# NEW COMPLEX ANALYTIC METHODS IN THE THEORY OF MINIMAL SURFACES: A SURVEY

@article{Alarcn2018NEWCA, title={NEW COMPLEX ANALYTIC METHODS IN THE THEORY OF MINIMAL SURFACES: A SURVEY}, author={A. Alarc{\'o}n and F. Forstneri{\vc}}, journal={Journal of the Australian Mathematical Society}, year={2018}, volume={106}, pages={287 - 341} }

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… Expand

#### 19 Citations

Carleman approximation by conformal minimal immersions and directed holomorphic curves

- Mathematics
- 2020

Abstract 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… Expand

Proper superminimal surfaces of given conformal types in the hyperbolic four-space

- Mathematics
- 2020

Let $H^4$ denote the hyperbolic four-space. Given a bordered Riemann surface, $M$, we prove that every smooth conformal superminimal immersion $\overline M\to H^4$ can be approximated uniformly on… Expand

A strong parametric h-principle for complete minimal surfaces

- Mathematics
- 2021

We prove a parametric h-principle for complete nonflat conformal minimal immersions of an open Riemann surface M into Rn, n ≥ 3. It follows that the inclusion of the space of such immersions into the… Expand

Algebraic approximation and the Mittag-Leffler theorem for minimal surfaces

- Mathematics
- 2019

In this paper, we prove a uniform approximation theorem with interpolation for complete conformal minimal surfaces with finite total curvature in the Euclidean space $\mathbb{R}^n$ $(n\ge 3)$. As… Expand

The Calabi–Yau problem for Riemann surfaces with finite genus and countably many ends

- Mathematics
- 2019

In this paper, we show that if $R$ is a compact Riemann surface and $M=R\setminus\,\bigcup_i D_i$ is a domain in $R$ whose complement is a union of countably many pairwise disjoint smoothly bounded… Expand

Holomorphic Legendrian curves in projectivised cotangent bundles

- Mathematics
- 2018

We study holomorphic Legendrian curves in the standard complex contact structure on the projectivised cotangent bundle $X=\mathbb P(T^*Z)$ of a complex manifold $Z$ of dimension at least $2$. We… Expand

HOLOMORPHIC LEGENDRIAN CURVES IN CP AND SUPERMINIMAL SURFACES IN S4

- 2019

We obtain a Runge approximation theorem for holomorphic Legendrian curves and immersions in the complex projective 3-space CP, both from open and compact Riemann surfaces, and we prove that the space… Expand

Tropical approach to legendrian curves in $\mathbb{C}P^3$

- Mathematics
- 2016

The famous twistor construction tells us that harmonic surfaces in $S^4$ can be obtained as projections of complex legendrian curves in $\mathbb{C}P^3$. A mostly computational attempt to study the… Expand

A foliation of the ball by complete holomorphic discs

- Mathematics
- 2019

We show that the open unit ball $\mathbb{B}^n$ of $\mathbb{C}^n$ $(n>1)$ admits a nonsingular holomorphic foliation by complete properly embedded holomorphic discs.

Holomorphic approximation: the legacy of Weierstrass, Runge, Oka-Weil, and Mergelyan

- Mathematics
- 2018

In this paper we survey the theory of holomorphic approximation, from the classical nineteenth century results of Runge and Weierstrass, continuing with the twentieth century work of Oka and Weil,… Expand

#### References

SHOWING 1-10 OF 160 REFERENCES

New Complex Analytic Methods in the Study of Non-Orientable Minimal Surfaces in ℝⁿ

- Mathematics
- 2020

The aim of this work is to adapt the complex analytic methods originating in modern Oka theory to the study of non-orientable conformal minimal surfaces in $\mathbb{R}^n$ for any $n\ge 3$. These… Expand

Conformal properties in classical minimal surface theory

- Mathematics
- 2004

This is a survey of recent developments in the classical theory of minimal surfaces in R3 with an emphasis on the conformal properties of these surfaces such as recurrence and parabolicity. We cover… Expand

THE GAUSS MAP OF MINIMAL SURFACES

- Mathematics
- 2002

We give a new approach to the study of relations between the Gauss map and compactness properties for families of minimal surfaces in the Euclidean three space. In particular, we give a simple and… Expand

Holomorphic curves in complex spaces

- Mathematics
- 2006

We study the existence of topologically closed complex curves normalized by bordered Riemann surfaces in complex spaces. Our main result is that such curves abound in any noncompact complex space… Expand

A survey on classical minimal surface theory

- Mathematics
- 2012

Meeks and Perez present a survey of recent spectacular successes in classical minimal surface theory. The classification of minimal planar domains in three-dimensional Euclidean space provides the… Expand

Every bordered Riemann surface is a complete conformal minimal surface bounded by Jordan curves

- Mathematics
- 2015

In this paper we find approximate solutions of certain Riemann-Hilbert boundary value problems for minimal surfaces in $\mathbb{R}^n$ and null holomorphic curves in $\mathbb{C}^n$ for any $n\ge 3$.… Expand

Interpolation by conformal minimal surfaces and directed holomorphic curves

- Mathematics
- 2017

Let $M$ be an open Riemann surface and $n\ge 3$ be an integer. We prove that on any closed discrete subset of $M$ one can prescribe the values of a conformal minimal immersion $M\to\mathbb{R}^n$. Our… Expand

Modified defect relations for the gauss map of minimal surfaces, III

- Mathematics
- 1991

In [5], the author proved that the Gauss map of a nonflat complete minimal surface immersed in R 3 can omit at most four points of the sphere, and in [7] he revealed some relations between this… Expand

Null curves and directed immersions of open Riemann surfaces

- Mathematics
- 2014

In this paper we study holomorphic immersions of open Riemann surfaces into C^n whose derivative lies in a conical algebraic subvariety A of C^n that is smooth away from the origin. Classical… Expand

A Course in Minimal Surfaces

- Mathematics
- 2011

Minimal surfaces date back to Euler and Lagrange and the beginning of the calculus of variations. Many of the techniques developed have played key roles in geometry and partial differential… Expand