Area minimizing surfaces in 3-manifolds
It is known that the least area incompressible surface in a free homotopy class in a 3-manifold has the minimal possible self intersection [\textit{M. Freedman, J. Hass}, and \textit{P. Scott}, Invent. Math. 71, 609-642 (1983; Zbl 0482.53045)]. The author analyzes the situation for surfaces with least area among all immersed incompressible surfaces. He obtains a positive result for some Seifert fiber spaces and shows that absolutely least area surfaces do not have to intersect least in general. A corollary of his positive result is that any absolutely least area surface in an orientable Seifert fiber space with more than three singular fibers is embedded. He establishes that any smoothly immersed incompressible surface in general position in a closed 3-manifold is a least area surface in a regular neighborhood for some Riemannian metric. In addition, the absolutely least area surface will be in this regular neighborhood. He uses this result to prove that any abstract immersion of a surface \(F\) with only simple, essential, disjoint, non separating double curves may be realized as an absolutely least area surface in some Haken manifold. His proof requires a creative and intricate geometric construction. This paper ends with three open questions about least area surfaces.
scientific article; zbMATH DE number 1505951