First eigenvalue of the $p$-Laplace operator along the Ricci flow
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arXiv
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| Format: | Preprint |
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2009
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| _version_ | 1866910058281959424 |
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| author | Wu, Jia-Yong Wang, Er-Min Zheng, Yu |
| author_facet | Wu, Jia-Yong Wang, Er-Min Zheng, Yu |
| contents | In this paper, we mainly investigate continuity, monotonicity and differentiability for the first eigenvalue of the $p$-Laplace operator along the Ricci flow on closed manifolds. We show that the first $p$-eigenvalue is strictly increasing and differentiable almost everywhere along the Ricci flow under some curvature assumptions. In particular, for an orientable closed surface, we construct various monotonic quantities and prove that the first $p$-eigenvalue is differentiable almost everywhere along the Ricci flow without any curvature assumption, and therefore derive a $p$-eigenvalue comparison-type theorem when its Euler characteristic is negative. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_0912_4775 |
| institution | arXiv |
| publishDate | 2009 |
| record_format | arxiv |
| spellingShingle | First eigenvalue of the $p$-Laplace operator along the Ricci flow Wu, Jia-Yong Wang, Er-Min Zheng, Yu Differential Geometry Analysis of PDEs 58C40, 53C44 In this paper, we mainly investigate continuity, monotonicity and differentiability for the first eigenvalue of the $p$-Laplace operator along the Ricci flow on closed manifolds. We show that the first $p$-eigenvalue is strictly increasing and differentiable almost everywhere along the Ricci flow under some curvature assumptions. In particular, for an orientable closed surface, we construct various monotonic quantities and prove that the first $p$-eigenvalue is differentiable almost everywhere along the Ricci flow without any curvature assumption, and therefore derive a $p$-eigenvalue comparison-type theorem when its Euler characteristic is negative. |
| title | First eigenvalue of the $p$-Laplace operator along the Ricci flow |
| topic | Differential Geometry Analysis of PDEs 58C40, 53C44 |
| url | https://arxiv.org/abs/0912.4775 |