Compact complete minimal immersions in R^3
Abstract
Description
In this paper we find, for any arbitrary finite topological type, a compact Riemann surface $\mathcal{M},$ an open domain $M\subset\mathcal{M}$ with the fixed topological type, and a conformal complete minimal immersion $X:M\to\R^3$ which can be extended to a continuous map $X:\bar{M}\to\R^3,$ such that $X_{|\partial M}$ is an embedding and the Hausdorff dimension of $X(\partial M)$ is $1.$
We also prove that complete minimal surfaces are dense in the space of minimal surfaces spanning a finite set of closed curves in $\R^3$, endowed with the topology of the Hausdorff distance.
16 pages. Main theorem improved. To appear in Trans. Amer. Math. Soc
16 pages. Main theorem improved. To appear in Trans. Amer. Math. Soc