Collapsing construction with nilpotent structures

Qingsong Cai, Xiaochun Rong

Research output: Contribution to journalArticlepeer-review

5 Scopus citations

Abstract

A fundamental result concerning collapsed manifolds with bounded sectional curvature is the existence of compatible local nilpotent symmetry structures whose orbits capture all collapsed directions of the local geometry [CFG]. The underlying topological structure is called an N-structure of positive rank. We show that if a manifold M admits such an N-structure N, then M admits a one-parameter family of metrics g with curvature bounded in absolute value while injectivity radii and the diameters of N -orbits away from the singular set of N uniformly converge to zero as ∈ → 0. Moreover, g is N -invariant away from the singular set. This result extends collapsing results in [CG1], [Fu3] and [G].

Original languageEnglish (US)
Pages (from-to)1503-1524
Number of pages22
JournalGeometric and Functional Analysis
Volume18
Issue number5
DOIs
StatePublished - Feb 2009

All Science Journal Classification (ASJC) codes

  • Analysis
  • Geometry and Topology

Keywords

  • Collapsing construction
  • Invariant metrics
  • Nilpotent structure

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