Minimal surfaces in geometric 3-manifolds. (English) Zbl 1119.53042

Hoffman, David (ed.), Global theory of minimal surfaces. Proceedings of the Clay Mathematics Institute 2001 summer school, Berkeley, CA, USA, June 25–July 27, 2001. Providence, RI: American Mathematical Society (AMS). Cambridge, MA: Clay Mathematics Institute (ISBN 0-8218-3587-4/pbk). Clay Mathematics Proceedings 2, 725-746 (2005).
This article looks more like a survey of results concerning existence and topology of minimal surfaces imbedded in 3-manifolds carrying one of Thurston geometries. Sketchy proofs of several theorems are included. Some results concern also minimal hypersurfaces in 4-manifolds of positive Ricci curvature. For example (Thm. 2.3): if \(W\) is a compact orientable 4-manifold of positive Ricci curvature and \(M\) is a closed orientable embedded minimal 3-manifold, then both of the complementary domains for \(M\) in \(W\) have 2-dimensional spines, or dual handle presentation with only 2-, 3- and 4-dimensional handles attached to a thickened copy of \(M\).
For the entire collection see [Zbl 1078.53002].


53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)