Holomorphic differentials on open Riemann surfaces

Kusunoki, Yukio; Sainouchi, Yoshikazu

doi:10.1215/kjm/1250523693

Cited by 14 publications

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“…Among them, we emphasize the following ones:• A Gunning-Narasimhan type theorem for non-orientable conformal surfaces.• An existence theorem for non-orientable minimal surfaces in R 3 , with arbitrary conformal structure, properly projecting into a plane. • An existence result for non-orientable minimal surfaces in R 3 with arbitrary conformal structure and Gauss map omitting one projective direction.Finally, in a different line of applications, we prove an extension of the classical Gunning-Narasimhan theorem [18] (see also [21]); more specifically, we show that, for any open Riemann surface N and any antiholomorphic involution I : N → N without fixed points, there exist holomorphic 1-forms ϑ on N with I * ϑ = ϑ and prescribed periods and canonical divisor (see Theorem 6.4).Outline of the paper. The necessary notation and background on non-orientable minimal surfaces in R 3 is introduced in Sec.…”

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confidence: 86%

“…Finally, in a different line of applications, we prove an extension of the classical Gunning-Narasimhan theorem [18] (see also [21]); more specifically, we show that, for any open Riemann surface N and any antiholomorphic involution I : N → N without fixed points, there exist holomorphic 1-forms ϑ on N with I * ϑ = ϑ and prescribed periods and canonical divisor (see Theorem 6.4).…”

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Approximation theory for nonorientable minimal surfaces and applications

Alarcón

López

2015

Geom. Topol.

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We prove a version of the classical Runge and Mergelyan uniform approximation theorems for non-orientable minimal surfaces in Euclidean 3-space R 3 . Then, we obtain some geometric applications. Among them, we emphasize the following ones:• A Gunning-Narasimhan type theorem for non-orientable conformal surfaces.• An existence theorem for non-orientable minimal surfaces in R 3 , with arbitrary conformal structure, properly projecting into a plane. • An existence result for non-orientable minimal surfaces in R 3 with arbitrary conformal structure and Gauss map omitting one projective direction.Finally, in a different line of applications, we prove an extension of the classical Gunning-Narasimhan theorem [18] (see also [21]); more specifically, we show that, for any open Riemann surface N and any antiholomorphic involution I : N → N without fixed points, there exist holomorphic 1-forms ϑ on N with I * ϑ = ϑ and prescribed periods and canonical divisor (see Theorem 6.4).Outline of the paper. The necessary notation and background on non-orientable minimal surfaces in R 3 is introduced in Sec. 2. In Sec. 3 we describe the compact subsets involved in the Mergelyan type approximation, and define the notion of conformal non-orientable minimal immersion from such a subset into R 3 . In Sec. 4 we prove several preliminary approximation results that flatten the way to the proof of the main theorem in Sec. 5. Finally, the applications are derived in Sec. 6. PreliminariesLet · denote the Euclidean norm in K n (K = R or C). Given a compact topological space K and a continuous map f : K → K n , we denote bythe maximum norm of f on K. The corresponding space of continuous functions on K will be endowed with the C 0 topology associated to · 0,K . Given a topological surface N, we denote by bN the (possibly non-connected) 1dimensional topological manifold determined by its boundary points. Given a subset A ⊂ N, we denote by A • and A the interior and the closure of A in N , respectively. Open connected subsets of N \ bN will be called domains of N , and those proper connected Proof. We begin with the following assertion.Claim 5.7. There exists a connected I-admissible subsetŜ ⊂ N such that RŜ = R S and CŜ ⊃ C S ; that is to say,Ŝ is constructed by adding a finite family of Jordan arcs to S.Proof. If S is connected chooseŜ = S.Assume that S is not connected. We distinguish the following two cases. (See Remark 3.1 and Def. 2.2 and 2.3 for notation.) Case 1. S is a connected subset of the non-orientable Riemann surface N . In this situation, any tubular neighborhood of S is an orientable surface. Then, take any Jordan arc γ ⊂ N with end points in bR S and otherwise disjoint from S , such that any tubular neighborhood

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Approximation theory for nonorientable minimal surfaces and applications

Alarcón

López

2015

Geom. Topol.

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“…Our goal in this paper is to show that a closed holomorphic 1-form without zeros can be chosen in every cohomology class. (See also [14] for open Riemann surfaces.) Theorem 1.…”

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confidence: 99%

Closed holomorphic 1-forms without zeros on Stein manifolds

Majcen

2007

Math. Z.

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“…(b) Take Y = C * ⊂ C, let θ be a holomorphic 1-form on X with divisor D, and apply Corollary 2(b). (The existence of such a form does not rely on [10]. It is an immediate consequence of [4,Satz 4].…”

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“…Gunning and Narasimhan showed that the zero class contains a holomorphic form with no zeros [9]. In other words, there is a holomorphic immersion X → C. Kusunoki and Sainouchi generalised these two theorems and proved that both the periods and the divisor of a holomorphic form on X can be prescribed arbitrarily [10,Theorem 1].…”

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Representing de Rham cohomology classes on an open Riemann surface by holomorphic forms

Alarcón

Lárusson

2017

Int. J. Math.

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Abstract. Let X be a connected open Riemann surface. Let Y be an Oka domain in the smooth locus of an analytic subvariety of C n , n ≥ 1, such that the convex hull of Y is all of C n . 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 ) → H 1 (X, C n ) 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 Forstnerič and Lárusson in 2016.We start by recalling three theorems from the early decades of modern Riemann surface theory. Behnke and Stein proved that the periods of a holomorphic form on an open Riemann surface X can be prescribed arbitrarily [3, Satz 10] (see also [5, Theorem 28.6]). In other words, every class in the cohomology group H 1 (X, C) contains a holomorphic form. Gunning and Narasimhan showed that the zero class contains a holomorphic form with no zeros [9]. In other words, there is a holomorphic immersion X → C. Kusunoki and Sainouchi generalised these two theorems and proved that both the periods and the divisor of a holomorphic form on X can be prescribed arbitrarily [10, Theorem 1].Our main result subsumes the theorem of Kusunoki and Sainouchi as a very special case. It also subsumes different and much more recent results from the theory of minimal surfaces, which we shall now describe. Let M * (X, R n ) denote the space (with the compact-open topology) of conformal minimal immersions X → R n , n ≥ 3, that are nonflat in the sense that the image of X is not contained in any affine 2-plane in R n . Some such immersions are obtained as the real parts of holomorphic null curves, that is, holomorphic immersions X → C n directed by the null quadric A = {(z 1 , . . . , z n ) ∈ C n : z 2 1 + · · · + z 2 n = 0}. We denote by N * (X, C n ) the space of holomorphic null curves that are nonflat, meaning that the image of X is not contained in any affine complex line in C n .

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Holomorphic differentials on open Riemann surfaces

Cited by 14 publications

References 5 publications

Approximation theory for nonorientable minimal surfaces and applications

Approximation theory for nonorientable minimal surfaces and applications

Closed holomorphic 1-forms without zeros on Stein manifolds

Representing de Rham cohomology classes on an open Riemann surface by holomorphic forms

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