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 $u:M\to {\mathbb{R}}^n$ is non-proper, almost proper, and $g$-complete with respect to any given Riemannian metric $g$ 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 ${\mathbb{R}}^3$.

To appear in Proc. Roy. Soc. Edinburgh Sect. A