Critical sinh-Gordon flow with non-negative weight functions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913014493478912 |
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| author | Fei, Qiang Jevnikar, Aleks Moon, Sang-Hyuck |
| author_facet | Fei, Qiang Jevnikar, Aleks Moon, Sang-Hyuck |
| contents | The aim of this article is twofold: one one side we introduce and study the properties of a critical sinh-Gordon type flow \begin{equation*} {\frac{\partial}{\partial t}}e^u=Δ_gu+8π\left({\frac{h_1e^u}{\int_Σh_1e^udV_g}}-1\right)-ρ_2\left({\frac{h_2e^{-u}}{\int_Σh_2e^{-u}dV_g}}-1\right), \end{equation*} where $ρ_2<8π$, $h_1,h_2$ are non-negative weight functions and $Σ$ is a closed Riemannian surface. Secondly, under suitable geometric conditions, we prove the convergence of the flow to a solution of the critical sinh-Gordon equation, extending the result of Zhou (2008) to the case of non-negative weights. The argument is based on a careful blow-up analysis. Some remarks about a Toda flow are also given. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_03624 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Critical sinh-Gordon flow with non-negative weight functions Fei, Qiang Jevnikar, Aleks Moon, Sang-Hyuck Analysis of PDEs 35J20, 35J61, 53C44 The aim of this article is twofold: one one side we introduce and study the properties of a critical sinh-Gordon type flow \begin{equation*} {\frac{\partial}{\partial t}}e^u=Δ_gu+8π\left({\frac{h_1e^u}{\int_Σh_1e^udV_g}}-1\right)-ρ_2\left({\frac{h_2e^{-u}}{\int_Σh_2e^{-u}dV_g}}-1\right), \end{equation*} where $ρ_2<8π$, $h_1,h_2$ are non-negative weight functions and $Σ$ is a closed Riemannian surface. Secondly, under suitable geometric conditions, we prove the convergence of the flow to a solution of the critical sinh-Gordon equation, extending the result of Zhou (2008) to the case of non-negative weights. The argument is based on a careful blow-up analysis. Some remarks about a Toda flow are also given. |
| title | Critical sinh-Gordon flow with non-negative weight functions |
| topic | Analysis of PDEs 35J20, 35J61, 53C44 |
| url | https://arxiv.org/abs/2511.03624 |