Variation of the modulus of a foliation

TL;DR

This paper investigates how the p-modulus of a foliation on a Riemannian manifold varies, extending the concept of extremal length, and examines the product of moduli for orthogonal foliations.

Contribution

It introduces the study of the variation of p-modulus for foliations and explores the behavior of the product of moduli for orthogonal foliations on Riemannian manifolds.

Findings

Derived formulas for the variation of p-modulus.

Analyzed the product of moduli for orthogonal foliations.

Extended extremal length concepts to higher-dimensional foliations.

Abstract

The $p$--modulus ${\rm mod}_p(\mathcal{F})$ of a foliation $\mathcal{F}$ on a Riemannian manifold $M$ is a generalization of extremal length of plane curves introduced by L. Ahlfors. We study the variation $t\mapsto{\rm mod}_p(\mathcal{F}_t)$ of the modulus. In particular, we consider product of moduli of orthogonal foliations.