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.

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

References in corpus (2)

• Schwarz-Pick lemma for harmonic maps which are conformal at a point
• Hyperbolic domains in real Euclidean spaces

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