Archiv der Mathematik

, Volume 88, Issue 1, pp 71–76 | Cite as

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

Open Access


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 53C21 Secondary 58J50 


Scalar curvature Ricci flow conformal geometry Perelman invariant Yamabe problem 

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

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