Blow-up analysis of Large conformal metrics with prescribed Gaussian and geodesic curvatures
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| Format: | Preprint |
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2024
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| author | Caju, Rayssa Cruz, Tiarlos Santos, Almir Silva |
| author_facet | Caju, Rayssa Cruz, Tiarlos Santos, Almir Silva |
| contents | Consider a compact Riemannian surface $(M,g)$ with nonempty boundary and negative Euler characteristic. Given two smooth non-constant functions $f$ in $M$ and $h$ in $\partial M$ with $\max f= \max h= 0$, under a suitable condition on the maximum points of $f$ and $h$, we prove that for sufficiently small positive constants $λ$ and $μ$, there exist at least two distinct conformal metrics $g_{λ,μ}=e^{2u_{μ,λ}}g$ and $g^{λ,μ}=e^{2u^{μ,λ}}g$ with prescribed sign-changing Gaussian and geodesic curvature equal to $f + μ$ and $h + λ,$ respectively. Additionally, we employ the method used in Borer et al. (2015) to study the blowing up behavior of the large solution $u^{μ,λ}$ when $μ\downarrow 0$ and $λ\downarrow 0$. Finally, we derive a new Liouville-type result for the half-space, eliminating one of the potential blow-up profiles. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_02467 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Blow-up analysis of Large conformal metrics with prescribed Gaussian and geodesic curvatures Caju, Rayssa Cruz, Tiarlos Santos, Almir Silva Differential Geometry Analysis of PDEs 35B44, 58J32, 35J20, 35J60 Consider a compact Riemannian surface $(M,g)$ with nonempty boundary and negative Euler characteristic. Given two smooth non-constant functions $f$ in $M$ and $h$ in $\partial M$ with $\max f= \max h= 0$, under a suitable condition on the maximum points of $f$ and $h$, we prove that for sufficiently small positive constants $λ$ and $μ$, there exist at least two distinct conformal metrics $g_{λ,μ}=e^{2u_{μ,λ}}g$ and $g^{λ,μ}=e^{2u^{μ,λ}}g$ with prescribed sign-changing Gaussian and geodesic curvature equal to $f + μ$ and $h + λ,$ respectively. Additionally, we employ the method used in Borer et al. (2015) to study the blowing up behavior of the large solution $u^{μ,λ}$ when $μ\downarrow 0$ and $λ\downarrow 0$. Finally, we derive a new Liouville-type result for the half-space, eliminating one of the potential blow-up profiles. |
| title | Blow-up analysis of Large conformal metrics with prescribed Gaussian and geodesic curvatures |
| topic | Differential Geometry Analysis of PDEs 35B44, 58J32, 35J20, 35J60 |
| url | https://arxiv.org/abs/2402.02467 |