Collapsing vs. Positive Pinching

Let M be a closed simply connected manifold and 0 < δ ≤ 1. Klingenberg and Sakai conjectured that there exists a constant i0 = i0(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 i0. We study this question by collapsing and Alexandrov space techniques. In particular we establish a bounded version of the KlingenbergSakai conjecture: Given any metric d0 on M , there exists a constant i0 = i0(M,d0, δ) > 0, such that the injectivity radius of any δ-pinched d0-bounded Riemannian metric g on M (i.e., distg ≤ d0 and δ ≤ Kg ≤ 1) can be estimated from below by i0. 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.