Generic properties of minimal surfaces

TL;DR

This paper investigates generic properties of conformal minimal immersions from open Riemann surfaces into Euclidean spaces, revealing that most such immersions are chaotic, dense, and have infinite geometric complexity.

Contribution

It establishes that a generic conformal minimal immersion exhibits chaotic behavior and infinite geometric complexity, expanding understanding of minimal surface properties in higher dimensions.

Findings

A generic immersion is non-proper and almost proper.

The image is dense in Euclidean space and disjoint from certain rational subsets.

Such immersions have infinite area, total curvature, and unbounded curvature.

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 $g$-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 $\mathbb{R}^3$.