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MiS Preprint
11/1999
Collapsing vs. positive pinching
Anton Petrunin, Xiaochun Rong and Wilderich Tuschmann
Abstract
Let M be a closed simply connected manifold and $0<\delta \leq 1$. 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 $\delta \leq K_g \leq 1$ 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., $dist_g \leq d_0$ and $\delta \leq K_g \leq 1$) 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 can not converge to a metric space of strictly lower dimension.
