Interpolation by conformal minimal surfaces and directed holomorphic curves

Alarcón, Antonio; Castro-Infantes, Ildefonso

doi:10.2140/apde.2019.12.561

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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 $n \geq 3$ 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 $n \geq 5$ 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

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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

Vol. 12, No. 2, 2019