Complete Embedded Minimal N-submanifolds in C

A classical problem in the theory of minimal submanifolds of Euclidean spaces is to study the existence of a minimal submanifold with a prescribed behavior at infinity, or to determine from the asymptotes the geometry of the whole submanifold. Beyond the intrinsic interest of these questions, they are also of crucial importance when studying the possible singularities of minimal submanifolds in general Riemannian manifold. When studying minimal surfaces, i.e. two dimensional submanifolds, the standard tool to solve these problems is given by the Weierstrass representation formula which relates the geometry of the minimal surface to complex analytic properties of holomorphic one-forms on Riemann surfaces. Recently, gluing technics have been developed and have provided an abundant number of new examples of minimal hypersurfaces in Euclidean space.