Aproksimacija v kompleksni kontaktni geometriji

In this thesis we treat several new results, obtained by the author, regarding the approximation properties of holomorphic Legendrian curves lying in the complex Euclidean space equipped with the standard contact structure. We first generalise the Runge-type approximation result by Alarcón, Forstnerič and López to include jet-interpolation at a prescribed closed discrete set of points in the given open Riemann surface, showing in particular that any open Riemann surface may be properly embedded as a Legendrian curve in the standard complex Euclidean space hitting every point in a prescribed divergent sequence. We next treat a Carleman-type approximation result for Legendrian curves, defined on a certain type of closed sets in open Riemann surfaces, to show in particular that any smooth proper isotropic embedding of the real number line into the standard complex Euclidean space may be approximated by a proper holomorphic Legendrian embedding of a chosen open Riemann surface into the said Euclidean space. We also apply the interpolation result to Legendrian curves in the complex special linear group of degree two. We next turn our attention to the case when the image of a given Legendrian curve, defined on a compact bordered Riemann surface, lies in a convex domain in standard complex Euclidean space. We show that the curve in question may be approximated with a proper and complete one, albeit the approximant may only be defined on the interior of the given Riemann surface in the case when the convex domain is either unbounded or has a nonsmooth boundary. Using the same methods, we show that any convex domain contains an injectively immersed almost proper holomorphic Legendrian curve with dense image. Lastly, we treat regular Legendrian curves in the complex Euclidean space defined on smooth affine curves. We show that the curves of this type satisfy a Mergelyan-type approximation property, moreover, we are also able to make the approximants proper.