Collapsing vs. positive pinching

Anton Petrunin, Xiaochun Rong, Wilderich Tuschmann

Research output: Contribution to journalArticle

10 Scopus citations

Abstract

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.

Original languageEnglish (US)
Pages (from-to)699-735
Number of pages37
JournalGeometric and Functional Analysis
Volume9
Issue number4
DOIs
StatePublished - Jan 1 1999

All Science Journal Classification (ASJC) codes

  • Analysis
  • Geometry and Topology

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