Partial regularity of mass-minimizing rectifiable sections

Abstract

Let B be a fiber bundle with compact fiber F over a compact Riemannian n-manifold M n . There is a natural Riemannian metric on the total space B consistent with the metric on M. With respect to that metric, the volume of a rectifiable section σ : M → B is the mass of the image σ (M) as a rectifiable n-current in B. (2000): 49F20, 49F22, 49F10, 58A25, 53C42, 53C65. Theorem 1. For any homology class of sections of B, there is a mass-minimizing rectifiable current T representing that homology class which is the graph of a C 1 section on an open dense subset of M. Mathematics Subject Classifications

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].…”

“…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.…”

“…Proof Let S be a mass-minimizing section which is continuous over an open, dense subset, as guaranteed to exist by [4]. Assume that S has a degree-zero isolated singularity x 0 , with the entire fiber contained in the support of S. There is a h-cone S 0 of S at x 0 in Sect.…”

“…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 ).…”

“…We provide here a more direct proof of this fact in the case we need. Note that the existence of h-cones is established only for the mass-minimizing currents (with good partial-regularity) shown to exist in [4], which are limits of a sequence of minimizers of functionals with an additional penalty term. It is not known whether other mass-minimizers exist, without the required partial regularity.…”