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The geometry and conformal structure of properly embedded minimal surfaces of finite topology in \(\mathbb{R}^ 3\). (English) Zbl 0803.53007

The authors study properly embedded minimal surfaces \(M\) in \(\mathbb{R}^ 3\) of finite topology and allow \(M\) to have infinite total curvature. Their main theorem analyses the behavior of the possible ends of \(M\): Let \(M\) have more than one end. If \(A\) is an annular end of \(M\), then (after rotation of \(M\)) either \(A\) is smoothly asymptotic to a horizontal plane or \(x3\) restricted to \(A\) is a proper harmonic function on \(A\). In particular, every such \(A\) is conformally diffeomorphic to the punctured disk.

MSC:

53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
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References:

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