Geometric approach to the metric appeared in the function theory

Research Project

Project/Area Number	60540101
Research Category	Grant-in-Aid for General Scientific Research (C)
Allocation Type	Single-year Grants
Research Field	解析学
Research Institution	Shiga University.
Principal Investigator	HIROSHI YAMAGUCHI Shiga university, Faculty of Educations Professor, 教育学部, 教授 (20025406)
Co-Investigator(Kenkyū-buntansha)	KATSUMI Okubo Shiga University, Faculty of Educations Asist. Professor, 教育学部, 助教授 (80024933)
Project Period (FY)	1985 – 1986
Project Status	Completed (Fiscal Year 1986)
Budget Amount *help	¥1,000,000 (Direct Cost: ¥1,000,000) Fiscal Year 1986: ¥500,000 (Direct Cost: ¥500,000) Fiscal Year 1985: ¥500,000 (Direct Cost: ¥500,000)
Keywords	Complex manifold / Green's function / Robin constant / Pseudoconvex / Subharmonic function / 擬凸状域
Research Abstract	Let M be a complex manifold of dimension n, and B a disk in the complex t plane. Consider a domain <about!D> of the space BXM such that D <CONTAINS> BX(O) where O is a fixed point in M. For each t <Member of> B, we put D(t) = <about!D> <INTERSECTION> ( (t) X M). We then regard <about!D> as a variation of domains D(t) of M with parameter t in B. We write it <about!D> : t <->> D(t) (t <Member of> B). Assume that M has a koehler metric <ds^2> . Then each D(t) carries the Green's function G(t,z) for <ds^2> with pole at O. Let L(z) denote an elementary solution for <ds^2> at O. Then we have in a neighborhood of O, G(t,z) = L(z) + <lambda> (t) + H(t,z) where <lambda> (t) is a constant and H(t,O) = 0. <lambda> (t) is called the Robin constant for (D(t), (O)). Under these notations, we get the following Theorem: If D is a pseudoconvex domain in BXM, then <lambda> (t) is a superharmonic function on B. The Robin constant plays an important role in the electromagnetism.