On eigenvalue pinching in positive Ricci curvature
On eigenvalue pinching in positive Ricci curvature
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DOI：
10.1007/s002220050339
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发表时间：
1999-10
影响因子：
3.1
通讯作者：
P. Petersen
中科院分区：
文献类型：
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作者：
P. Petersen
我们将证明对于Ric≥n−1的流形，半径接近π当且仅当第（n+1）个特征值接近ton。这推广了Cheng和Croke的结果，即当第一特征值接近ton时，直径接近π.我们还将给出一个新的证明的一个重要定理的冷的效果，如果半径接近π，那么体积接近的领域和流形是Gromov-Hausdorff接近的领域。从工作的Cheeger和Colding这些条件意味着，流形是一个领域的同胚。
We shall show that for manifolds with Ric≥n−1 the radius is close to π iff the (n+1)st eigenvalue is close ton. This extends results of Cheng and Croke which show that the diameter is close to π iff the first eigenvalue is close ton. We shall also give a new proof of an important theorem of Colding to the effect that if the radius is close to π, then the volume is close to that of the sphere and the manifold is Gromov-Hausdorff close to the sphere. From work of Cheeger and Colding these conditions imply that the manifold is diffeomorphic to a sphere.