Work

Holomorphic null curves in the special linear group

Year: 2025

Type: article

Abstract: In this paper we develop the theory of approximation for holomorphic null curves in the special linear group SL 2 ( C ) {\operatorname {SL}}_2(\mathbb {C}) . In particular, we establish Runge, Mergelyan, Mittag-Leffler, and Carleman type theorems for the family of holomorphic null immersions M → SL 2 ( C ) M\to {\operatorname {SL}}_2(\mathbb {C}) from any open Riemann surface M M . Our results include jet interpolation of Weierstrass type and approximation by embeddings, as well as global conditions on the approximating curves. As application, we show that every open Riemann surface admits a proper holomorphic null embedding into SL 2 ( C ) {\operatorname {SL}}_2(\mathbb {C}) , and hence also a proper conformal immersion of constant mean curvature 1 1 into hyperbolic 3-space. This settles a problem posed by Alarcón and Forstnerič in 2015.

Source: Transactions of the American Mathematical Society

DOI: 10.1090/tran/9595

Authors: Antonio Alarcón, Jorge Hidalgo

Institution: Universidad de Granada

Language: English

Topic: Geometry and complex manifolds

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funder: Agencia Estatal de Investigación

Awards: G3878682347, G4571679087, G567365627