, Volume 88, Issue 1, pp 71-76,
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


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.

Received: 6 October 2006; Revised: 17 October 2006