Collapsing vs. positive pinching

Anton Petrunin, Xiaochun Rong, and Wilderich Tuschmann

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Submission date: 07. Nov. 1999 (revised version: November 1999)
Pages: 25
published in: Geometric and functional analysis, 9 (1999) 4, p. 699-735 
DOI number (of the published article): 10.1007/s000390050100
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Abstract:
Let M be a closed simply connected manifold and . Klingenberg and Sakai conjectured that there exists a constant such that the injectivity radius of any Riemannian metric g on M with can be estimated from below by . We study this question by collapsing and Alexandrov space techniques. In particular we establish a bounded version of the Klingenberg-Sakai conjecture: Given any metric on M, there exists a constant , such that the injectivity radius of any -pinched -bounded Riemannian metric g on M (i.e., and ) can be estimated from below by . We also establish a continuous version of the Klingenberg-Sakai conjecture, saying that a continuous family of metrics on M with positively uniformly pinched curvature can not converge to a metric space of strictly lower dimension.