Liouville Theorem with Boundary Conditions from Chern--Gauss--Bonnet Formula
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| Format: | Preprint |
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2024
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| _version_ | 1866914983050215424 |
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| author | Chu, BaoZhi Li, YanYan Li, Zongyuan |
| author_facet | Chu, BaoZhi Li, YanYan Li, Zongyuan |
| contents | The $σ_k(A_g)$ curvature and the boundary $\mathcal{B}^k_g$ curvature arise naturally from the Chern--Gauss--Bonnet formula for manifolds with boundary. In this paper, we prove a Liouville theorem for the equation $σ_k(A_g)=1$ in $\overline{\mathbb{R}^n_+}$ with the boundary condition $\mathcal{B}^k_g=c$ on $\partial\mathbb{R}^n_+$, where $g=e^{2v}|dx|^2$ and $c$ is some nonnegative constant. This extends an earlier result of Wei, which assumes the existence of $\lim_{|x|\to\infty}(v(x)+2\log|x|)$. In addition, we establish a local gradient estimate for solutions of such equations, assuming an upper bound on the solution $v$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_16384 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Liouville Theorem with Boundary Conditions from Chern--Gauss--Bonnet Formula Chu, BaoZhi Li, YanYan Li, Zongyuan Analysis of PDEs Differential Geometry 35J60 (Primary) 53C21, 58J05 (Secondary) The $σ_k(A_g)$ curvature and the boundary $\mathcal{B}^k_g$ curvature arise naturally from the Chern--Gauss--Bonnet formula for manifolds with boundary. In this paper, we prove a Liouville theorem for the equation $σ_k(A_g)=1$ in $\overline{\mathbb{R}^n_+}$ with the boundary condition $\mathcal{B}^k_g=c$ on $\partial\mathbb{R}^n_+$, where $g=e^{2v}|dx|^2$ and $c$ is some nonnegative constant. This extends an earlier result of Wei, which assumes the existence of $\lim_{|x|\to\infty}(v(x)+2\log|x|)$. In addition, we establish a local gradient estimate for solutions of such equations, assuming an upper bound on the solution $v$. |
| title | Liouville Theorem with Boundary Conditions from Chern--Gauss--Bonnet Formula |
| topic | Analysis of PDEs Differential Geometry 35J60 (Primary) 53C21, 58J05 (Secondary) |
| url | https://arxiv.org/abs/2410.16384 |