$\mathrm{C}^2$ estimates for general $p$-Hessian equations on closed Riemannian manifolds
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908529620680704 |
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| author | Qiao, Yuxiang |
| author_facet | Qiao, Yuxiang |
| contents | We study the $\mathrm{C}^2$ estimates for $p$-Hessian equations with general left-hand and right-hand terms on closed Riemannian manifolds of dimension $n$. To overcome the constraints of closed manifolds, we advance a new kind of "subsolution", called pseudo-solution, which generalizes "$\mathcal{C}$-subsolution" to some extent and is well-defined for fully general $p$-Hessian equations. Based on pseudo-solutions, we prove the $\mathrm{C}^1$ estimates for general $p$-Hessian equations, and the corresponding second-order estimates when $p\in\{2, n-1, n\}$, under sharp conditions -- we don't impose curvature restrictions, convexity conditions or "MTW condition" on our main results. Some other conclusions related to a priori estimates and different kinds of "subsolutions" are also given, including estimates for "semi-convex" solutions and when there exists a pseudo-solution. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_10773 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $\mathrm{C}^2$ estimates for general $p$-Hessian equations on closed Riemannian manifolds Qiao, Yuxiang Analysis of PDEs Differential Geometry 35B45, 58J05 (Primary) 35J60, 35R01 (Secondary) We study the $\mathrm{C}^2$ estimates for $p$-Hessian equations with general left-hand and right-hand terms on closed Riemannian manifolds of dimension $n$. To overcome the constraints of closed manifolds, we advance a new kind of "subsolution", called pseudo-solution, which generalizes "$\mathcal{C}$-subsolution" to some extent and is well-defined for fully general $p$-Hessian equations. Based on pseudo-solutions, we prove the $\mathrm{C}^1$ estimates for general $p$-Hessian equations, and the corresponding second-order estimates when $p\in\{2, n-1, n\}$, under sharp conditions -- we don't impose curvature restrictions, convexity conditions or "MTW condition" on our main results. Some other conclusions related to a priori estimates and different kinds of "subsolutions" are also given, including estimates for "semi-convex" solutions and when there exists a pseudo-solution. |
| title | $\mathrm{C}^2$ estimates for general $p$-Hessian equations on closed Riemannian manifolds |
| topic | Analysis of PDEs Differential Geometry 35B45, 58J05 (Primary) 35J60, 35R01 (Secondary) |
| url | https://arxiv.org/abs/2508.10773 |