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 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.