Complete minimal surfaces densely lying in
arbitrary domains of ℝn

Alarcón, Antonio; Castro-Infantes, Ildefonso

doi:10.2140/gt.2018.22.571

Cited by 10 publications

(27 citation statements)

References 54 publications

(86 reference statements)

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“…Another recent application of the Riemann-Hilbert method is the construction of complete minimal surfaces lying densely in arbitrary domains of R n . The following result is due to Alarcón and Castro-Infantes [6].…”

Section: 2mentioning

confidence: 82%

New Complex Analytic Methods in the Theory of Minimal Surfaces: A Survey

Alarcón¹,

Forstnerič²

2018

J. Aust. Math. Soc.

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In this paper we survey recent developments in the classical theory of minimal surfaces in Euclidean spaces which have been obtained as applications of both classical and modern complex analytic methods; in particular, Oka theory, period dominating holomorphic sprays, gluing methods for holomorphic maps, and the Riemann-Hilbert boundary value problem. Emphasis is on results pertaining to the global theory of minimal surfaces, in particular, the Calabi-Yau problem, constructions of properly immersed and embedded minimal surfaces in R n and in minimally convex domains of R n , results on the complex Gauss map, isotopies of conformal minimal immersions, and the analysis of the homotopy type of the space of all conformal minimal immersions from a given open Riemann surface.

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Section: 2mentioning

confidence: 82%

New Complex Analytic Methods in the Theory of Minimal Surfaces: A Survey

Alarcón¹,

Forstnerič²

2018

J. Aust. Math. Soc.

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“…If n ≥ 5, then by Theorem 6.1 we may ensure that X is an embedding. Theorem 6.1 also yields a new proof of the following result due to Alarcón and the first author [1], which proves the existence of complete minimal surfaces densely lying in R n . Recall that the compact-open topology is the coarsest topology on C (R, R n ) containing all sets of the form [K, U] where K ⊂ R is compact and U ⊂ R n is open.…”

Section: Examples/applicationsmentioning

confidence: 76%

Carleman approximation by conformal minimal immersions and directed holomorphic curves

Castro-Infantes¹,

Chenoweth²

2020

Journal of Mathematical Analysis and Applications

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Let R be an open Riemann surface. In this paper we prove that every continuous function M → R n , n ≥ 3, defined on a divergent Jordan arc M ⊂ R can be approximated in the Carleman sense by conformal minimal immersions; thus providing a new generalization of Carleman's theorem. In fact, we prove that this result remains true for null curves and many other classes of directed holomorphic immersions for which the directing variety satisfies a certain flexibility property. Furthermore, the constructed immersions may be chosen to be complete or proper under natural assumptions on the variety and the continuous map.As a consequence we give an approximate solution to a Plateau problem for divergent Jordan curves in the Euclidean spaces.

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“…Their application led to the following result, which is a summary of several individual theorems. Parts (i), (ii) and (iv) are due to Alarcón, López, and myself [6,13,12] (the special case of (i) for = 3 was obtained beforehand in [19]), while (iii) was proved by Alarcón and Castro-Infantes [2,3]. Related results for conformal minimal surfaces of finite total curvature were given by Alarcón and López [18].…”

Section: Approximation Interpolation and General Position Theoremsmentioning

confidence: 79%

Minimal surfaces in Euclidean spaces by way of complex analysis

Forstnerič¹

2021

Preprint

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This is an expanded version of my plenary lecture at the 8th European Congress of Mathematics in Portorož on 23 June 2021. The main part of the paper is a survey of recent applications of complex-analytic techniques to the theory of conformal minimal surfaces in Euclidean spaces. New results concern approximation, interpolation, and general position properties of minimal surfaces, existence of minimal surfaces with a given Gauss map, and the Calabi-Yau problem for minimal surfaces. To be accessible to a wide audience, the article includes a self-contained elementary introduction to the theory of minimal surfaces in Euclidean spaces.

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Complete minimal surfaces densely lying in arbitrary domains of ℝn

Cited by 10 publications

References 54 publications

New Complex Analytic Methods in the Theory of Minimal Surfaces: A Survey

New Complex Analytic Methods in the Theory of Minimal Surfaces: A Survey

Carleman approximation by conformal minimal immersions and directed holomorphic curves

Minimal surfaces in Euclidean spaces by way of complex analysis

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