Quantitative volume space form rigidity under lower Ricci curvature bound I

Lina Chen, Xiaochun Rong, Shicheng Xu

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6 Scopus citations

Abstract

Let M be a compact n-manifold of RicM ≥ (n − 1)H (H is a constant). We are concerned with the following space form rigidity: M is isometric to a space form of constant curvature H under either of the following conditions: (i) There is ρ > 0 such that for any x ∈ M, the open ρ-ball at x in the (local) Riemannian universal covering space, (Uρ , x) → (Bρ(x), x), has the maximal volume, i.e., the volume of a ρ-ball in the simply connected n-space form of curvature H. (ii) For H = −1, the volume entropy of M is maximal, i.e., n − 1 ([LW1]). The main results of this paper are quantitative space form rigidity, i.e., statements that M is diffeomorphic and close in the Gromov–Hausdorff topology to a space form of constant curvature H, if M almost satisfies, under some additional condition, the above maximal volume condition. For H = 1, the quantitative spherical space form rigidity improves and generalizes the diffeomorphic sphere theorem in [CC2].

Original languageEnglish (US)
Pages (from-to)227-272
Number of pages46
JournalJournal of Differential Geometry
Volume113
Issue number2
DOIs
StatePublished - 2019

All Science Journal Classification (ASJC) codes

  • Analysis
  • Algebra and Number Theory
  • Geometry and Topology

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