The prescribed curvature flow on the disc
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912169596026880 |
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| author | Struwe, Michael |
| author_facet | Struwe, Michael |
| contents | For given functions $f$ and $j$ on the disc $B$ and its boundary $\partial B=S^1$, we study the existence of conformal metrics $g=e^{2u}g_0$ with prescribed Gauss curvature $K_g=f$ and boundary geodesic curvature $k_g=j$. Using the variational characterization of such metrics obtained by Cruz-Blazquez and Ruiz (2018), we show that there is a canonical negative gradient flow of such metrics, either converging to a solution of the prescribed curvature problem, or blowing up to a spherical cap. In the latter case, similar to our work Struwe (2005) on the prescribed curvature problem on the sphere, we are able to exhibit a $2$-dimensional shadow flow for the center of mass of the evolving metrics from which we obtain existence results complementing the results recently obtained by Ruiz (2021) by degree-theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_13377 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The prescribed curvature flow on the disc Struwe, Michael Analysis of PDEs Differential Geometry 53E99 (Primary) 35K55 (Secondary) For given functions $f$ and $j$ on the disc $B$ and its boundary $\partial B=S^1$, we study the existence of conformal metrics $g=e^{2u}g_0$ with prescribed Gauss curvature $K_g=f$ and boundary geodesic curvature $k_g=j$. Using the variational characterization of such metrics obtained by Cruz-Blazquez and Ruiz (2018), we show that there is a canonical negative gradient flow of such metrics, either converging to a solution of the prescribed curvature problem, or blowing up to a spherical cap. In the latter case, similar to our work Struwe (2005) on the prescribed curvature problem on the sphere, we are able to exhibit a $2$-dimensional shadow flow for the center of mass of the evolving metrics from which we obtain existence results complementing the results recently obtained by Ruiz (2021) by degree-theory. |
| title | The prescribed curvature flow on the disc |
| topic | Analysis of PDEs Differential Geometry 53E99 (Primary) 35K55 (Secondary) |
| url | https://arxiv.org/abs/2401.13377 |