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arXiv · 2508.14494

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

Xi-Nan Ma·Tian Wu·Xiao Zhou
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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.

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Xi-Nan Ma, Tian Wu, Xiao Zhou. 2025-08-20. The Liouville-type equation and an Onofri-type inequality on closed 4-manifolds. https://arxiv.org/abs/2508.14494

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