Compact complete minimal immersions in R^3

Details Compact complete minimal immersions in R^3 Compact complete minimal immersions in R^3

2007-11-15

arxiv.org/abs/0711.2394v4

Mathematics

Differential Geometry

16 pages. Main theorem improved. To appear in Trans. Amer. Math. Soc

Scientific paper

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.

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