Minimal Surfaces from a Complex Analytic Viewpoint

Alarcón, Antonio; Forstnerič, Franc; López, Francisco J.

doi:10.1007/978-3-030-69056-4

Cited by 38 publications

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“…A similar version of such an approximation theorem on admissible sets was recently obtained for conformal minimal immersions, null curves [10,11] and for holomorphic Legendrian curves [11,12].…”

Section: Introductionmentioning

confidence: 56%

Approximation theorems for Pascali systems

Drnovsek

Kuzman

2021

Complex Variables and Elliptic Equations

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Section: Introductionmentioning

confidence: 56%

Approximation theorems for Pascali systems

Drnovsek

Kuzman

2021

Complex Variables and Elliptic Equations

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“…We refer to [81,Subsect. 3.3] and [8,Sect. 3.10] for the discussion of this topic, also in the context of minimal surfaces, and for references to the earlier counterexamples for the disc X = D.…”

Section: Stein Manifolds With the Density Property And Oka Manifoldsmentioning

confidence: 99%

“…Yang's problem is an analogue of the Calabi-Yau problem for minimal surfaces in R n . We refer to [8,Chapter 7] for background and a discussion of recent results on this subject.…”

Section: Complete Complex Submanifoldsmentioning

confidence: 99%

“…In Section 9 we survey recent results concerning the problem of Paul Yang from 1977, asking whether there are bounded (metrically) complete complex submanifolds of C n . (This is holomorphic analogue of the Calabi-Yau problem concerning minimal surfaces in R n for n ≥ 3; see [8,Chapter 7] for the latter.) It has been discovered fairly recently that the ball of C n and, more generally, any pseudoconvex Runge domain in C n can be foliated by complete complex submanifolds of any codimension and with partial control of the topology of the leaves.…”

Section: Introductionmentioning

confidence: 99%

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The first thirty years of Andersen-Lempert theory

Forstnerič¹,

Kutzschebauch²

2021

Preprint

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In this paper we expose the impact of the fundamental discovery, made by Erik Andersén and László Lempert in 1992, that the group generated by shears is dense in the group of holomorphic automorphisms of a complex Euclidean space of dimensions n > 1. In three decades since its publication, their groundbreaking work led to the discovery of several new phenomena and to major new results in complex analysis and geometry involving Stein manifolds and affine algebraic manifolds with many automorphisms. The aim of this survey is to present the focal points of these developments, with a view towards the future. Dedicated to LászlóLempert in honour of his 70th birthday CONTENTS 1. Introduction 2. Stein manifolds with density properties 2.1. Density property 2.2. Volume density property 2.3. Relative density properties 2.4. Fibred density properties 2.5. Symplectic density property 3. Automorphisms with given jets 4. Fatou-Bieberbach domains 5. Twisted complex lines in C n and nonlinearizable automorphisms 6. Embedding open Riemann surfaces in C 2 7. Complex manifolds exhausted by Euclidean spaces 8. Stein manifolds with the density property and Oka manifolds 9. Complete complex submanifolds 10. An application in 3-dimensional topology 11. The recognition problem for complex Euclidean spaces References

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“…This led to an essentially optimal solution of the Calabi-Yau problem for minimal surfaces, originating in conjectures of Eugenio Calabi from 1965; see Theorems 3.5 and 3.6. This technique was also used in the construction of complete proper minimal surfaces in minimally convex domains of R (see [16,Chapter 8]).…”

mentioning

confidence: 99%

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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Minimal Surfaces from a Complex Analytic Viewpoint

Cited by 38 publications

References 0 publications

Approximation theorems for Pascali systems

Approximation theorems for Pascali systems

The first thirty years of Andersen-Lempert theory

Minimal surfaces in Euclidean spaces by way of complex analysis

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