Bounds in capacity inequalities for two sheeted spheres
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- Nakai Mitsuru
- Department of Mathematics Nagoya Institute of Technology
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Take a pair of two disjoint nonpolar compact subsets A and B of the complex plane C = $¥widehat{¥bf C}¥setminus${∞}, the complex sphere less the point at infinity, with connected complement $¥widehat{¥bf C}¥setminus$(A ∪ B) and a simple arc γ in $¥widehat{¥bf C}¥setminus$(A ∪ B). We form the two sheeted covering surface $¥widehat{¥bf C}$γ of $¥widehat{¥bf C}$ by pasting $¥widehat{¥bf C}¥setminus$γ with another copy $¥widehat{¥bf C}¥setminus$γ crosswise along γ. Embed A and B in $¥widehat{¥bf C}$γ either in the same sheet or in the different sheets and consider the variational 2-capacity cap(A, $¥widehat{¥bf C}$γ$¥setminus$B) of A contained in the open subset $¥widehat{¥bf C}$γ$¥setminus$B of $¥widehat{¥bf C}$γ. Concerning the relation between the above capacity and the variational 2-capacity cap(A, $¥widehat{¥bf C}¥setminus$B) of A contained in the open subset $¥widehat{¥bf C}¥setminus$B of $¥widehat{¥bf C}$, we will establish the following capacity inequality for the two sheeted cover and its base:<br>0 < cap(A, ¥widehat{¥bf C}$γ$¥setminus$B) < 2 · cap(A, ¥widehat{¥bf C}¥setminus$B),<br>where the bound 2 in the above inequality is the best possible in the sense that, for any 0<τ <2, there is a triple of A, B, and γ such that cap(A, $¥widehat{¥bf C}$γ$¥setminus$B) > τ · cap(A, $¥widehat{¥bf C}¥setminus$B), where A and B may in the same sheet or in the different sheets.
収録刊行物
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- ＫＯＤＡＩ ＭＡＴＨＥＭＡＴＩＣＡＬ ＪＯＵＲＮＡＬ
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ＫＯＤＡＩ ＭＡＴＨＥＭＡＴＩＣＡＬ ＪＯＵＲＮＡＬ 30 (2), 223-236, 2007
国立大学法人 東京工業大学理学院数学系
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詳細情報 詳細情報について
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- CRID
- 1390001205271532672
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- NII論文ID
- 130004948984
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- ISSN
- 18815472
- 03865991
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- en
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