$p$-Eigenvalue pinching sphere theorems

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.