Generic properties of minimal surfaces

2024/12/16 by Alarcon, Antonio, Lopez, Francisco J. · 1 citation
#Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #General Topology (math.GN)

paper · doi:10.48550/arxiv.2412.11563

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\colon 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.

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