Regularity of volume-minimizing graphs

Abstract

Abstract. Let M be a compact, oriented Riemannian nmanifold, and let B → M be a fiber bundle over M , with compact fiber F . Given a section of B, its volume is defined as the n-dimensional Hausdorff measure (or mass) of the graph of ϕ. We show that a volume-minimizing graph in a given homology class, which is in general a rectifiable section, is a continuous graph over all but a set of Hausdorff codimension 3 in the base M , unless the fiber F admits stable 2-dimensional minimal currents (without boundary), in which case there could be a codimension 2 set of singular points with poles (support of the current over a singular point) consisting of such 2-dimensional currents. Locally, we also show that a volume-minimizing graph of a map ϕ : Ω → F for Ω ⊂ R n with prescribed boundary ϕ |∂Ω = ψ : ∂Ω → F has codimension-3 poles in Ω, under the assumption that Ω is convex and that F is a compact, convex subset of Euclidean space.This work grew out of a question posed by Herman Gluck and Wolfgang Ziller regarding the "volume of flows", volumes of sections of the unit tangent bundle of a compact manifold, viewed as one-dimensional foliations on M . They established that the standard Hopf fibration, as a foliation on S 3 , minimizes volume among all such sections. However, in higher dimensions the Hopf fibrations are not volume-minimizing, and it is likely that volume-minimizing flows on these manifolds are singular, due to examples of Sharon Pedersen.In this setting our results show that, except in dimension 3, outside of a codimension-3 set of possible "poles", a minimizing foliation will be C 1 . In dimension 3 there can be a codimension-2 set of poles, almost all of which involve the whole fiber in that for each unit tangent vector at almost-all poles (in the Hausdorff measure on the set of singular points), there is a limit of tangents of a leaf converging to that vector.

Volume-Minimizing Foliations on Spheres

“…This notion of a singular foliation is similar to, but more general than, that studied by the secondnamed author and Smith in [6]. In that article, the singular sections of arbitrary vector bundles that are considered are those in the weak closure of the space of smooth sections.…”

Regularity of volume-minimizing flows on 3-manifolds

“…A simple modification of the Federer-Flemming closure and compactness theorems shows the following result: [4,5].…”

“…This definition can be extended to sections σ of any Riemannian fiber-bundle B → M with compact fiber F , as defined in [5,4]. The volume functional is essentially the same, except that the highest-degree term in the square root is the minimum of the dimension of M or that of the fiber, V(σ ) = M 1 + ∇σ 2 + · · · + ∇σ ∧n 2 dV M , with terms ∇σ i 2 being 0 for i > dim(F ).…”

“…The h-cone is then defined on the Euclidean product B(x 0 , r) × F ⊂ R n × F . It was shown in [5] that, for mass-minimizing rectifiable sections as constructed in [4], h-cones always exist for some sequence of dilations, since a simple monotonicity result shows that the set of dilations T λ will have equibounded mass. We provide here a more direct proof of this fact in the case we need.…”

“…However, in higher dimensions the Hopf fibrations are not volume-minimizing, and it is likely that volume-minimizing flows on these manifolds are singular. In her thesis [9], Sharon Pedersen illustrated a stable, singular foliation which has much less mass than the Hopf fibration of S 5 .…”