Sharp pinching theorems for complete submanifolds in the sphere
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909347131424768 |
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| author | Magliaro, Marco Mari, Luciano Roing, Fernanda Savas-Halilaj, Andreas |
| author_facet | Magliaro, Marco Mari, Luciano Roing, Fernanda Savas-Halilaj, Andreas |
| contents | We prove that every complete, minimally immersed submanifold $f\: M^n \to \mathbb{S}^{n+p}$ whose second fundamental form satisfies $|A|^2 \le np/(2p-1)$, is either totally geodesic, or (a covering of) a Clifford torus or a Veronese surface in $\mathbb{S}^4$, thereby extending the well-known results by Simons, Lawson and Chern, do Carmo & Kobayashi from compact to complete $M^n$. We also obtain the corresponding result for complete hypersurfaces with nonvanishing constant mean curvature, due to Alencar & do Carmo in the compact case, under the optimal bound on the umbilicity tensor. In dimension $n \le 6$, a pinching theorem for complete higher-codimensional submanifolds with non-vanishing parallel mean curvature is proved, partly generalizing previous work of Santos. Our approach is inspired by the conformal method of Fischer-Colbrie, Shen & Ye and Catino, Mastrolia & Roncoroni. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_17861 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sharp pinching theorems for complete submanifolds in the sphere Magliaro, Marco Mari, Luciano Roing, Fernanda Savas-Halilaj, Andreas Differential Geometry We prove that every complete, minimally immersed submanifold $f\: M^n \to \mathbb{S}^{n+p}$ whose second fundamental form satisfies $|A|^2 \le np/(2p-1)$, is either totally geodesic, or (a covering of) a Clifford torus or a Veronese surface in $\mathbb{S}^4$, thereby extending the well-known results by Simons, Lawson and Chern, do Carmo & Kobayashi from compact to complete $M^n$. We also obtain the corresponding result for complete hypersurfaces with nonvanishing constant mean curvature, due to Alencar & do Carmo in the compact case, under the optimal bound on the umbilicity tensor. In dimension $n \le 6$, a pinching theorem for complete higher-codimensional submanifolds with non-vanishing parallel mean curvature is proved, partly generalizing previous work of Santos. Our approach is inspired by the conformal method of Fischer-Colbrie, Shen & Ye and Catino, Mastrolia & Roncoroni. |
| title | Sharp pinching theorems for complete submanifolds in the sphere |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2401.17861 |