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On the solution of the harmonic-divgrad PDEs system

Federico Manzoni

TL;DR

The paper analyzes a coupled harmonic-divgrad PDE system on space forms, employing the Killing-Hopf theorem to lift the problem to the sphere and leverage constant-curvature geometry. It proves that on positive-curvature space forms ($K>0$) of dimension $D-2$ the only solutions are trivial, $f=0$ and $h_i=0$, by deriving a biharmonic equation and applying uniqueness results, then pulling back along the Riemannian covering. The main contribution is a rigorous trivialization theorem for the system under $K>0$, with edge-case considerations for $k_1$ or $k_2$ vanishing. This result has implications for asymptotic symmetry analyses in $p$-form and mixed-symmetry tensor gauge theories, connecting PDE behavior to geometric topology and potential constraints on asymptotic charges in certain spacetimes.

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.

On the solution of the harmonic-divgrad PDEs system

TL;DR

The paper analyzes a coupled harmonic-divgrad PDE system on space forms, employing the Killing-Hopf theorem to lift the problem to the sphere and leverage constant-curvature geometry. It proves that on positive-curvature space forms () of dimension the only solutions are trivial, and , by deriving a biharmonic equation and applying uniqueness results, then pulling back along the Riemannian covering. The main contribution is a rigorous trivialization theorem for the system under , with edge-case considerations for or vanishing. This result has implications for asymptotic symmetry analyses in -form and mixed-symmetry tensor gauge theories, connecting PDE behavior to geometric topology and potential constraints on asymptotic charges in certain spacetimes.

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 -form gauge theories and in mixed symmetry tensor gauge theories.
Paper Structure (4 sections, 2 theorems, 24 equations)

This paper contains 4 sections, 2 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. Space form and Killing–Hopf theorem
  3. A result on the harmonic-divgrad PDEs system
  4. Conclusions

Key Result

Theorem 2.1

Let $(M,\boldsymbol{g})$ be a complete $D$-dimensional Riemannian manifold of constant sectional curvature $K$, then the Riemannian universal cover $(\Tilde{M},\tilde{\boldsymbol{g}})$ of $(M,\boldsymbol{g})$ is

Theorems & Definitions (6)

  • Definition 2.1: Space form
  • Theorem 2.1: Killing-Hopf
  • proof
  • Definition 3.1: Harmonic-divgrad equations
  • Theorem 3.1: Trivialization of harmonic-divgrad equations on a positive sectional curvature space form
  • proof