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Type: Journal article
Title: Representing de Rham cohomology classes on an open Riemann surface by holomorphic forms
Author: Alarcón, A.
Lárusson, F.
Citation: International Journal of Mathematics, 2017; 28(9):1740004-1-1740004-12
Publisher: World Scientific Publishing
Issue Date: 2017
ISSN: 0129-167X
Statement of
Antonio Alarcón, Finnur Lárusson
Abstract: Let X be a connected open Riemann surface. Let Y be an Oka domain in the smooth locus of an analytic subvariety of Cn, n ≥ 1, such that the convex hull of Y is all of Cn. Let O∗(X, Y ) be the space of nondegenerate holomorphic maps X → Y. Take a holomorphic 1-form θ on X, not identically zero, and let π : O∗(X, Y ) → H1(X, Cn) send a map g to the cohomology class of gθ. Our main theorem states that π is a Serre fibration. This result subsumes the 1971 theorem of Kusunoki and Sainouchi that both the periods and the divisor of a holomorphic form on X can be prescribed arbitrarily. It also subsumes two parametric h-principles in minimal surface theory proved by Forstneriˇc and L´arusson in 2016.
Keywords: Riemann surface; de Rham cohomology; minimal surface; holomorphic null curve; Serre fibration; convex integration; period-dominating spray
Rights: © World Scientific Publishing Company
DOI: 10.1142/S0129167X17400043
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Mathematical Sciences publications

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