Collapsing vs. Positive Pinching

Abstract.

Let M be a closed simply connected manifold and 0 < $\delta \le 1$. Klingenberg and Sakai conjectured that there exists a constant ${i_0} = {i_0}(M,\delta) > 0$ such that the injectivity radius of any Riemannian metric g on M with $\delta \le {K_g} \le 1$ can be estimated from below by i ₀. 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 d ₀ on M, there exists a constant ${i_0} = {i_0}(M,{d_0},\delta) > 0$, such that the injectivity radius of any $\delta$-pinched d ₀-bounded Riemannian metric g on M (i.e., ${\rm dist}_g \le d_0$ and $\delta \le K_g \le 1)$ can be estimated from below by i ₀. 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.