The $σ_{2}$-curvature equation on a compact manifold with boundary
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866911333763514368 |
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| author | Chen, Xuezhang Wei, Wei |
| author_facet | Chen, Xuezhang Wei, Wei |
| contents | We first establish local $C^2$ estimates of solutions to the $σ_2$-curvature equation with nonlinear Neumann boundary condition. Then, under assumption that the mean curvature of a background metric is nonnegative on totally non-umbilic boundary, for dimensions three and four there exists a conformal metric having a prescribed positive $σ_2$-curvature and a prescribed nonnegative boundary mean curvature. The local estimates play an important role in the blow up analysis for the latter existence result. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_13942 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The $σ_{2}$-curvature equation on a compact manifold with boundary Chen, Xuezhang Wei, Wei Differential Geometry Analysis of PDEs We first establish local $C^2$ estimates of solutions to the $σ_2$-curvature equation with nonlinear Neumann boundary condition. Then, under assumption that the mean curvature of a background metric is nonnegative on totally non-umbilic boundary, for dimensions three and four there exists a conformal metric having a prescribed positive $σ_2$-curvature and a prescribed nonnegative boundary mean curvature. The local estimates play an important role in the blow up analysis for the latter existence result. |
| title | The $σ_{2}$-curvature equation on a compact manifold with boundary |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2307.13942 |