Rigidity of vector valued harmonic maps of linear growth

TL;DR

This paper investigates vector valued harmonic maps with linear growth on non-compact manifolds with non-negative Ricci curvature, establishing maximum principles and equalities related to the pull-back volume form's behavior.

Contribution

It introduces a strong maximum principle and identifies key equalities for the pull-back volume form of harmonic maps with linear growth.

Findings

Strong maximum principle for the pull-back volume form

Equalities among supremum, asymptotic average, and heat evolution

Insights into the structure of harmonic maps with linear growth

Abstract

Consider vector valued harmonic maps of at most linear growth, defined on a complete non-compact Riemannian manifold with non-negative Ricci curvature. For the norm square of the pull-back of the target volume form by such maps, we report a strong maximum principle, and equalities among its supremum, its asymptotic average, and its large-time heat evolution.