Existence of minimal surfaces of bounded topological type in three- manifolds. (English) Zbl 0602.49028

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

Geometry and partial differential equations, Miniconf. Canberra/Aust. 1985, Proc. Cent. Math. Anal. Aust. Natl. Univ. 10, 163-176 (1986).

[For the entire collection see Zbl 0583.00013.]
Using the methods of geometric measure theory, this paper investigates the existence of smooth embedded minimal surfaces with special topological properties in three-manifolds. The conclusions of the theorems refer to closed minimal surfaces in compact three-manifolds and mininal surfaces with boundary lying in the boundary of a uniformly convex subset of \(R^ 3\). On the basis of these conclusions the authors give a number of new examples in which minimal surfaces are realized in three-manifolds in topologically interesting ways.

Reviewer: C.Udrişte

MSC:

49Q05	Minimal surfaces and optimization
49Q20	Variational problems in a geometric measure-theoretic setting
53A10	Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53C20	Global Riemannian geometry, including pinching
57N10	Topology of general \(3\)-manifolds (MSC2010)

Keywords:

Heegard genus; existence of smooth embedded minimal surfaces; three- manifolds

Citations:

Zbl 0583.00013