On extremal function of a modulus of a foliation

TL;DR

This paper studies the extremal function of a foliation's modulus on Riemannian manifolds, providing conditions for existence, properties, and integral formulas linking manifold and leaf integrals.

Contribution

It introduces necessary and sufficient conditions for extremal functions and explores their relation to the geometry orthogonal to the foliation.

Findings

Derived integral formulas connecting manifold and leaf integrals.

Established conditions for the existence of extremal functions.

Analyzed the relationship between extremal functions and orthogonal distribution geometry.

Abstract

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