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 ≥ 1 is sufficiently close to the first non-zero p -eigenvalue of S n then M is homeomorphic to S n . The second states that if the first non-zero p -eigenvalue of a closed Riemannian n -manifold with Ricci curvature R i c M ≥ ( n − 1 ) and injectivity radius inj M ≥ i 0 > 0 is sufficiently close to the first non-zero p -eigenvalue of S n then M is diffeomorphic to S n . Our results extend sphere theorems originally settled for the Laplacian by S. Croke [1] and G.P. Bessa [2] respectively.