Abstract
We study the uniformization conjecture of Yau by using the Gromov-Hausdorff convergence. As a consequence, we confirm Yau’s finite generation conjecture. More precisely, on a complete noncompact Kähler manifold with nonnegative bisectional curvature, the ring of polynomial growth holomorphic functions is finitely generated. During the course of the proof, we prove if $M^n$ is a complete noncompact Kähler manifold with nonnegative bisectional curvature and maximal volume growth, then $M$ is biholomorphic to an affine algebraic variety. We also confirm a conjecture of Ni on the existence of polynomial growth holomorphic functions on Kähler manifolds with nonnegative bisectional curvature.
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[[Y2]]
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@article{[Y2], mrkey = {0896385},
author = {Yau, Shing-Tung},
title = {Nonlinear analysis in geometry},
journal = {Enseign. Math.},
fjournal = {L'Enseignement Mathématique. Revue Internationale. IIe Série},
volume = {33},
year = {1987},
number = {1-2},
pages = {109--158},
issn = {0013-8584},
coden = {ENMAAR},
mrclass = {58-02 (58G30)},
mrnumber = {0896385},
zblnumber = {0631.53002},
URL = {http://www.e-periodica.ch/digbib/view?pid=ens-001:1987:33#242},
}