Flexible domains for minimal surfaces in Euclidean spaces

Abstract

In this paper we introduce and investigate a new notion of flexibility for domains in Euclidean spaces ${\mathbb{R}}^n$ for $n\ge 3$ in terms of minimal surfaces which they contain. A domain $\Omega$ in ${\mathbb{R}}^n$ is said to be flexible if every conformal minimal immersion $U\to \Omega$ from a Runge domain $U$ in an open conformal surface $M$ can be approximated uniformly on compacts, with interpolation on any given finite set, by conformal minimal immersion $M\to \Omega$. Together with hyperbolicity phenomena considered in recent works, this extends the dichotomy between flexibility and rigidity from complex analysis to minimal surface theory.

arXiv commentJ. Math. Anal. Appl., to appear. https://www.sciencedirect.com/science/article/pii/S0022247X22006679

References in corpus (7)

• Every bordered Riemann surface is a complete conformal minimal surface bounded by Jordan curves
• Oka properties of ball complements
• The Calabi-Yau problem, null curves, and Bryant surfaces
• Schwarz-Pick lemma for harmonic maps which are conformal at a point
• The parametric h-principle for minimal surfaces in ${\mathbb{R}}^n$ and null curves in ${\mathbb{C}}^n$
• Hyperbolic domains in real Euclidean spaces
• Proper holomorphic curves attached to domains

Cited by in corpus (4)

• Proper holomorphic maps in Euclidean spaces avoiding unbounded convex sets
• Families of proper holomorphic maps
• Oka-1 manifolds
• Domains without parabolic minimal submanifolds and weakly hyperbolic domains