Carleman approximation by conformal minimal immersions and directed holomorphic curves

Abstract

Let R be an open Riemann surface. In this paper we prove that every continuous function M → R n , n ≥ 3, defined on a divergent Jordan arc M ⊂ R can be approximated in the Carleman sense by conformal minimal immersions; thus providing a new generalization of Carleman's theorem. In fact, we prove that this result remains true for null curves and many other classes of directed holomorphic immersions for which the directing variety satisfies a certain flexibility property. Furthermore, the constructed immersions may be chosen to be complete or proper under natural assumptions on the variety and the continuous map.As a consequence we give an approximate solution to a Plateau problem for divergent Jordan curves in the Euclidean spaces.

Approximation theorems for Pascali systems

“…An analogue of this statement was recently proved for conformal minimal immersions [13]. We provide a version for solutions of Pascali systems.…”

Approximation theorems for Pascali systems

“…An analogue of this statement was recently proved for conformal minimal immersions [5]. We provide a version for solutions of Pascali systems.…”

A strong parametric h-principle for complete minimal surfaces

“…• The space CMI c nf (M, R n ) of complete nonflat conformal minimal immersions M → R n is dense (with respect to the compact-open topology) in the space CMI nf (M, R n ) of all nonflat conformal minimal immersions ([8, Theorem 7.1]; the case of n = 3 follows from [13,Theorem 5.6], which slightly predates the introduction of Oka theory in minimal surface theory). In more recent work, the density theorem has been strengthened to Mergelyan and Carleman approximation theorems including Weierstrass interpolation and other additional features (see [1], [10,Section 3.9], and [17]).…”