The Liouville-type equation and an Onofri-type inequality on closed 4-manifolds

TL;DR

This paper investigates a specific Liouville-type equation on closed 4-manifolds with positive Ricci curvature, classifies solutions using invariant tensors, and establishes an Onofri-type inequality with rigidity results.

Contribution

It introduces a new classification method for solutions of a fourth-order PDE on 4-manifolds and derives a novel Onofri-type inequality with rigidity on the 4-sphere.

Findings

Classification of solutions within certain parameter ranges

Derivation of a new Onofri-type inequality on the 4-sphere

Rigidity result for the inequality

Abstract

In this paper, we study the Liouville-type equation \[\Delta ^2 u-\lambda_1\kappa\Delta u+\lambda_2\kappa^2(1-\mathrm e^{4u})=0\] on a closed Riemannian manifold \((M^4,g)\) with \(\operatorname{Ric}\geqslant 3\kappa g\) and \(\kappa>0\). Using the method of invariant tensors, we derive a differential identity to classify solutions within certain ranges of the parameters \(\lambda_1,\lambda_2\). A key step in our proof is a second-order derivative estimate, which is established via the continuity method. As an application of the classification results, we derive an Onofri-type inequality on the 4-sphere and prove its rigidity.