The topology of minimal surfaces in Seifert fiber spaces. (English) Zbl 0866.57011

A basic question in the theory of minimal surfaces in 3-dimensional manifolds is to decide which embeddings of surfaces can be realized by minimal surfaces. Fundamental results were obtained in the case of Riemannian metrics of positive curvatures. Seifert fiber spaces [cf. P. Orlik, Seifert manifolds, Lect. Notes Math. 291 (1972; Zbl 0263.57001)] are an important class of examples of 3-dimensional manifolds that admit 1-dimensional foliations by circles. J. Hass studied the topology of \(\pi_1\)-injective minimal surfaces in Seifert fiber spaces [Topology Appl. 18, 145-151 (1984; Zbl 0559.57005)].
In this paper, the authors obtain the following topological classification of arbitrary embedded minimal surfaces in such 3-manifolds, extending the result of Hass. If \(\Sigma\) is a closed connected embedded minimal surface in a Seifert fiber space \(M\), then the embedding of \(\Sigma\) is one of the following three (naturally exclusive) types: (1) \(\Sigma\) is a vertical torus or a vertical Klein bottle. (2) \(\Sigma\) is horizontal and \(\pi_1\)-injective. As noted by J. Hass, there are three distinct subcases. (3) \(\Sigma\) is neither vertical nor horizontal. There is a (possibly empty) disjoint family \(\mathfrak F\) of vertical minimal tori and Klein bottles in \(M-\Sigma\). The closure of \(M-\Sigma-\bigcup {\mathfrak F}\) has \(j\) components that are handlebodies or hollow handlebodies; the rest are Seifert fiber spaces with boundaries on \(\bigcup {\mathfrak F}\). Here \(j=2\) if \(\Sigma\) is 2-sided in \(M\) and \(j=1\) if \(\Sigma\) is 1-sided in \(M\). Finally, the authors construct some examples of interesting minimal surfaces in Seifert fiber spaces by means of the minimax technique developed by other authors.


57M50 General geometric structures on low-dimensional manifolds
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
Full Text: DOI