Generic properties of minimal surfaces

Abstract Let M be an open Riemann surface and $n\ge 3$ be an integer. In this paper, we establish some generic properties (in Baire category sense) in the space of all conformal minimal immersions $M\to{\mathbb{R}}^n$ endowed with the compact-open topology, pointing out that a generic such immersion is chaotic in many ways. For instance, we show that a generic conformal minimal immersion $u\colon M\to {\mathbb{R}}^n$ is non-proper, almost proper, and ${\mathfrak{g}}$ -complete with respect to any given Riemannian metric ${\mathfrak{g}}$ in ${\mathbb{R}}^n$ . Further, its image u ( M ) is dense in ${\mathbb{R}}^n$ and disjoint from ${\mathbb{Q}}^3\times {\mathbb{R}}^{n-3}$ , and has infinite area, infinite total curvature, and unbounded curvature on every open set in ${\mathbb{R}}^n$ . In case n = 3, we also prove that a generic conformal minimal immersion $M\to {\mathbb{R}}^3$ has infinite index of stability on every open set in ${\mathbb{R}}^3$ .