Corpus ID: 218502279

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

@article{Forstneri2020ProperSS,
title={Proper superminimal surfaces of given conformal types in the hyperbolic four-space},
author={F. Forstneri{\vc}},
journal={arXiv: Differential Geometry},
year={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 compacts in $M$ by proper conformal superminimal immersions $M\to H^4$. In particular, $H^4$ contains properly immersed conformal superminimal surfaces normalised by any given open Riemann surface of finite topological type without punctures. The proof uses the analysis of holomorphic Legendrian… Expand
1 Citations
Developments in Oka theory since 2017.
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#### References

SHOWING 1-10 OF 43 REFERENCES
NEW COMPLEX ANALYTIC METHODS IN THE THEORY OF MINIMAL SURFACES: A SURVEY
• Mathematics
• Journal of the Australian Mathematical Society
• 2018
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 analyticExpand
Minimal surfaces in minimally convex domains
• Mathematics
• 2015
In this paper, we prove that every conformal minimal immersion of a compact bordered Riemann surface $M$ into a minimally convex domain $D\subset \mathbb{R}^3$ can be approximated, uniformly onExpand
Darboux charts around holomorphic Legendrian curves and applications
• Mathematics
• 2017
In this paper we find a holomorphic Darboux chart along any immersed noncompact holomorphic Legendrian curve in a contact complex manifold $(X,\xi)$. As an application we show that every holomorphicExpand
The Calabi–Yau Property of Superminimal Surfaces in Self-Dual Einstein Four-Manifolds
In this paper, we show that if ( X ,  g ) is an oriented four-dimensional Einstein manifold which is self-dual or anti-self-dual then superminimal surfaces in X of appropriate spin enjoy theExpand
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 boundedExpand
HOLOMORPHIC LEGENDRIAN CURVES IN CP AND SUPERMINIMAL SURFACES IN S4
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 spaceExpand
On superminimal surfaces
Using the Cartan method O. Boruvka (see [B1], [B2]) studied superminimal surfaces in four-dimensional space forms. In particular, he described locally the family of all superminimal surfaces andExpand
Bounded holomorphic functions on finite Reimann surfaces
In ?2, we use this theorem to establish an analogous result in the setting of finite open Riemann surfaces. ??3 and 4 consider certain questions which arise naturally in the course of the proof ofExpand
Every bordered Riemann surface is a complete proper curve in a ball
• Mathematics
• 2013
We prove that every bordered Riemann surface admits a complete proper holomorphic immersion into a ball of $$\mathbb C ^2$$, and a complete proper holomorphic embedding into a ball of \mathbb CExpand
BOUNDED HOLOMORPHIC FUNCTIONS ON FINITE RIEMANN SURFACES
In §2, we use this theorem to establish an analogous result in the setting of finite open Riemann surfaces. §§3 and 4 consider certain questions which arise naturally in the course of the proof ofExpand