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

摘要

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.