$p$-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 $K_M\ge 1$ 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 ${\mathrm{R}\mathrm{i}\mathrm{c}}_M\ge (n-1)$ and injectivity radius ${\mathrm{i}\mathrm{n}\mathrm{j}}_M\ge i_0>0$ 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.