New Complex Analytic Methods in the Theory of Minimal Surfaces: A Survey

Abstract

In this paper we survey recent developments in the classical theory of minimal surfaces in Euclidean spaces which have been obtained as applications of both classical and modern complex analytic methods; in particular, Oka theory, period dominating holomorphic sprays, gluing methods for holomorphic maps, and the Riemann-Hilbert boundary value problem. Emphasis is on results pertaining to the global theory of minimal surfaces, in particular, the Calabi-Yau problem, constructions of properly immersed and embedded minimal surfaces in R n and in minimally convex domains of R n , results on the complex Gauss map, isotopies of conformal minimal immersions, and the analysis of the homotopy type of the space of all conformal minimal immersions from a given open Riemann surface.

Approximation theorems for Pascali systems

“…Our first result is the following Mergelyan-type theorem. A similar version of such an approximation theorem on admissible sets was recently obtained for conformal minimal immersions, null curves [1,2] and for holomorphic Legendrian curves [7,2].…”

A foliation of the ball by complete holomorphic discs

“…A related result in higher dimension was obtained by Drinovec Drnovšek [10]. Parallel developments were made in minimal surface theory where the corresponding circle of questions is known as the Calabi-Yau problem; see the survey [4] and the preprint [5].…”

Approximation theorems for Pascali systems

“…A similar version of such an approximation theorem on admissible sets was recently obtained for conformal minimal immersions, null curves [10,11] and for holomorphic Legendrian curves [11,12].…”