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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2508.14494 |
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| _version_ | 1866916909836926976 |
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| author | Ma, Xi-Nan Wu, Tian Zhou, Xiao |
| author_facet | Ma, Xi-Nan Wu, Tian Zhou, Xiao |
| contents | In this paper, we study the Liouville-type equation
\[Δ^2 u-λ_1κΔu+λ_2κ^2(1-\mathrm e^{4u})=0\]
on a closed Riemannian manifold \((M^4,g)\) with \(\operatorname{Ric}\geqslant 3κg\) and \(κ>0\). Using the method of invariant tensors, we derive a differential identity to classify solutions within certain ranges of the parameters \(λ_1,λ_2\). 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_14494 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Liouville-type equation and an Onofri-type inequality on closed 4-manifolds Ma, Xi-Nan Wu, Tian Zhou, Xiao Analysis of PDEs In this paper, we study the Liouville-type equation \[Δ^2 u-λ_1κΔu+λ_2κ^2(1-\mathrm e^{4u})=0\] on a closed Riemannian manifold \((M^4,g)\) with \(\operatorname{Ric}\geqslant 3κg\) and \(κ>0\). Using the method of invariant tensors, we derive a differential identity to classify solutions within certain ranges of the parameters \(λ_1,λ_2\). 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. |
| title | The Liouville-type equation and an Onofri-type inequality on closed 4-manifolds |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2508.14494 |