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

Kazuo Akutagawa; Masashi Ishida; Claude LeBrun

doi:10.1007/s00013-006-2181-0

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

Kazuo Akutagawa
, Masashi Ishida
, Claude LeBrun

Mathematics

Tokyo University of Science
Stony Brook University
Sophia University

Research output: Contribution to journal › Article › peer-review

23 Scopus citations

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.

Original language	English
Pages (from-to)	71-76
Number of pages	6
Journal	Archiv der Mathematik
Volume	88
Issue number	1
DOIs	https://doi.org/10.1007/s00013-006-2181-0
State	Published - Jan 2007

Keywords

Conformal geometry
Perelman invariant
Ricci flow
Scalar curvature
Yamabe problem

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10.1007/s00013-006-2181-0

Cite this

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title = "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.",

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AU - Ishida, Masashi

AU - LeBrun, Claude

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AB - 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.

KW - Conformal geometry

KW - Perelman invariant

KW - Ricci flow

KW - Scalar curvature

KW - Yamabe problem

UR - https://www.scopus.com/pages/publications/33846981854

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

Abstract

Keywords

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