, Volume 88, Issue 1, pp 71-76,
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Date: 18 Dec 2006

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