$p$-Eigenvalue pinching sphere theorems
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866918387486031872 |
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| author | Silva, Paulo Henryque C. |
| author_facet | Silva, Paulo Henryque C. |
| contents | In this paper, we establish two $p$-eigenvalue pinching sphere theorems, for the \( p \)-Laplacian, $p>1$. The first result states that if the first non-zero $p$-eigenvalue of a closed Riemannian $n$-manifold with sectional curvature $K_{M}\geq 1$ is sufficiently close to the first non-zero $p$-eigenvalue of $\mathbb{S}^{n}$ then $M$ is homeomorphic to $\mathbb{S}^{n}$. The second states that if the first non-zero $p$-eigenvalue of a closed Riemannian $n$-manifold with Ricci curvature ${\rm Ric}_{M}\geq (n-1)$ and injectivity radius ${\rm inj}_{M}\geq i_0>0$ is sufficiently close to the first non-zero $p$-eigenvalue of $\mathbb{S}^{n}$ then $M$ is diffeomorphic to $\mathbb{S}^{n}$. Our results extend sphere theorems originally settled for the Laplacian by S. Croke~\cite{Croke1982} and G.P. Bessa~\cite{bessa} respectively. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_22962 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $p$-Eigenvalue pinching sphere theorems Silva, Paulo Henryque C. Differential Geometry In this paper, we establish two $p$-eigenvalue pinching sphere theorems, for the \( p \)-Laplacian, $p>1$. The first result states that if the first non-zero $p$-eigenvalue of a closed Riemannian $n$-manifold with sectional curvature $K_{M}\geq 1$ is sufficiently close to the first non-zero $p$-eigenvalue of $\mathbb{S}^{n}$ then $M$ is homeomorphic to $\mathbb{S}^{n}$. The second states that if the first non-zero $p$-eigenvalue of a closed Riemannian $n$-manifold with Ricci curvature ${\rm Ric}_{M}\geq (n-1)$ and injectivity radius ${\rm inj}_{M}\geq i_0>0$ is sufficiently close to the first non-zero $p$-eigenvalue of $\mathbb{S}^{n}$ then $M$ is diffeomorphic to $\mathbb{S}^{n}$. Our results extend sphere theorems originally settled for the Laplacian by S. Croke~\cite{Croke1982} and G.P. Bessa~\cite{bessa} respectively. |
| title | $p$-Eigenvalue pinching sphere theorems |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2506.22962 |