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Some global properties of complete minimal surfaces of finite topology in \({\mathbb{R}{}}^ 3\). (English) Zbl 0737.53010

The authors try to answer a question in the theory of classical minimal surfaces. They want to know if the helicoid is the only properly embedded minimal surface \(M\) in \(\mathbb{R}^ 3\) with finite topology and infinite total curvature. In a former paper D. Hoffman and W. H. Meeks III [J. Am. Math. Soc. 2, No. 4, 667-682 (1989; Zbl 0683.53005)] proved that \(M\) can have at most two annular ends of infinite total curvature. Here the authors go further into the geometry of these objects getting nice results. For example: If \(M\) is an annular end with compact boundary properly immersed in the region above the catenoid then it has finite total curvature. Very interesting is the result about the standard position of \(M\) with two ends of infinite total curvature: These ends lie in two disjoint half spaces and the others have asymptotic planes parallel to the boundary of those half spaces. If \(M\) has smooth compact boundary and an end of the catenoid type then it has at most one end of infinite total curvature. The authors also apply their techniques to understand the well-known examples of minimal planes, annuli in a slab or the behaviour of the Gauss map of a properly immersed minimal surface in a half space.
Reviewer: L.P.Jorge

MSC:

53A20 Projective differential geometry

Citations:

Zbl 0683.53005
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