The Weyl problem of isometric immersions revisited
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2020
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| _version_ | 1866914340130521088 |
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| author | Li, Siran |
| author_facet | Li, Siran |
| contents | We revisit the classical problem due to Weyl, as well as its generalisations, concerning the isometric immersions of $\mathbb{S}^2$ into simply-connected $3$-dimensional Riemannian manifolds with non-negative Gauss curvature. A sufficient condition is exhibited for the existence of global $C^{1,1}$-isometric immersions. Our developments are based on the framework à la Labourie (Immersions isométriques elliptiques et courbes pseudo-holomorphes, J. Diff. Geom. 30 (1989), 395--424) of studying isometric immersions using $J$-holomorphic curves. We obtain along the way a generalisation of a classical theorem due to Heinz and Pogorelov. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2004_05532 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | The Weyl problem of isometric immersions revisited Li, Siran Differential Geometry Analysis of PDEs 35J60, 53C23, 53C42, 53C21, 53C20 We revisit the classical problem due to Weyl, as well as its generalisations, concerning the isometric immersions of $\mathbb{S}^2$ into simply-connected $3$-dimensional Riemannian manifolds with non-negative Gauss curvature. A sufficient condition is exhibited for the existence of global $C^{1,1}$-isometric immersions. Our developments are based on the framework à la Labourie (Immersions isométriques elliptiques et courbes pseudo-holomorphes, J. Diff. Geom. 30 (1989), 395--424) of studying isometric immersions using $J$-holomorphic curves. We obtain along the way a generalisation of a classical theorem due to Heinz and Pogorelov. |
| title | The Weyl problem of isometric immersions revisited |
| topic | Differential Geometry Analysis of PDEs 35J60, 53C23, 53C42, 53C21, 53C20 |
| url | https://arxiv.org/abs/2004.05532 |