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A new proof of a theorem of Petersen

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Abstract

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,S n) → 0; 2) the volume of M Vol(M) → Vol(S n); 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.

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Correspondence to YiHu Yang.

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Yang, Y., Zhang, Y. A new proof of a theorem of Petersen. Sci. China Math. 59, 935–944 (2016). https://doi.org/10.1007/s11425-015-5091-4

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  • DOI: https://doi.org/10.1007/s11425-015-5091-4

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