A conformal minimal immersion from an open Riemann surface into the Euclidean space ${\mathbb R}^{n}$, $n \geq 3$, is minimal if and only if it is harmonic. This fundamental condition characterizes minimal surfaces, formally defined as stationary points of the area functional. The simplest examples, known since the 18th century, are catenoid and helicoid, the real and imaginary parts of the holomorphic null curve called helicatenoid. The idea behind approximation and interpolation of minimal surfaces, our main goal, are classical theorems for holomorphic functions, although they need to be suitably adapted. Period dominating sprays, Morse theory and Gromov’s convex integration theory concerning the existence of paths with prescribed integrals enable us to find nearby maps with vanishing real periods, which define minimal surfaces by the Enneper-Weierstrass formula. It turns out that theorems of Mergelyan, Weierstrass and Mittag-Leffler type hold for conformal minimal immersions as well as more general holomorphic null curves. Additionally, such immersions can be chosen to be proper.
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