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

Abstract

In his study of Ricci flow, Perelman introduced a smooth-manifold invariant called λ. 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. © 2006 Birkhäuser Verlag.