Archiv der Mathematik

, Volume 88, Issue 1, pp 71-76

First online:

Open Access This content is freely available online to anyone, anywhere at any time.

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

  • Kazuo AkutagawaAffiliated withDept. Mathematics, Tokyo Univ. of Science
  • , Masashi IshidaAffiliated withDepartment of Mathematics, SUNYDepartment of Mathematics, Sophia University
  • , Claude LeBrunAffiliated withDepartment of Mathematics, SUNY Email author 


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