Least area surfaces can have excess triple points
It is known by results of \textit{R. Schoen} and \textit{S.-T. Yau} [Ann. Math., II. Ser. 110, 127-142 (1979; Zbl 0431.53051)] that in the homotopy class of any map f of a closed surface to a Riemannian 3-manifold which induces a monomorphism on fundamental groups there is an area minimizing immersion. By results of \textit{M. Freedman, J. Hass}, and \textit{P. Scott} [Invent. Math. 71, 609-642 (1983; Zbl 0482.53045)] such a least area map will minimize, over the homotopy class of f, a certain invariant D(f) which is related to the number, d(f), of double curves of f. In general D(f)\(\neq d(f)\) and examples show that d(f) need not be minimized by a least area map. However in the case of a torus \(D(f)=d(f).\) In the present paper an example is given of a least area map of a torus which has triple points but which is homotopic to a general position immersion without triple points. This shows that the proper measure of complexity in this subject is necessarily different from the conventional triple point, double curve count.
scientific article; zbMATH DE number 4036618