Interpolation by conformal minimal surfaces and directed holomorphic curves

Abstract

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 → 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 ≥ 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 → C n , including null curves.

New Complex Analytic Methods in the Theory of Minimal Surfaces: A Survey

“…Proceeding as before, we obtain a new spray of the same type with h j,k ∈ O(M ). It is elementary to see that the map (p, t) → φ j,k g(p)h j,k (p)t (f (p)) is tangent to f to order r at every point p ∈ P (see [5,Lemma 2.2]); hence the same holds for their composition Φ f . Remark 3.3.…”

“…This is for instance the case of the López-Ros deformation for minimal surfaces X : M → R 3 (see [80]) which amounts to multiplying the complex Gauss map g X by a nowhere vanishing holomorphic function, subject to suitable period vanishing conditions on the Weierstrass data. Its main shortcoming is that one needs the initial conformal minimal immersion X already defined everywhere on M ; let us point out that, at that time, few open Riemann surfaces were known to be the underlying complex structure of a minimal surface in Theorem 4.1 is a compilation of results from the paper [23] by Alarcón and López where the existence and approximation was proved for conformal minimal immersions into R 3 , the paper [18] by the authors and López where the same was done in any dimension n ≥ 3, and the paper [5] by Alarcón and Castro-Infantes where interpolation was added.…”

“…Theorem 4.1 also admits a version in which all but two components of the initial immersion are preserved. The next theorem is a compilation of results from [5,11,18,23]. ) Assume in addition that X = (X 1 , .…”

Carleman approximation by conformal minimal immersions and directed holomorphic curves

“…This section is devoted to proving Theorem 1.4 concerning properness of the approximating maps. Here we adapt the ideas that appear in [8,2] to our particular case. Theorem 1.4 is going to be proven using a recursive process similar to that used to prove our previous results.…”

“…For every S-immersion F : S → C n (see Def. 2.3) and every positive continuous function ǫ : S → R + , there is an injective S-immersion F : R → C n such that needed to prove our gluing lemma, Lemma 3.1, are provided by [2,3,6]. In particular, we find [2, Lemma 3.3] especially useful.…”