Applications of minimax to minimal surfaces and the topology of 3- manifolds. (English) Zbl 0639.49030

Geometry and partial differential equations, 2nd Miniconf., Canberra/Aust. 1986, Proc. Cent. Math. Anal. Aust. Natl. Univ. 12, 137-170 (1987).
[For the entire collection see Zbl 0626.00019.]
The authors prove a very general existence theorem for minimal surfaces in 3-manifolds which is based on a minimax-argument. This minimax- argument within the framework of geometric measure theory (especially varifolds) was first used by J. T. Pitts [Existence and regularity of minimal surfaces on Riemannian manifolds (1981; Zbl 0462.58003)] and has turned out to be extremely useful. The applications given here include triply periodic minimal surfaces in \({\mathbb{R}}^ 3,\) complete minimal surfaces of finite area in complete noncompact hyperbolic 3- manifolds of finite volume, equivariant constructions in arbitrary dimensions, complete minimal surfaces in \({\mathbb{R}}^ 3 \)having genus \(\geq 1\) and three ends, as well as a scheme for classifying 3-manifolds of positive Ricci curvature.
Reviewer: M.Grüter


49Q05 Minimal surfaces and optimization
49Q15 Geometric measure and integration theory, integral and normal currents in optimization
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
49J35 Existence of solutions for minimax problems
57M99 General low-dimensional topology
57N10 Topology of general \(3\)-manifolds (MSC2010)