Archiv der Mathematik

, Volume 88, Issue 1, pp 71–76

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


  • Kazuo Akutagawa
    • Dept. MathematicsTokyo Univ. of Science
  • Masashi Ishida
    • Department of MathematicsSUNY
    • Department of MathematicsSophia University
    • Department of MathematicsSUNY
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 53C21 Secondary 58J50


Scalar curvature Ricci flow conformal geometry Perelman invariant Yamabe problem

Copyright information

© Birkhäuser Verlag, Basel/Switzerland 2006