×

Minimal measured laminations in geometric 3-manifolds. (English) Zbl 0727.57027

Let M be a 3-manifold and L a lamination of M [J. W. Morgan and P. B. Shalen, Ann. Math., II. Ser. 127, 403-456 (1988; Zbl 0656.57003)]. The lamination L in the Riemannian manifold M is said to be minimal if all its leaves are minimal surfaces. A measured lamination is a lamination equipped with an invariant transverse measure. Let \({\mathcal S}\) be the set of free homotopy classes of loops in M and \({\mathbb{R}}^{{\mathcal S}}\) the set of all functions from \({\mathcal S}\) to the set of real numbers \({\mathbb{R}}\). The author characterizes incompressible branched surfaces in Seifert fibered manifolds with \(SL_ 2{\mathbb{R}}\) or \(H^ 2\times E\)-structures and he proves the following Theorem: Let M be a 3-manifold with a fixed \(SL_ 2{\mathbb{R}}\)-structure g. Let m be a class of \({\mathbb{R}}^{{\mathcal S}}\) represented by an incompressible measured lamination in M. Then there exists a unique incompressible minimal measured lamination L in (M,g) which represents m. Moreover L is vertical with respect to the geometric fibration of (M,g). A corresponding Theorem for Seifert fibered manifolds with \(H^ 2\times E\)-structures is given.

MSC:

57R30 Foliations in differential topology; geometric theory
53C10 \(G\)-structures
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature

Citations:

Zbl 0656.57003
PDFBibTeX XMLCite
Full Text: DOI