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

Matsuura, Shozo

doi:10.1090/S0002-9947-1975-0372255-3

The generalized Martin’s minimum problem and its applications in several complex variables
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by Shozo Matsuura

Trans. Amer. Math. Soc. 208 (1975), 273-307

DOI: https://doi.org/10.1090/S0002-9947-1975-0372255-3

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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.

References

Über die kernfunktion eines Bereiches und ihr Verhalten am Rande

169

172

Sur la fonction-noyau d’un domaine et ses application dans la théorie des transformations pseudoconformes

11

Stefan Bergman, The kernel function and conformal mapping, Second, revised edition, Mathematical Surveys, No. V, American Mathematical Society, Providence, R.I., 1970. MR 0507701
Bruce L. Chalmers, On boundary behavior of the Bergman kernel function and related domain functionals, Pacific J. Math. 29 (1969), 243–250. MR 247133, DOI 10.2140/pjm.1969.29.243
Bruce L. Chalmers, Subspace kernels and minimum problems in Hilbert spaces with kernel function, Pacific J. Math. 31 (1969), 619–628. MR 344864, DOI 10.2140/pjm.1969.31.619

Über geodasische Mannigfaltigkeiten einer invarianten Geometrie

2

B. A. Fuks, Spetsial′nye glavy teorii analiticheskikh funktsiĭ mnogikh kompleksnykh peremennykh, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1963 (Russian). MR 0174786
B. A. Fuks, The Ricci curvature of the Bergman metric invariant with respect to biholomorphic mappings, Dokl. Akad. Nauk SSSR 167 (1966), 996–999 (Russian). MR 0196768
Lars Hörmander, $L^{2}$ estimates and existence theorems for the $\bar \partial$ operator, Acta Math. 113 (1965), 89–152. MR 179443, DOI 10.1007/BF02391775

On the estimation of the unitary curvature of the space of several complex variables

4

Sadao Kató, Canonical domains in several complex variables, Pacific J. Math. 21 (1967), 279–291. MR 214810, DOI 10.2140/pjm.1967.21.279
Keiz\B{o} Kikuchi, Canonical domains and their geometry in $C^{n}$, Pacific J. Math. 38 (1971), 681–696. MR 304704, DOI 10.2140/pjm.1971.38.681
Shoshichi Kobayashi, Geometry of bounded domains, Trans. Amer. Math. Soc. 92 (1959), 267–290. MR 112162, DOI 10.1090/S0002-9947-1959-0112162-5
K. H. Look, Schwarz lemma and analytic invariants, Sci. Sinica 7 (1958), 453–504. MR 106294
W. T. Martin, On a minimum problem in the theory of analytic functions of several variables, Trans. Amer. Math. Soc. 48 (1940), 351–357. MR 2619, DOI 10.1090/S0002-9947-1940-0002619-5

On the normal domains and the geodesics in the bounded symmetric spaces and the projective space

15

Shozo Matsuura, Bergman kernel functions and the three types of canonical domains, Pacific J. Math. 33 (1970), 363–384. MR 274800, DOI 10.2140/pjm.1970.33.363
Harold S. Shapiro, Reproducing kernels and Beurling’s theorem, Trans. Amer. Math. Soc. 110 (1964), 448–458. MR 159006, DOI 10.1090/S0002-9947-1964-0159006-5
J. M. Stark, Minimum problems in the theory of pseudo-conformal transformations and their application to estimation of the curvature of the invariant metric, Pacific J. Math. 10 (1960), 1021–1038. MR 121499, DOI 10.2140/pjm.1960.10.1021
Teruo Tsuboi and Syozo Matsuura, Some canonical domains in $C^{n}$ and moment of inertia theorems, Duke Math. J. 36 (1969), 517–536. MR 255849
Wilhelm Wirtinger, Über eine Minimalaufgabe im Gebiete der analytischen Funktionen von mehreren Veränderlichen, Monatsh. Math. Phys. 47 (1939), 426–431 (German). MR 56, DOI 10.1007/BF01695512
K. Yano and S. Bochner, Curvature and Betti numbers, Annals of Mathematics Studies, No. 32, Princeton University Press, Princeton, N. J., 1953. MR 0062505