Archiv der Mathematik

, Volume 88, Issue 1, pp 71–76

Perelman’s invariant, Ricci flow, and the Yamabe invariants of smooth manifolds

Open AccessArticle

DOI: 10.1007/s00013-006-2181-0

Cite this article as:
Akutagawa, K., Ishida, M. & LeBrun, C. Arch. Math. (2007) 88: 71. doi:10.1007/s00013-006-2181-0


In his study of Ricci flow, Perelman introduced a smooth-manifold invariant called \(\bar{\lambda}\). We show here that, for completely elementary reasons, this invariant simply equals the Yamabe invariant, alias the sigma constant, whenever the latter is non-positive. On the other hand, the Perelman invariant just equals +∞ whenever the Yamabe invariant is positive.

Mathematics Subject Classification (2000).

Primary 53C21Secondary 58J50


Scalar curvatureRicci flowconformal geometryPerelman invariantYamabe problem
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Copyright information

© Birkhäuser Verlag, Basel/Switzerland 2006

Authors and Affiliations

  • Kazuo Akutagawa
    • 1
  • Masashi Ishida
    • 2
    • 3
  • Claude LeBrun
    • 2
  1. 1.Dept. MathematicsTokyo Univ. of ScienceNodaJapan
  2. 2.Department of MathematicsSUNYStony BrookUSA
  3. 3.Department of MathematicsSophia UniversityTokyoJapan