TY - JOUR

T1 - Collapsing vs. positive pinching

AU - Petrunin, Anton

AU - Rong, Xiaochun

AU - Tuschmann, Wilderich

N1 - Funding Information: During the preparation of this paper, the rst and the third author enjoyed the hospitality of the following institutions: The Max-Planck Institutes for Mathematics at Bonn and Leipzig, the Institute of Mathematical Sciences at Stony Brook, the Euler Mathematical Institute at St. Petersburg. Thanks to all of them! The second author is supported partially by NSF Grant DMS 9626252 and Alfred P. Sloan Research Fellowship.

PY - 1999

Y1 - 1999

N2 - Let M be a closed simply connected manifold and 0 < δ ≤ 1. Klingenberg and Sakai conjectured that there exists a constant io = io(M, δ) > 0 such that the injectivity radius of any Riemannian metric g on M with δ ≤ Kg ≤ 1 can be estimated from below by io. 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 do on M, there exists a constant io = io(M, do, δ) > 0, such that the injectivity radius of any δ-pinched do-bounded Riemannian metric g on M (i.e., distg ≤ do and δ ≤ Kg ≤ 1) can be estimated from below by io. 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 cannot converge to a metric space of strictly lower dimension.

AB - Let M be a closed simply connected manifold and 0 < δ ≤ 1. Klingenberg and Sakai conjectured that there exists a constant io = io(M, δ) > 0 such that the injectivity radius of any Riemannian metric g on M with δ ≤ Kg ≤ 1 can be estimated from below by io. 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 do on M, there exists a constant io = io(M, do, δ) > 0, such that the injectivity radius of any δ-pinched do-bounded Riemannian metric g on M (i.e., distg ≤ do and δ ≤ Kg ≤ 1) can be estimated from below by io. 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 cannot converge to a metric space of strictly lower dimension.

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U2 - https://doi.org/10.1007/s000390050100

DO - https://doi.org/10.1007/s000390050100

M3 - Article

VL - 9

SP - 699

EP - 735

JO - Geometric and Functional Analysis

JF - Geometric and Functional Analysis

SN - 1016-443X

IS - 4

ER -