Existence and uniqueness of periodic pseudospherical surfaces emanating from Cauchy problems
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866908078298890240 |
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| author | Mutlubas, Nilay Duruk Freire, Igor Leite |
| author_facet | Mutlubas, Nilay Duruk Freire, Igor Leite |
| contents | We study implications and consequences of well-posed solutions of Cauchy problems of a Novikov equation describing pseudospherical surfaces. We show that if the co-frame of dual one-forms satisfies certain conditions for a given periodic initial datum, then there exists exactly two families of periodic one-forms satisfying the structural equations for a surface. Each pair then defines a metric of constant Gaussian curvature and a corresponding Levi-Civita connection form. We prove the existence of universal connection forms giving rise to second fundamental forms compatible with the metric. The main tool to prove our geometrical results is the Kato's semi-group approach, which is used to establish well-posedness of solutions of the Cauchy problem involved and ensure $C^1$ regularity for the first fundamental form and the Levi-Civita connection form. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2309_06291 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Existence and uniqueness of periodic pseudospherical surfaces emanating from Cauchy problems Mutlubas, Nilay Duruk Freire, Igor Leite Differential Geometry Mathematical Physics Analysis of PDEs 35B10, 53A05, 58J60, 35A30 We study implications and consequences of well-posed solutions of Cauchy problems of a Novikov equation describing pseudospherical surfaces. We show that if the co-frame of dual one-forms satisfies certain conditions for a given periodic initial datum, then there exists exactly two families of periodic one-forms satisfying the structural equations for a surface. Each pair then defines a metric of constant Gaussian curvature and a corresponding Levi-Civita connection form. We prove the existence of universal connection forms giving rise to second fundamental forms compatible with the metric. The main tool to prove our geometrical results is the Kato's semi-group approach, which is used to establish well-posedness of solutions of the Cauchy problem involved and ensure $C^1$ regularity for the first fundamental form and the Levi-Civita connection form. |
| title | Existence and uniqueness of periodic pseudospherical surfaces emanating from Cauchy problems |
| topic | Differential Geometry Mathematical Physics Analysis of PDEs 35B10, 53A05, 58J60, 35A30 |
| url | https://arxiv.org/abs/2309.06291 |