Regularity of mass-minimizing one-dimensional foliations (Q2771406)

Let \({\mathcal F}\) be a one-dimensional transversely oriented foliation on a compact oriented Riemannian manifold \(M\) and \(T_1(M)\) the unit tangent bundle of \(M\). The volume of \({\mathcal F}\) is defined as the Hausdorff \(n\)-dimensional measure of the image of the Gauss map \(\xi:M\to T_1(M)\) mapping \(x\in M\) to a unit vector \(\xi(x)\in T_1(M,x)\) tangent to \({\mathcal F}\). Let \(\text{Cart} ({\mathcal F}^1)\) be the set of rectifiable one-dimensional foliations on \(M\). The regular points of a regular foliation \(S\) correspond to points where the Gauss map is continuous and pole points (singularities) are points \(x\in M\) where the Gauss map is discontinuous. The aim of this paper is to study the existence of a one-dimensional rectifiable foliation which minimizes the volume functional.NEWLINENEWLINENEWLINEThe authors prove the following two theorems: 1) The set of pole points \(P\) of a volume-minimizing one-dimensional rectifiable foliation \(S\in \text{Cart} ({\mathcal F}^1)\) over a compact Riemannian manifold \(M\) of dimension \(n\neq 3\) has codimension at least 3 in \(M\). 2) Let \(M\) be a compact, oriented three-manifold. There is a volume-minimizing one-dimensional rectifiable foliation \(S\) on \(M\) with only a finite number of pole points. The support of \(S\) includes the entire fiber over each pole point. The space of rectifiable foliations is a natural completion of the set of smooth foliations to a space of rectifiable currents; in this space the existence of a mass-minimizing 1-dimension foliation is a consequence of a compactness result.NEWLINENEWLINEFor the entire collection see [Zbl 0953.00036].