Interpolation by conformal minimal surfaces and directed holomorphic curves

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 $M\to {\mathbb{R}}^n$. 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 ${\mathbb{R}}^n$ if the prescription of values is proper, and one-to-one if $n\ge 5$ and the prescription of values is one-to-one. We may also prescribe the flux map of the examples. We also show analogous results for a large family of directed holomorphic immersions $M\to {\mathbb{C}}^n$, including null curves.

To appear in Analysis & PDE

References in corpus (2)

• Complete minimal surfaces densely lying in arbitrary domains of ${\mathbb{R}}^n$
• Every meromorphic function is the Gauss map of a conformal minimal surface

Cited by in corpus (5)

• New complex analytic methods in the theory of minimal surfaces: a survey
• Carleman approximation by conformal minimal immersions and directed holomorphic curves
• Holomorphic Legendrian curves in ${\mathbb{CP}}^3$ and superminimal surfaces in ${\mathbb{S}}^4$
• Complete CMC-1 surfaces in hyperbolic space with arbitrary complex structure
• Interpolation by complete minimal surfaces whose Gauss map misses two points