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

Pitts, Jon T.; Rubinstein, J. H.

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

MSC:

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)

Keywords:

minimal surfaces in 3-manifolds; minimax-argument; geometric measure theory; triply periodic minimal surfaces; complete minimal surfaces; equivariant constructions

Citations:

Zbl 0626.00019; Zbl 0462.58003