Regularity of volume-minimizing flows on 3-manifolds

Abstract

In Johnson and Smith (Indiana Univ Math J 44:45–85, 1995; Ann Global Anal Geometry 30:239–287, 2006; Proceedings of the VII International Colloquium on Differential Geometry, 1994, World Scientific, pp. 81–98), the authors characterized the singular set (discontinuities of the graph) of a volume-minimizing rectifiable section of a fiber bundle, showing that, except under certain circumstances, there exists a volume-minimizing rectifiable section with the singular set lying over a codimension-3 set in the base space. In particular, it was shown that for 2-sphere bundles over 3-manifolds, a minimizer exists with a discrete set of singular points. In this article, we show that for a 2-sphere bundle over a compact 3-manifold, such a singular point cannot exist. As a corollary, for any compact 3-manifold, there is a C ¹ volume-minimizing one-dimensional foliation. In addition, this same analysis is used to show that the examples, due to Pedersen (Trans Am Math Soc 336:69–78, 1993), of potentially volume-minimizing rectifiable sections (rectifiable foliations) of the unit tangent bundle to S ²ⁿ⁺¹ are not, in fact, volume minimizing.