Classification Theory of Riemann Surfaces

Sario, Leo; Nakai, Mitsuru

doi:10.1007/978-3-642-48269-4

Cited by 260 publications

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“…e.g. [12], [17], [15], [9], [10], [6], [7], [8], etc.). However in the present paper we only state the following direct consequence of the formula as a byproduct: Theorem 1.1.…”

Section: Nakaimentioning

confidence: 99%

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The Dependence of Capacities on moving Branch points

Nakai

2007

Nagoya Mathematical Journal

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Abstract. We are concerned with the question how the capacity of the ideal boundary of a subsurface of a covering Riemann surface over a Riemann surface varies according to the variation of its branch points. In the present paper we treat the most primitive but fundamental situation that the covering surface is a two sheeted sphere with two branch points one of which is fixed and the other is moving and the subsurface is given as the complement of two disjoint continua each in different sheets of the covering surface whose projections are two disjoint continua in the base plane given in advance not touching the projections of branch points. We will derive a variational formula for the capacity and as one of its many useful consequences expected we will show that the capacity changes smoothly as one branch point moves in the subsurface.

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“…e.g. [12], [17], [15], [9], [10], [6], [7], [8], etc.). However in the present paper we only state the following direct consequence of the formula as a byproduct: Theorem 1.1.…”

Section: Nakaimentioning

confidence: 99%

“…e.g. [15], [2], etc. ), where in general we denote by H(X) the class of harmonic functions defined on an open subset X of some Riemann surface and L 1,2 indicates the Dirichlet space (cf.…”

Section: Nakaimentioning

confidence: 99%

The Dependence of Capacities on moving Branch points

Nakai

2007

Nagoya Mathematical Journal

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“…In passing we remark that the Theorem is invalid if the Martin compactification is replaced by the Royden compactification (cf., e.g., Sario and Nakai [6]). Concerning the other compactifications such as Kuramochi and Wiener compactifications the question seems to be entirely open (cf., e.g., Constantinescu and Cornea [3]).…”

Section: Theoremmentioning

confidence: 99%

Martin Compactifications and Quasiconformal Mappings

Segawa¹,

Tada²

1985

Proceedings of the American Mathematical Society

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“…e.g. [1], [15]) assures that lim^ #£ e HD(R). If X = B, then let u = u γ -u 2 with %eP5(β) + (i = 1, 2).…”

Section: P 2 X(r) Let W E Px(r) and Set V = T P2)pl U E P 2 X(r)mentioning

confidence: 99%

Order Comparisons on Canonical Isomorphisms

Nakai

1973

Nagoya Mathematical Journal

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MITSURU NAKAIConsider a nonnegative Holder continuous 2-f orm P(z)dxdy (z = x + iy) on a connected Riemann surface R. We denote by P(R) the linear space of solutions u of the equation Δu = Pu on R and by PX(R) the subspace of P(R) consisting of those u with a certain boundedness property X. We also use the standard notations H(R) and HX(R) for P(R) and PX(R) with P = 0. As for X we take B to mean the finiteness of the supremum norm \\u\\ -sup Λ |%|, D the finiteness of the Dirichlet integral D(u) = ί du Λ* du, E the finiteness of the energy integral E(u) = (d% Λ* cfot + u\z)P(z)dxdy), and their nontrivial combinations BD and .RE 1 . Let Q{z)dxdy be another 2-form of the same kind. We say that PX(R) is canonically isomorphic to QX(R) if there exists a linear isomorphism T of PX(R) onto QX(R) such that % and T% have the same ideal boundary values for every u in PX(R) in the sense that \u -2%| is dominated by a potential on R, i.e. a nonnegative superharmonic function whose greatest harmonic minorant is zero. In the pioneering work [14] concerning canonical isomorphisms, Royden proved the following order comparison theorem: If there exists a constant c > 1 such thaton hyperbolic R except possibly for a compact subset K of R, then PB(R) and QB(R) are canonically isomorphic. In this connection we wish to discuss the following two questions:1°. Is the condition (1) also sufficient for PX(R) and QX(R) to be canonically isomorphic for X = D, E, BD, and BEΊ 2°. In the affirmative case how large can we make the exceptional set K in (1) for X = B, D, E, BD, and BE?

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Classification Theory of Riemann Surfaces

Cited by 260 publications

References 0 publications

The Dependence of Capacities on moving Branch points

The Dependence of Capacities on moving Branch points

Martin Compactifications and Quasiconformal Mappings

Order Comparisons on Canonical Isomorphisms

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