Vol. 33, No. 2, 1970

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Bergman kernel functions and the three types of canonical domains

Shozo Matsuura

Vol. 33 (1970), No. 2, 363–384

The objects of this paper are to extend J. Mitchell’s theorems on minimal domains of moment of inertia for sufficiently wider class and to discuss the relations among the three types of canonical domains in the -equivalent class. Some of the results are that, (i) a domain D is the minimal domain of moment of inertia of the [0,In;0]D-equivalent class if and only if the following holds:

MD0In(Z,0) = Z for z ∈ D,

where MD0In(Z,0) is the minimizing function of the (0,In;0)D-class, and (ii) if A,B and C are the sets of Bergman’s minimal domains, Bergman’s representative domains and Mitchell’s minimal domains of moment of inertia with the same center in the -equivalent class respectively, and if any one of the three relations A Bϕ, A Cϕ and B Cϕ holds, then it follows that A B = C.

Mathematical Subject Classification
Primary: 32.25
Received: 22 May 1969
Published: 1 May 1970
Shozo Matsuura