Abstract
We show that immersed minimal surfaces in the euclidean 3-space with bounded curvature and proper self intersections are proper. We also showthat restricted to wide components the immersing map is always proper, regardless the map being proper or not. Prior to these results it was only known that injectively immersed minimal surfaces with bounded curvature were proper.
Highlights
Most of the results about the structure of complete minimal surfaces of R3 requires the hypothesis that the surfaces are proper
It would be an interesting problem to determine what geometries imply that a complete minimal surface of R3 is proper
There are examples of complete non proper minimal surfaces of R3 with bounded sectional curvature, whose closures are dense in large subsets of R3, (Andrade 2000)
Summary
Most of the results about the structure of complete minimal surfaces of R3 requires the hypothesis that the surfaces are proper. He proved that an injectively immersed complete minimal surface of R3 with bounded sectional curvature is proper. There are examples of complete non proper minimal surfaces of R3 with bounded sectional curvature, whose closures are dense in large subsets of R3, (Andrade 2000).
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