On evans potential

Nakai, Mitsuru

doi:10.3792/pja/1195523234

Cited by 44 publications

(38 citation statements)

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“…For the convenience of the reader, the proof of Sario's existence theorem of principal functions [RS] and Nakai's construction of the Evans-Selberg potential [Na1], [Na2], [SaNo] are provided in this section. These facts were applied in [NR1].…”

Section: Principal Functions and The Evans-selberg Potentialmentioning

confidence: 99%

Filtered Ends, Proper Holomorphic Mappings of Kähler Manifolds to Riemann Surfaces, and Kähler Groups

Napier

Ramachandran

2007

GAFA Geom. funct. anal.

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Abstract. The main result of this paper is that a connected bounded geometry complete Kähler manifold which has at least 3 filtered ends admits a proper holomorphic mapping onto a Riemann surface. As an application, it is also proved that any properly ascending HNN extension with finitely generated base group, as well as Thompson's groups V, T, and F, are not Kähler. The results and techniques also yield a different proof of the theorem of Gromov and Schoen that, for a connected compact Kähler manifold whose fundamental group admits a proper amalgamated product decomposition, some finite unramified cover admits a surjective holomorphic mapping onto a curve of genus at least 2.This version of this paper contains details not in the version

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Section: Principal Functions and The Evans-selberg Potentialmentioning

confidence: 99%

Filtered Ends, Proper Holomorphic Mappings of Kähler Manifolds to Riemann Surfaces, and Kähler Groups

Napier

Ramachandran

2007

GAFA Geom. funct. anal.

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“…As in the proof of Theorem 2.6 of [NR1], one may apply a theorem of Nakai [Na1], [Na2] and a theorem of Sullivan [Sul] to obtain a continuous function ρ on E 0 such that ρ is harmonic on E 0 , ρ = 0 on ∂E 0 , ρ(x) → ∞ at infinity, and the L 2 norm of ∇ρ on a ball of radius R is equal to o(R). Therefore ρ is pluriharmonic by Lemma 6.4 and ρ has a (nonempty) compact fiber in E 0 .…”

Section: Castelnuovo-de Franchis For An Endmentioning

confidence: 99%

L² CASTELNUOVO–DE FRANCHIS, THE CUP PRODUCT LEMMA, AND FILTERED ENDS OF KÄHLER MANIFOLDS

Napier

Ramachandran

2009

J. Topol. Anal.

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Abstract. Simple approaches to the proofs of the L 2 Castelnuovo-de Franchis theorem and the cup product lemma which give new versions are developed. For example, suppose ω 1 and ω 2 are two linearly independent closed holomorphic 1-forms on a bounded geometry connected complete Kähler manifold X with ω 2 in L 2 . According to a version of the L 2 Castelnuovo-de Franchis theorem obtained in this paper, if ω 1 ∧ ω 2 ≡ 0, then there exists a surjective proper holomorphic mapping of X onto a Riemann surface for which ω 1 and ω 2 are pull-backs. Previous versions required both forms to be in L 2 . IntroductionAccording to the classical theorem of Castelnuovo and de Franchis (see [Be], [BarPV]), if, on a connected compact complex manifold X, there exist linearly independent closed holomorphic 1-forms ω 1 and ω 2 with ω 1 ∧ ω 2 ≡ 0, then there exist a surjective holomorphic mapping Φ of X onto a curve C of genus g ≥ 2 and holomorphic 1-forms θ 1 and θ 2 on C such that ω j = Φ * θ j for j = 1, 2. The main point is that the meromorphic function f ≡ ω 1 /ω 2 actually has no points of indeterminacy, so one may Stein factor the holomorphic map f : X → P 1 .Remark. The requirement that the forms be closed is superfluous if the compact manifold X is a surface or if X is Kähler. For, if η = √ −1g ij dz i ∧ dz j is the Kähler form for aKähler metric g and ω is a holomorphic 1-form, then, by Stokes' theorem, we havewhere n = dim X. Since the integrand is a nonnegative 2n-form, the form must vanish and it follows that dω = 0. For X a surface, the same argument with the factor η n−2 removed again yields dω = 0. In general, given a connected complex manifold X and linearly independent closed holomorphic 1-forms ω 1 and ω 2 on X with ω 1 ∧ ω 2 ≡ 0, the meromorphic function f ≡ ω 1 /ω 2 has no points of indeterminacy, f is locally constant on the analytic set Z = { x ∈ X | (ω 1 ) x = 0 or (ω 2 ) x = 0 }, and f is constant on each leaf of the holomorphic foliation determined by ω 1 and ω 2 in X \ Z (see, for example, [NR2] for an elementary proof). In particular, if the levels of the holomorphic map f : X → P 1 are compact, thenStein factorization gives a surjective proper holomorphic mapping of X onto a Riemann surface.We will say that a complete Hermitian manifold (X, g) has bounded geometry of order k if, for some constant C > 0 and for every point p ∈ X, there is a biholomorphism Ψ of the unit ball B = B(0; 1) ⊂ C n onto a neighborhood of p in X such that Ψ(0) = p and, on B, C −1 g C n ≤ Ψ * g ≤ Cg C n and |D m Ψ * g| ≤ C for m = 0, 1, 2, . . . , k.For k = 0, we will simply say that (X, g) has bounded geometry. Gromov [Gro2] observed that, for f = ω 1 /ω 2 as above, one gets compact levels if X is a bounded geometry complete Kähler manifold and the 1-forms are in L 2 and have exact real parts; thus giving an L 2 version of the Castelnuovo-de Franchis theorem. He also introduced his so-called cup product lemma, according to which, two L 2 holomorphic 1-forms ω 1 and ω 2 with exact real parts on a bounded geometry complete Kähler m...

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“…Since 7 has zero capacity we can see that there exists an Evans potential e 0 for 7, i.e., a function e o eH(R -ζ) satisfying the following conditions (Nakai [4] Needless to say e o eF.…”

Section: ( J) {Pn} Converges Uniformly To P On Any Compact K Withmentioning

confidence: 99%

On unicity of capacity functions

Osada¹

1969

Pacific J. Math.

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On evans potential

Cited by 44 publications

References 0 publications

Filtered Ends, Proper Holomorphic Mappings of Kähler Manifolds to Riemann Surfaces, and Kähler Groups

Filtered Ends, Proper Holomorphic Mappings of Kähler Manifolds to Riemann Surfaces, and Kähler Groups

L² CASTELNUOVO–DE FRANCHIS, THE CUP PRODUCT LEMMA, AND FILTERED ENDS OF KÄHLER MANIFOLDS

On unicity of capacity functions

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On evans potential

Cited by 44 publications

References 0 publications

Filtered Ends, Proper Holomorphic Mappings of Kähler Manifolds to Riemann Surfaces, and Kähler Groups

Filtered Ends, Proper Holomorphic Mappings of Kähler Manifolds to Riemann Surfaces, and Kähler Groups

L2 CASTELNUOVO–DE FRANCHIS, THE CUP PRODUCT LEMMA, AND FILTERED ENDS OF KÄHLER MANIFOLDS

On unicity of capacity functions

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L² CASTELNUOVO–DE FRANCHIS, THE CUP PRODUCT LEMMA, AND FILTERED ENDS OF KÄHLER MANIFOLDS