MiS Preprint Repository

We have decided to discontinue the publication of preprints on our preprint server end of 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

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<δ1. 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 δKg1 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 Klingenberg-Sakai 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., distgd0 and δKg1) 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 can not converge to a metric space of strictly lower dimension.

Received:
07.11.99
Published:
07.11.99

Related publications

inJournal
1999 Repository Open Access
Anton Petrunin, Xiaochun Rong and Wilderich Tuschmann

Collapsing vs. positive pinching

In: Geometric and functional analysis, 9 (1999) 4, pp. 699-735