2012
DOI: 10.4310/jdg/1335273387
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Minimal surfaces in $\mathbb{R}^3$ properly projecting into $\mathbb{R}^2$

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Cited by 40 publications
(84 citation statements)
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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%
“…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%
“…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%
“…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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