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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2401.02087 |
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| _version_ | 1866913558591176704 |
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| author | Chen, Xuezhang Shi, Yalong |
| author_facet | Chen, Xuezhang Shi, Yalong |
| contents | We derive explicit representation formulae of Green functions for GJMS operators on $n$-spheres, including the fractional ones. These formulae have natural geometric interpretations concerning the extrinsic geometry of the round sphere. Conversely, we discover that this special feature uniquely characterizes spheres among closed embedded hypersurfaces in $\mathbb{R}^{n+1}$. Furthermore, for $n=3,4,5$ we prove a strong rigidity theorem for Green functions of hypersurfaces in $\mathbb{R}^{n+1}$ using the Positive Mass Theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_02087 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Green functions for GJMS operators on spheres, Gegenbauer polynomials and rigidity theorems Chen, Xuezhang Shi, Yalong Differential Geometry Analysis of PDEs 35J08, 53C24 We derive explicit representation formulae of Green functions for GJMS operators on $n$-spheres, including the fractional ones. These formulae have natural geometric interpretations concerning the extrinsic geometry of the round sphere. Conversely, we discover that this special feature uniquely characterizes spheres among closed embedded hypersurfaces in $\mathbb{R}^{n+1}$. Furthermore, for $n=3,4,5$ we prove a strong rigidity theorem for Green functions of hypersurfaces in $\mathbb{R}^{n+1}$ using the Positive Mass Theorem. |
| title | Green functions for GJMS operators on spheres, Gegenbauer polynomials and rigidity theorems |
| topic | Differential Geometry Analysis of PDEs 35J08, 53C24 |
| url | https://arxiv.org/abs/2401.02087 |