A new proof of a theorem of Petersen

Let M be an n-dimensional complete Riemannian manifold with Ricci curvature ? n - 1. By developing some new techniques, Colding (1996) proved that the following three conditions are equivalent: 1) d_GH(M,Sⁿ) → 0; 2) the volume of M Vol(M) → Vol(Sⁿ); 3) the radius of M rad(M) → π. By developing a different technique, Petersen (1999) gave the 4-th equivalent condition, namely he proved that the n + 1-th eigenvalue of M, λ_n+1(M) → n, is also equivalent to the radius of M, rad(M) → π, and hence the other two. In this paper, we use Colding’s techniques to give a new proof of Petersen’s theorem. We expect our estimates will have further applications.