Existence of $p$-energy minimizers in homotopy classes and lifts of Newtonian maps

TL;DR

This paper investigates the existence of energy-minimizing Newtonian maps within homotopy classes, establishing conditions under which such minimizers exist when the target space has specific geometric properties.

Contribution

It introduces a link between $p$-quasihomotopy and lifts of Newtonian maps, proving minimizer existence in classes with hyperbolic fundamental groups.

Findings

Every $p$-quasihomotopy class contains a $p$-energy minimizer under certain conditions.

The results connect geometric group properties with variational problems in Newtonian spaces.

The work extends understanding of energy minimization in nonpositively curved target spaces.

Abstract

We study the notion of $p$-quasihomotopy in Newtonian classes of mappings and link it to questions concerning lifts of Newtonian maps, under the assumption that the target space is nonpositively curved. Using this connection we prove that every $p$-quasihomotopy class of Newtonian maps contains a minimizer of the $p$-energy if the target has hyperbolic fundamental group.