On the extremal function of the modulus of a foliation

Abstract

Abstract.We investigate the properties of the modulus of a foliation on a Riemannian manifold. We give necessary and sufficient conditions for the existence of the extremal function and state some of its properties. We obtain an integral formula which, in a sense, combines the integral over the manifold with the integral over the leaves. We state a relation between the extremal function and the geometry of the distribution orthogonal to a foliation.Mathematics Subject Classification. 53C12, 58C35.

Rodin's formula in arbitrary codimension

“…Denote by Σ a family of level sets σ x , x ∈ U, or, more precisely, the family of m-Hausdorff measures on these level sets. Then the following fact holds [7] (see also [5]).…”

“…From this representation, it is clear, that the integral of the extremal function f Γ on any curve from the family Γ equals one (see also [5] for more general approach). Let us now concentrate on the quantity |ċx| J f .…”