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The generalized Martin's minimum problem and its applications in several complex variables


Author: Shozo Matsuura
Journal: Trans. Amer. Math. Soc. 208 (1975), 273-307
MSC: Primary 32H05
DOI: https://doi.org/10.1090/S0002-9947-1975-0372255-3
MathSciNet review: 0372255
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Abstract: The objectives of this paper are to generalize the Martin's $ {\mathfrak{L}^2}$-minimum problem under more general additional conditions given by bounded linear functionals in a bounded domain $ D$ in $ {C^n}$ and to apply this problem to various directions.

We firstly define the new $ i$th biholomorphically invariant Kähler metric and the $ i$th representative domain $ (i = 0,1,2, \ldots )$, and secondly give estimates on curvatures with respect to the Bergman metric and investigate the asymptotic behaviors via an $ A$-approach on the curvatures about a boundary point having a sort of pseudoconvexity.

Further, we study (i) the extensions of some results recently obtained by K. Kikuchi on the Ricci scalar curvature, (ii) a minimum property on the reproducing subspace-kernel in $ \mathfrak{L}_{(m)}^2(D)$, and (iii) an extension of the fundamental theorem of K. H. Look.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1975-0372255-3
Keywords: Bergman kernel function, Bergman metric, biholomorphically invariant, representative domain, bounded linear functional, holomorphic bisectional curvature, Ricci tensor, strictly pseudoconvex, classical Cartan domains
Article copyright: © Copyright 1975 American Mathematical Society

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