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| Main Author: | |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2506.22962 |
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Table of 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.