Curvature bounds for the spectrum of closed Einstein spaces and Simon conjecture
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866913377179140096 |
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| author | Guan, ShanLin Guo, Zhen |
| author_facet | Guan, ShanLin Guo, Zhen |
| contents | Let $(M^{n}, g)$ be a closed connected Einstein space, $n=dim M ,$ and $κ_{0} $ be the lower bound of the sectional curvature. In this paper, we prove Udo Simon's conjecture: on closed Einstein spaces, $n\geq 3,$ there is no eigenvalue $λ$ such that $$nκ_{0} < λ< 2(n + 1)κ_{0},$$ and both bounds are the best possible. Furthermore, we develop Simon's conjecture to the next gap of eigenvalue $λ:$ on closed Einstein spaces, there is no $λ$ such that $$
2(n + 1)κ_{0}< λ< 2(n+2)κ_{0}, $$ and both bounds are the best possible. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_10425 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Curvature bounds for the spectrum of closed Einstein spaces and Simon conjecture Guan, ShanLin Guo, Zhen Differential Geometry 53C24, 53C25, 53C21, 53C80 Let $(M^{n}, g)$ be a closed connected Einstein space, $n=dim M ,$ and $κ_{0} $ be the lower bound of the sectional curvature. In this paper, we prove Udo Simon's conjecture: on closed Einstein spaces, $n\geq 3,$ there is no eigenvalue $λ$ such that $$nκ_{0} < λ< 2(n + 1)κ_{0},$$ and both bounds are the best possible. Furthermore, we develop Simon's conjecture to the next gap of eigenvalue $λ:$ on closed Einstein spaces, there is no $λ$ such that $$ 2(n + 1)κ_{0}< λ< 2(n+2)κ_{0}, $$ and both bounds are the best possible. |
| title | Curvature bounds for the spectrum of closed Einstein spaces and Simon conjecture |
| topic | Differential Geometry 53C24, 53C25, 53C21, 53C80 |
| url | https://arxiv.org/abs/2304.10425 |