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A generalization of an inequality of Li and Zhong, and its geometric application. (English) Zbl 0532.53032
The authors generalize the pinching theorem of Li, Zhong, Treibergs [P. Li and J. Zhong, Invent. Math. 65, 221-225 (1981; Zbl 0496.53031); P. Li and A. E. Treibergs, ibid. 66, 35-38 (1982; Zbl 0496.53032)] which is in terms of the first eigenvalue of the Laplacian, from \(n\leq 4\) to arbitrary dimension n. The generalization is under the additional hypothesis that the first eigenfunction u satisfies \(| \quad \inf u/\sup u\quad |^ 2+| \quad \inf u/\sup u\quad |< 1\).
Reviewer: P.Buser
53C20 Global Riemannian geometry, including pinching
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
58J50 Spectral problems; spectral geometry; scattering theory on manifolds