseudoconvex manifolds , Russian Math. Surveys 32, 3 (1977), 59-130. · Zbl 0382.35038 [9] L. Hörmander , L2-estimates and existence theorems for the \partial -operator , Acta Math. 113 (1965), 89-152. · Zbl 0158.11002 [10] A. Huber , On subharmonic functions and differential geometry in the large , Comm. Math. Helv. 32 (1957), 13-72. · Zbl 0080.15001 [11] P. Lelong , Fonctions entières (n variables) et fonctions plurisousharmoniques d’ordre fini dans Cn , J. Anal. Math. 12 (1964), 365-407. · Zbl 0126.29602 [12] J. Moser , On Harnack’s theorem for elliptic differential equations , Comm. Pure and Appl. Math 14 (1961), 577-591. · Zbl 0111.09302 [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. · Zbl 0358.32006 [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. France 104 (1976), 225-299. · Zbl 0351.31007 [15] G. Stampacchia , Equations elliptiques du second ordre à coefficients discontinués , 1966 (Séminaire de Mathématiques Supérieures 16). · Zbl 0151.15501 [16] S.-T. Yau , Harmonic functions on complete Riemannian manifolds , Comm. Pure Appl. Math. 28 (1975), 201-228. · Zbl 0291.31002 [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. · Zbl 0125.39403 [18] S.I. Goldberg and S. Kobayashi , Holomorphic bisectional curvature , J. Diff. Geom. 1 (1967), 225-233. · Zbl 0169.53202 [19] R. Greene and H.-H. Wu , On a new gap phenomenon in Riemannian geometry , preprint. · Zbl 0487.53034
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