On the convergence and collapsing of Kahler metrics

In this paper we consider the convergence and collapsing of Kahler manifolds. While the convergence and collapsing of Riemannian manifolds have been discussed by many people and applied to many fields, how to generalize it to Kahler case is not apriorily clear. Our paper is an attempt in this direction. We discussed the corresponding concepts of convergence and collapsing for Kahler manifolds. We proved that when a sequence of Kahler manifolds with the fixed background complex compact manifold is not collapsing, it will converge to a complete Kahler manifold which is biholomorphic to a Zariski open set of the original background complex manifold with some possible "bubbling" on the complement of that Zariski open set. We also discussed the structure of collapsing. Especially we show the resulting Monge-Ampere foliation is holomorphic, produce some holomorphic Vector fields with respect to the foliation, and also give some applications of our results. The main methods we are using are estimates from the theories of harmonic maps and partial differential equations, some results from several complex variables, and ideas from Riemannian geometry.
Publisher
LEHIGH UNIV
Issue Date
1999-05
Language
ENG
Keywords

RIEMANNIAN-MANIFOLDS; RICCI CURVATURE; LELONG NUMBERS; ANALYTICITY; THEOREMS

Citation

JOURNAL OF DIFFERENTIAL GEOMETRY, v.52, no.1, pp.1 - 40

ISSN
0022-040X
URI
http://hdl.handle.net/10203/75032
Appears in Collection
MA-Journal Papers(저널논문)
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