A new proof of a theorem of Petersen

Let $M$ be an $n$-dimensional complete Riemannian manifold with Ricci curvature $\ge n-1$. In \cite{colding1, colding2}, Tobias Colding, by developing some new techniques, proved that the following three condtions: 1) $d_{GH}(M, S^n)\to 0$; 2) the volume of $M$ ${\text{Vol}}(M)\to{\text{Vol}}(S^n)$; 3) the radius of $M$ ${\text{rad}}(M)\to\pi$ are equivalent. In \cite{peter}, Peter Petersen, by developing a different technique, gave the 4-th equivalent condition, namely he proved that the $n+1$-th eigenvalue of $M$ $\lambda_{n+1}(M)\to n$ is also equivalent to the radius of $M$ ${\text{rad}}(M)\to\pi$, and hence the other two. In this note, we give a new proof of Petersen's theorem by utilizing Colding's techniques.