NEW COMPLEX ANALYTIC METHODS IN THE THEORY OF MINIMAL SURFACES: A SURVEY

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 $\mathbb{R}^{n}$ and in minimally convex domains of $\mathbb{R}^{n}$ , 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.

Every conformal minimal surface in ℝ3 is isotopic to the real part of a holomorphic\break null curve

We show that for every conformal minimal immersion {u:M\to\mathbb{R}^{3}} from an open Riemann surface M to {\mathbb{R}^{3}} there exists a smooth isotopy {u_{t}:M\to\mathbb{R}^{3}} ( {t\in[0,1]} ) of conformal minimal immersions, with {u_{0}=u} , such that {u_{1}} is the real part of a holomorphic null curve {M\to\mathbb{C}^{3}} (i.e. {u_{1}} has vanishing flux). If furthermore u is nonflat, then {u_{1}} can be chosen to have any prescribed flux and to be complete.