Every bordered Riemann surface is a complete conformal minimal surface bounded by Jordan curves

Abstract

In this paper we find approximate solutions of certain Riemann-Hilbert boundary value problems for minimal surfaces in ${\mathbb{R}}^n$ and null holomorphic curves in ${\mathbb{C}}^n$ for any $n\ge 3$. With this tool in hand we construct complete conformally immersed minimal surfaces in ${\mathbb{R}}^n$ which are normalized by any given bordered Riemann surface and have Jordan boundaries. We also furnish complete conformal proper minimal immersions from any given bordered Riemann surface to any smoothly bounded, strictly convex domain of ${\mathbb{R}}^n$ which extend continuously up to the boundary; for $n\ge 5$ we find embeddings with these properties.

Proc. London Math. Soc., in press

References in corpus (1)

• Bordered Riemann surfaces in C^2

Cited by in corpus (13)

• Holomorphic Legendrian curves
• New complex analytic methods in the theory of minimal surfaces: a survey
• Darboux charts around holomorphic Legendrian curves and applications
• The Calabi-Yau property of superminimal surfaces in self-dual Einstein four-manifolds
• Complete minimal surfaces densely lying in arbitrary domains of ${\mathbb{R}}^n$
• Holomorphic Legendrian curves in ${\mathbb{CP}}^3$ and superminimal surfaces in ${\mathbb{S}}^4$
• Proper holomorphic maps in Euclidean spaces avoiding unbounded convex sets
• The Oka principle for holomorphic Legendrian curves in ${\mathbb{C}}^{2n+1}$
• Embedded complex curves in the affine plane
• Flexible domains for minimal surfaces in Euclidean spaces
• Interpolation by complete minimal surfaces whose Gauss map misses two points
• Minimal surfaces with symmetries
• On the principal eigenvalue of the truncated Laplacian, and submanifolds with bounded mean curvature