Analysis & PDE

  • Anal. PDE
  • Volume 12, Number 2 (2019), 561-604.

Interpolation by conformal minimal surfaces and directed holomorphic curves

Antonio Alarcón and Ildefonso Castro-Infantes

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Let M be an open Riemann surface and n 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 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 n if the prescription of values is proper, and injective if n 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 n , including null curves.

Article information

Anal. PDE, Volume 12, Number 2 (2019), 561-604.

Received: 1 March 2018
Accepted: 1 May 2018
First available in Project Euclid: 9 October 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53A10: Minimal surfaces, surfaces with prescribed mean curvature [See also 49Q05, 49Q10, 53C42] 32E30: Holomorphic and polynomial approximation, Runge pairs, interpolation 32H02: Holomorphic mappings, (holomorphic) embeddings and related questions 53A05: Surfaces in Euclidean space

minimal surface directed holomorphic curve Weierstrass theorem Riemann surface Oka manifold


Alarcón, Antonio; Castro-Infantes, Ildefonso. Interpolation by conformal minimal surfaces and directed holomorphic curves. Anal. PDE 12 (2019), no. 2, 561--604. doi:10.2140/apde.2019.12.561.

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