Generic properties of minimal surfaces

Abstract 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 MRⁿ 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 R^n-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.