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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.)
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