2017/11/30 by Antonio Alarcón, Antonio Alarcon, Franc Forstnerič +1
Mathematics · #Algebra over a field #Algebraic and Geometric Analysis #Conformal map #Euclidean geometry #Euclidean space #Geometric Analysis and Curvature Flows #Geometry #Holomorphic and Operator Theory #Holomorphic function #Homotopy #Mathematical analysis #Mathematics #Minimal surface #Pure mathematics #Riemann surface #Surface (topology) #math.CV #math.DG
paper · pdf · doi:10.1017/s1446788718000125
published as J. Aust. Math. Soc., 106:3 (2019), 287-341 · To appear in J. Aust. Math. Soc
arxiv created 2018/03/18 · openalex publication_date 2018/08/23 · arxiv updated 2019/10/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 ℝn and in minimally convex domains of ℝn , 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.