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

Rubinstein, J. Hyam

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].

Reviewer: Paweł Walczak (Łódz)

MSC:

53C42

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

Keywords:

minimal surface; Thurston geometry