Smooth structures, normalized Ricci flows, and finite cyclic groups

Masashi Ishida, Ioana Şuvaina

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)


A solution to the normalized Ricci flow is called non-singular if it exists for all time with uniformly bounded sectional curvature. By using the techniques developed by the present authors [Ishida, The normalized Ricci flow on four-manifolds and exotic smooth structures; Şuvaina, Einstein metrics and smooth structures on non-simply connected 4-manifolds] we prove that for any finite cyclic group Zd, where d > 1, there exist infinitely many compact topological 4-manifolds, with fundamental group Zd, which admit at least one smooth structure for which non-singular solutions of the normalized Ricci flow exist, but also admit infinitely many distinct smooth structures for which no non-singular solution of the normalized Ricci flow exists. We show that there are no non-singular ℤd -equivariant, d > 1, solutions to the normalized Ricci flow on appropriate connected sums of ℂℙ2s and d > 1/ℂℙ2s.

Original languageEnglish
Pages (from-to)267-275
Number of pages9
JournalAnnals of Global Analysis and Geometry
Issue number3
Publication statusPublished - 2009 May


  • Exotic smooth structures
  • Finite cyclic fundamental groups
  • Normalized Ricci flows
  • Seiberg-Witten theory

ASJC Scopus subject areas

  • Analysis
  • Political Science and International Relations
  • Geometry and Topology


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