Stable foliations with respect to Fuglede modulus and level sets of $p$--harmonic functions

TL;DR

This paper investigates the stability of foliations with respect to the $p$-modulus, deriving second variation formulas, analyzing $p$-stability, and exploring the relationship with $q$-harmonic functions, including examples like distance function foliations.

Contribution

It derives the second variation formula for the $p$-modulus of foliations and studies $p$-stability, linking it to $q$-harmonicity and providing explicit examples.

Findings

Second variation formula for $p$-modulus of foliations.

Characterization of $p$-stable foliations, especially in codimension one.

Distance function foliations are critical points of the $p$-modulus functional.

Abstract

We continue the study of the variation of the $p$--modulus of a foliation initiated by the first author. We derive the formula for the second variation which allows to study $p$--stable foliations. We obtain some results concerning codimension one $p$--stable foliations. Moreover, we derive the equation for the critical point of the $p$--modulus functional of a foliation given by the level sets of smooth function. We show the correlation with the $q$--harmonicity. We give some examples. In particular, we show that foliations given by the distance function are critical points of $p$--modulus functional.