Abstract.
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.
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Received: 6 October 2006; Revised: 17 October 2006
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License ( https://creativecommons.org/licenses/by-nc/2.0 ), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Akutagawa, K., Ishida, M. & LeBrun, C. Perelman’s invariant, Ricci flow, and the Yamabe invariants of smooth manifolds. Arch. Math. 88, 71–76 (2007). https://doi.org/10.1007/s00013-006-2181-0
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DOI: https://doi.org/10.1007/s00013-006-2181-0