Vol. 12, No. 2, 2019

 Download this article For screen For printing
 Recent Issues
 The Journal About the Journal Editorial Board Editors’ Interests Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN: 1948-206X (e-only) ISSN: 2157-5045 (print) Author Index To Appear Other MSP Journals
Interpolation by conformal minimal surfaces and directed holomorphic curves

Antonio Alarcón and Ildefonso Castro-Infantes

Vol. 12 (2019), No. 2, 561–604
DOI: 10.2140/apde.2019.12.561
Abstract

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 $M\to {ℝ}^{n}$. Our result also ensures jet-interpolation of given finite order, and hence, in particular, one may in addition prescribe the values of the generalized Gauss map. Furthermore, the interpolating immersions can be chosen to be complete, proper into ${ℝ}^{n}$ if the prescription of values is proper, and injective if $n\ge 5$ and the prescription of values is injective. We may also prescribe the flux map of the examples.

We also show analogous results for a large family of directed holomorphic immersions $M\to {ℂ}^{n}$, including null curves.

 Dedicated to Franc Forstnerič on the occasion of his sixtieth birthday
Keywords
minimal surface, directed holomorphic curve, Weierstrass theorem, Riemann surface, Oka manifold
Mathematical Subject Classification 2010
Primary: 53A10, 32E30, 32H02, 53A05
Milestones
Received: 1 March 2018
Accepted: 1 May 2018
Published: 14 August 2018
Authors
 Antonio Alarcón Departamento de Geometría y Topología Instituto de Matemáticas (IEMath-GR) Universidad de Granada Campus de Fuentenueva Granada Spain Ildefonso Castro-Infantes Departamento de Geometría y Topología Instituto de Matemáticas (IEMath-GR) Universidad de Granada Campus de Fuentenueva Granada Spain