Every bordered Riemann surface is a complete conformal minimal surface bounded by Jordan curves: Figure. 5.1.

International Mathematics Research Notices

Forstnerič

2017

In this paper we find a holomorphic Darboux chart around any immersed noncompact holomorphic Legendrian curve in a complex contact manifold (X, ξ). By using such a chart, we show that every holomorphic Legendrian immersion R → X from an open Riemann surface can be approximated on relatively compact subsets of R by holomorphic Legendrian embeddings, and every holomorphic Legendrian immersion M → X from a compact bordered Riemann surface is a uniform limit of topological embeddings M ֒→ X such thatM ֒→ X is a complete holomorphic Legendrian embedding. We also establish a contact neighborhood theorem for isotropic Stein submanifolds in complex contact manifolds.

Section: Introduction and Main Resultsmentioning

confidence: 99%

Darboux Charts Around Holomorphic Legendrian Curves and Applications

International Mathematics Research Notices

Forstnerič

2017

“…These curves are closely related to minimal surfaces in R n since the real and the imaginary part of a null curve M → C n are flux-vanishing conformal minimal immersions M → R n (see e. g. Osserman [32]). The required tools in order to prove analogous results to Theorems 1.1 and 1.2 for holomorphic null curves have been provided recently in [10,4,5,1]. In this framework, the general position is embedded for n ≥ 3.…”

Section: 2mentioning

confidence: 97%

Complete minimal surfaces densely lying in arbitrary domains of ℝn

Castro-Infantes

2017

Geom. Topol.

In this paper we prove that, given an open Riemann surface M and an integer n ≥ 3, the set of complete conformal minimal immersions M → R n with X(M ) = R n forms a dense subset in the space of all conformal minimal immersions M → R n endowed with the compact-open topology. Moreover, we show that every domain in R n contains complete minimal surfaces which are dense on it and have arbitrary orientable topology (possibly infinite); we also provide such surfaces whose complex structure is any given bordered Riemann surface.Our method of proof can be adapted to give analogous results for non-orientable minimal surfaces in R n (n ≥ 3), complex curves in C n (n ≥ 2), holomorphic null curves in C n (n ≥ 3), and holomorphic Legendrian curves in C 2n+1 (n ∈ N).

“…It is perhaps worth mentioning to this respect that, if S is as in assertion (I) and F | Λ : Λ → C n is not proper, Theorem 1.3 provides complete S-immersions M → C n which are not proper maps; these seem to be the first known examples of such apart from the case when S is the null quadric. Let us emphasize that the particular geometry of A has allowed to construct complete null holomorphic curves in C n and minimal surfaces in R n with a number of different asymptotic behaviors (other than proper in space); see [13,8,4,5,3] and the references therein.…”

Section: Furthermorementioning

confidence: 99%

Interpolation by conformal minimal surfaces and directed holomorphic curves

Castro-Infantes

2019

Analysis & PDE

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.