Construction of embedded area-minimizing surfaces via a topological type induction scheme
This paper presents a new scheme for constructing embedded area-minimizing surfaces in three-dimensional manifolds, based on a parametric approach. Denoting a two-dimensional integral homology class in a given three-manifold by \(A\), an existence and immersion theory is given that implies the existence of a surface of least area and lowest genus in \(A\). If the surface is embedded, the desired result is attained. Otherwise, a cut-and-paste argument gives a surface with lesser least area amongst surfaces in \(A\) with higher genus. Repeating this process produces a sequence of stable least area surfaces. Stability gives estimates for the genus of each surface in the sequence in terms of the surface's area, implying that there are only finitely many surfaces in the sequence. The last surface must then be embedded, and the desired result is attained. The author makes this argument in several different situations. First he considers the case of closed surfaces, proving a result already known using geometric measure theory, but by a very different parametric approach. The closed case then serves as a model for proving new results when the surfaces have boundary. Free boundary problems are considered.
scientific article; zbMATH DE number 2102287