Liouville properties for p-harmonic maps with finite q-energy

TL;DR

This paper establishes Liouville theorems for p-harmonic functions with finite energy on certain manifolds, revealing nonexistence of nontrivial solutions under specific curvature and energy conditions, and extends results to p-harmonic maps.

Contribution

It introduces new Liouville theorems for p-harmonic functions and maps with finite energy, using an approximate p-Laplace operator and Bochner's formula, expanding understanding of their behavior on manifolds.

Findings

Proves nonexistence of nontrivial p-harmonic functions with finite energy under curvature conditions.

Shows a manifold has at most one p-hyperbolic end given certain energy and curvature assumptions.

Extends Liouville theorems to p-harmonic morphisms and conformal maps between Riemannian manifolds.

Abstract

We introduce and study an approximate solution of the p-Laplace equation, and a linearlization $L_{\epsilon}$ of a perturbed p-Laplace operator. By deriving an $L_{\epsilon}$-type Bochner's formula and a Kato type inequality, we prove a Liouville type theorem for weakly p-harmonic functions with finite p-energy on a complete noncompact manifold M which supports a weighted Poincar\'{e} inequality and satisfies a curvature assumption. This nonexistence result, when combined with an existence theorem, yields in turn some information on topology, i.e. such an M has at most one p-hyperbolic end. Moreover, we prove a Liouville type theorem for strongly p-harmonic functions with finite q-energy on Riemannian manifolds, where the range for q contains p. As an application, we extend this theorem to some p-harmonic maps such as p-harmonic morphisms and conformal maps between Riemannian manifolds.