PL minimal surfaces in 3-manifolds. (English) Zbl 0652.57005

Starting with a fixed triangulation of a given 3-manifold M, the authors choose a metric on the 2-skeleton which is hyperbolic on every triangle in the sense of the hyperbolic plane. If a surface F in M meets the 2- skeleton of M transversally, then one can associate with F the total length of the intersection curves. The notion of a PL minimal surface F refers to a surface for which this length is stationary with respect to small variations of F. A mean curvature H can be introduced as the geodesic curvature of the intersection curves.
In the paper the authors carry over results due to W. Meeks, P. Scott, S.-T. Yau and others to the case of PL minimal surfaces rather than ordinary minimal surfaces. In particular they give existence theorems of PL minimal surfaces in given normal homotopy classes, they discuss the exchange-and-roundoff-trick, and they give many results about the topology of M using PL minimal surfaces. It is impossible to mention all these in detail here. As an example, Corollary 2 says that a closed orientable irreducible 3-manifold with infinite fundamental group which is finitely covered by a Seifert fiber space must be a Seifert fiber space. Another theorem says that any covering of a \(P^ 2\)-irreducible 3-manifold is itself \(P^ 2\)-irreducible.
Reviewer: W.Kühnel


57N10 Topology of general \(3\)-manifolds (MSC2010)
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
57Q35 Embeddings and immersions in PL-topology
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