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.

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).

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This article is motivated by a problem posed by David A. Singer in 1999 and by the classical Euler elastic curves. We study spacelike and timelike curves in the Lorentz-Minkowski plane 𝕃2 whose curvature is expressed in terms of the Lorentzian pseudodistance to fixed geodesics. In this way, we get a complete description of all the elastic curves in 𝕃2 and provide the Lorentzian versions of catenaries and grim-reaper curves. We show several uniqueness results for them in terms of their geometric linear momentum. In addition, we are able to get arc-length parametrizations of all the aforementioned curves and they are depicted graphically.