Multiply connected minimal surfaces and the geometric annulus theorem (Q1058155)

The main result is the geometric annulus theorem: ``Let \(M\) be a compact orientable 3-manifold with convex incompressible boundary and let \(A\) be a smooth annulus. Suppose that there is an essential smooth map \(f: (A,\partial A)\to (M,\partial M)\). Then: (1) There exists an essential immersion \(f^*: (A,\partial A)\to (M,\partial M)\) which has least area among all such essential smooth maps. (2) Any such immersion of least area is either an embedding, or a double covering map onto an embedded Möbius strip. (3) The images of any two such extremal maps either are disjoint, or are identical or intersect each other along a single essential arc. Furthermore the distinct images of the double covering maps, which happen to appear, are all mutually disjoint.''