Acylindrical surfaces in 3-manifolds. (English) Zbl 0862.57011

Hass, Joel

Mich. Math. J. 42, No. 2, 357-365 (1995).

A closed, incompressible surface in a 3-manifold is called acyclic if its complement has no incompressible and boundary-incompressible annulus. Using properties of minimal surfaces in hyperbolic manifolds the author shows that the genus of an acylindrical surface is bounded from above by a function which depends on the volume of the underlying manifold alone. As a corollary one obtains the result that the set of all acylindrical surfaces in a compact, orientable 3-manifold is finite (mod isotopy).

Reviewer: K.Johannson (Knoxville)

MSC:

57N10	Topology of general \(3\)-manifolds (MSC2010)
53A10	Minimal surfaces in differential geometry, surfaces with prescribed mean curvature

Keywords:

incompressible surface; 3-manifold; minimal surfaces; hyperbolic manifolds