Generic properties of minimal surfaces

Let M be an open Riemann surface and n≥ 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→Rⁿ 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→ Rⁿ is non-proper, almost proper, and g -complete with respect to any given Riemannian metric g in Rⁿ . Further, its image u ( M ) is dense in Rⁿ and disjoint from Q³× Rⁿ⁻³ , and has infinite area, infinite total curvature, and unbounded curvature on every open set in Rⁿ . In case n = 3, we also prove that a generic conformal minimal immersion M→ R³ has infinite index of stability on every open set in R³ .