Minimal surfaces in $\mathbb{R}^3$ properly projecting into $\mathbb{R}^2$

Alarcón, Antonio; López, Francisco J.

doi:10.4310/jdg/1335273387

Cited by 38 publications

(84 citation statements)

References 21 publications

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“…We remark that, for null curves in C 3 , Runge and Mergelyan theorems were proved by Alarcón and López [AL2]. Their analysis depends on the Weierstrass representation of a null curve, a tool that is not available in the general situation considered here.…”

Section: Resultsmentioning

confidence: 88%

Null curves and directed immersions of open Riemann surfaces

2013

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Let M be an open Riemann surface and A be the punctured cone in C n \ {0} on a smooth projective variety Y in P n−1 . Recently, Runge approximation theorems with interpolation for holomorphic immersions M → C n , directed by A, have been proved under the assumption that A is an Oka manifold. We prove analogous results in the algebraic setting, for regular immersions directed by A from a smooth affine curve M into C n . The Oka property is naturally replaced by the stronger assumption that A is algebraically elliptic, which it is if Y is uniformly rational. Under this assumption, a homotopy-theoretic necessary and sufficient condition for approximation and interpolation emerges. We show that this condition is satisfied in many cases of interest.

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Section: Resultsmentioning

confidence: 88%

Null curves and directed immersions of open Riemann surfaces

2013

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“…The following definition of a conformal minimal immersion of an admissible subset emulates the spirit of the concept of marked immersion [6] and provides the natural initial objects for the Mergelyan approximation by conformal minimal immersions. …”

Section: H-runge Approximation Theorem For Conformal Minimal Immersionsmentioning

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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“…The latter part of the former item in the above theorem was already proven in [8] where also complete bounded immersed null curves in SL 2 (C) with arbitrary topology were given. Complete bounded immersed simply connected null holomorphic curves in SL 2 (C), hence complete bounded simply-connected Bryant surfaces in H 3 , were provided in [40,70].…”

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

confidence: 87%

“…The first such examples were provided only very recently by Alarcón and López [9] who constructed complete null curves with arbitrary topology properly immersed in any given convex domain of C 3 ; this answers a question by Martín, Umehara, and Yamada [70,Problem 1]. Their method, which is different from Nadirashvili's one, relies on a RungeMergelyan type theorem for null curves in C 3 [8], a new and powerful tool that gave rise to a number of constructions of both minimal surfaces in R 3 and null curves in C 3 (see [8,2,9,10,12]). Very recently, Ferrer, Martín, Umehara, and Yamada [40] showed that Nadirashvili's method can be adapted to null curves, giving an alternative proof of the existence of complete bounded null discs in C 3 .…”

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

confidence: 99%

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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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Minimal surfaces in $\mathbb{R}^3$ properly projecting into $\mathbb{R}^2$

Cited by 38 publications

References 21 publications

Null curves and directed immersions of open Riemann surfaces

Null curves and directed immersions of open Riemann surfaces

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

Complex Geometry and Dynamics

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Minimal surfaces in $\mathbb{R}^3$ properly projecting into $\mathbb{R}^2$

Cited by 38 publications

References 21 publications

Null curves and directed immersions of open Riemann surfaces

Null curves and directed immersions of open Riemann surfaces

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

Complex Geometry and Dynamics

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