Compact complete minimal immersions in ℝ³

Abstract

In this paper we find, for any arbitrary finite topological type, a compact Riemann surface M , \mathcal {M}, an open domain M ⊂ M M\subset \mathcal {M} with the fixed topological type, and a conformal complete minimal immersion X : M → R 3 X:M\to \mathbb {R}^3 which can be extended to a continuous map X : M ¯ → R 3 , X:\overline {M}\to \mathbb {R}^3, such that X | ∂ M X_{|\partial M} is an embedding and the Hausdorff dimension of X ( ∂ M ) X(\partial M) is 1. 1. We also prove that complete minimal surfaces are dense in the space of minimal surfaces spanning a finite set of closed curves in R 3 \mathbb {R}^3 , endowed with the topology of the Hausdorff distance.