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 lambda-bar. 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 + infinity whenever the Yamabe invariant is positive. 
Kazuo Akutagawa, Masashi Ishida, and Claude LeBrun