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

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

20 Citations (Scopus)

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
Publication status	Published - 2007 Jan
Externally published	Yes

Keywords

Conformal geometry
Perelman invariant
Ricci flow
Scalar curvature
Yamabe problem

ASJC Scopus subject areas

Mathematics(all)

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10.1007/s00013-006-2181-0

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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.",

keywords = "Conformal geometry, Perelman invariant, Ricci flow, Scalar curvature, Yamabe problem",

author = "Kazuo Akutagawa and Masashi Ishida and Claude LeBrun",

note = "Funding Information: ∗Supported in part by NSF grant DMS-0604735.",

year = "2007",

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language = "English",

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AU - Akutagawa, Kazuo

AU - Ishida, Masashi

AU - LeBrun, Claude

N1 - Funding Information: ∗Supported in part by NSF grant DMS-0604735.

PY - 2007/1

Y1 - 2007/1

N2 - 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.

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

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ER -

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

Abstract

Keywords

ASJC Scopus subject areas

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