Energy of generalized distributions

TL;DR

This paper investigates the energy of smooth and singular generalized distributions on compact Riemannian manifolds, providing lower bounds and examples of energy-minimizing foliations.

Contribution

It introduces bounds for the energy of almost regular distributions and identifies examples that minimize this energy.

Findings

Derived lower bounds for the energy of q-dimensional distributions.

Constructed examples of energy-minimizing foliations.

Analyzed distributions with isolated singularities and disjoint submanifolds.

Abstract

We consider the energy of smooth generalized distributions and also of singular foliations on compact Riemannian manifolds for which the set of their singularities consists of a finite number of isolated points and of pairwise disjoint closed submanifolds. We derive a lower bound for the energy of all $q$-dimensional almost regular distributions, for each $q< \dim M,$ and find several examples of foliations which minimize the energy functional over certain sets of smooth generalized distributions.