Null curves and directed immersions of open Riemann surfaces

Alarcón, Antonio; Forstnerič, Franc

doi:10.1007/s00222-013-0478-8

Cited by 48 publications

(180 citation statements)

References 53 publications

(34 reference statements)

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“…The main idea is contained in the proof of [4,Lemma 5.1]; we outline the main idea for the sake of readability. Since σ is nondegenerate on I, there are points x 1 , x 2 , x 3 ∈I and holomorphic vector fields V 1 , V 2 , V 3 on C 3 which are tangential to the quadric A (2.2) such that the vectors V j (σ(x j )) ∈ C 3 for j = 1, 2, 3 are a complex basis of C 3 .…”

Section: Lemma 36 (Period Dominating Sprays Of Loops) Letmentioning

confidence: 99%

Every conformal minimal surface in ℝ³ is isotopic to the real part of a holomorphic\break null curve

Alarcón

Forstnerič

2015

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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We show that for every conformal minimal immersion u : M → R 3 from an open Riemann surface M to R 3 there exists a smooth isotopy u t : M → R 3 (t ∈ [0, 1]) of conformal minimal immersions, with u 0 = u, such that u 1 is the real part of a holomorphic null curve M → C 3 (i.e. u 1 has vanishing flux). If furthermore u is nonflat then u 1 can be chosen to have any prescribed flux and to be complete. Keywords Riemann surfaces, minimal surfaces, holomorphic null curves. MSC (2010):53C42; 32B15, 32H02, 53A10. The main resultsLet M be a smooth oriented surface. A smooth immersion u = (u 1 , u 2 , u 3 ) : M → R 3 is minimal if its mean curvature vanishes at every point. The requirement that an immersion u be conformal uniquely determines a complex structure on M . Finally, a conformal immersion is minimal if and only if it is harmonic:of an open Riemann surface to C 3 is said to be a null curve if its differential dF = (dF 1 , dF 2 , dF 3 ) satisfies the equationThe real and the imaginary part of a null curve M → C 3 are conformal minimal immersions M → R 3 . Conversely, the restriction of a conformal minimal immersion u : M → R 3 to any simply connected domain Ω ⊂ M is the real part of a holomorphic null curve Ω → C 3 ; u is globally the real part of a null curve if and only if its conjugate differential d c u satisfiesThis period vanishing condition means that u admits a harmonic conjugate v, and F = u + ıv : M → C 3 (ı = √ −1) is then a null curve.In this paper we prove the following result which further illuminates the connection between conformal minimal surfaces in R 3 and holomorphic null curves in C 3 . We shall systematically use the term isotopy instead of the more standard regular homotopy when talking of smooth 1-parameter families of immersions. The analogous result holds for minimal surfaces in R n for any n ≥ 3, and the tools used in the proof are available in that setting as well. On a compact bordered Riemann surface we also have an up to the boundary version of the same result (cf. Theorem 4.1).

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Section: Lemma 36 (Period Dominating Sprays Of Loops) Letmentioning

confidence: 99%

Every conformal minimal surface in ℝ³ is isotopic to the real part of a holomorphic\break null curve

Alarcón

Forstnerič

2015

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…Joining together the methods in the proof of the above result and the Mergelyan theorem for null curves [8] (see also [6]), we also get the following result. …”

Section: Theorem 18 ([7 Theorem 14]) Every Bordered Riemann Surfamentioning

confidence: 76%

“…The key to the proof is that the general position of null curves in C 3 is embedded. In fact, Theorem 1.1 easily follows from the existence of complete properly immersed null curves in convex domains of C 3 [9] and the following desingularization result from [6]. Since complex submanifolds of complex Euclidean spaces are area minimizing [38], the Calabi-Yau problem is closely related to a question, posed by Yang [98,99] in 1977, whether there exist complete bounded complex submanifolds of C n for n > 1.…”

Section: On Null Curves In C 3 and Minimal Surfaces In Rmentioning

confidence: 98%

“…In order to force the image to lie in SL 2 (C), one must add another equation expressing the condition that the tangent vector to the curve is also tangent to SL 2 (C) (as a submanifold of C 4 ). Unfortunately the resulting system of equations is no longer autonomous (the second equation depends on the point in space), and hence the methods of [6] do not apply.…”

Section: On Null Curves In Sl 2 (C) and Bryant Surfaces In Hmentioning

confidence: 99%

“…

Abstract In this paper we survey some recent contributions by the authors [5,6,7] to the theory of null holomorphic curves in the complex Euclidean space C 3 , as well as their applications to null holomorphic curves in the special linear group SL 2 (C), minimal surfaces in the Euclidean space R 3 , and constant mean curvature one surfaces (Bryant surfaces) in the hyperbolic space H 3 . The paper is an expanded version of the lecture given by the second named author at the Abel Symposium 2013 in Trondheim.

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Complex Geometry and Dynamics

Fornæss¹,

Irgens²,

Wold³

2015

Abel Symposia

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In this paper we survey some recent contributions by the authors [5,6,7] to the theory of null holomorphic curves in the complex Euclidean space C 3 , as well as their applications to null holomorphic curves in the special linear group SL 2 (C), minimal surfaces in the Euclidean space R 3 , and constant mean curvature one surfaces (Bryant surfaces) in the hyperbolic space H 3 . The paper is an expanded version of the lecture given by the second named author at the Abel Symposium 2013 in Trondheim.

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Complete Minimal Surfaces of Finite Total Curvature

Alarcón

Forstnerič

López

2021

Springer Monographs in Mathematics

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Let M be a Riemann surface biholomorphic to an affine algebraic curve. We show that the inclusion of the space ℜNC * (M, C n ) of real parts of nonflat proper algebraic null immersions M → C n , n ≥ 3, into the space CMI * (M, R n ) of complete nonflat conformal minimal immersions M → R n of finite total curvature is a weak homotopy equivalence. We also show that the (1, 0)-differential ∂, mapping CMI * (M, R n ) or ℜNC * (M, C n ) to the space A 1 (M, A) of algebraic 1-forms on M with values in the punctured null quadric A ⊂ C n \ {0}, is a weak homotopy equivalence. Analogous results are obtained for proper algebraic immersions M → C n , n ≥ 2, directed by a flexible or algebraically elliptic punctured cone in C n \ {0}.

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Null curves and directed immersions of open Riemann surfaces

Cited by 48 publications

References 53 publications

Every conformal minimal surface in ℝ³ is isotopic to the real part of a holomorphic\break null curve

Every conformal minimal surface in ℝ³ is isotopic to the real part of a holomorphic\break null curve

Complex Geometry and Dynamics

Complete Minimal Surfaces of Finite Total Curvature

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Null curves and directed immersions of open Riemann surfaces

Cited by 48 publications

References 53 publications

Every conformal minimal surface in ℝ3 is isotopic to the real part of a holomorphic\break null curve

Every conformal minimal surface in ℝ3 is isotopic to the real part of a holomorphic\break null curve

Complex Geometry and Dynamics

Complete Minimal Surfaces of Finite Total Curvature

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Product

Resources

About

Every conformal minimal surface in ℝ³ is isotopic to the real part of a holomorphic\break null curve

Every conformal minimal surface in ℝ³ is isotopic to the real part of a holomorphic\break null curve