On the Nirenberg problem on spheres: Arbitrarily many solutions in a perturbative setting
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866916337260953600 |
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| author | Ahmedou, Mohameden Ayed, Mohamed Ben Mehdi, Khalil El |
| author_facet | Ahmedou, Mohameden Ayed, Mohamed Ben Mehdi, Khalil El |
| contents | Given a smooth positive function $K$ on the standard sphere $(\mathbb{S}^n,g_0)$, we use Morse theoretical methods and counting index formulae to prove that, under generic conditions on the function $K$, there are arbitrarily many metrics $g$ conformally equivalent to $g_0$ and whose scalar curvature is given by the function $K$ provided that the function is sufficiently close to the scalar curvature of $g_0$. Our approach leverages a comprehensive characterization of blowing-up solutions of a subcritical approximation, along with various Morse relations involving their indices. Notably, this multiplicity result is achieved without relying on any symmetry or periodicity assumptions about the function $K$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_18622 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Nirenberg problem on spheres: Arbitrarily many solutions in a perturbative setting Ahmedou, Mohameden Ayed, Mohamed Ben Mehdi, Khalil El Analysis of PDEs Differential Geometry 35A01, 58J05, 58E05 Given a smooth positive function $K$ on the standard sphere $(\mathbb{S}^n,g_0)$, we use Morse theoretical methods and counting index formulae to prove that, under generic conditions on the function $K$, there are arbitrarily many metrics $g$ conformally equivalent to $g_0$ and whose scalar curvature is given by the function $K$ provided that the function is sufficiently close to the scalar curvature of $g_0$. Our approach leverages a comprehensive characterization of blowing-up solutions of a subcritical approximation, along with various Morse relations involving their indices. Notably, this multiplicity result is achieved without relying on any symmetry or periodicity assumptions about the function $K$. |
| title | On the Nirenberg problem on spheres: Arbitrarily many solutions in a perturbative setting |
| topic | Analysis of PDEs Differential Geometry 35A01, 58J05, 58E05 |
| url | https://arxiv.org/abs/2407.18622 |