On the solution of the harmonic-divgrad PDEs system

TL;DR

This paper investigates a specific PDE system involving harmonic, divergence, and gradient operators on Riemannian manifolds, establishing conditions for trivial solutions and motivated by applications in gauge theories.

Contribution

It provides new conditions under which the harmonic-divgrad PDE system admits only trivial solutions on positive curvature space forms.

Findings

Trivial solutions occur under certain curvature conditions.

The analysis applies Killing Hopf theorem to PDE systems.

Results are relevant for gauge theory symmetries.

Abstract

We study a particular system of partial differential equations in which the harmonic, the divergence and the gradient operators of the unknown functions appear (harmonic-divgrad system). Using the Killing Hopf theorem and leveraging the properties of Riemannian manifolds with constant sectional curvature we establish the conditions under which these equations admit only the trivial solutions proving their trivialization on positive curvature space forms. The analysis of this particular system is motivated by its occurrence in the study of asymptotic symmetries in $p$-form gauge theories and in mixed symmetry tensor gauge theories.