In his study of Ricci flow, Perelman introduced a smooth-manifold invariant
called lambda-bar. 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 + infinity whenever the Yamabe invariant is positive.Comment: LaTeX2e, 7 pages. To appear in Arch. Math. Revised version improves
  result to also cover positive cas

Akutagawa, Kazuo

Ishida, Masashi

LeBrun, Claude

English

arXiv

Kazuo KazuoAkutagawa

Masashi Ishida

Claude LeBrun

Springer

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

Kazuo Akutagawa

Springer - Publisher Connector

Arch. Math. 88 (2007), 71–76
c© 2006 Birkha¨user Verlag Basel/Switzerland
0003/889X/010071-6, published online 2006-12-18
DOI 10.1007/s00013-006-2181-0 Archiv der Mathematik
Perelman’s invariant, Ricci ﬂow, and the Yamabe invariants
of smooth manifolds
Kazuo Akutagawa, Masashi Ishida and Claude LeBrun∗
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.
Mathematics Subject Classiﬁcation (2000). Primary 53C21; Secondary 58J50.
Keywords. Scalar curvature, Ricci flow, conformal geometry, Perelman invari-
ant, Yamabe problem.
Let M be a smooth compact manifold of dimension n ≥ 3. Perelman’s cele-
brated work on Ricci ﬂow [12, 13] led him to consider the functional which asso-
ciates to every Riemannian metric g the least eigenvalue λg of the elliptic operator
4∆g +sg, where sg denotes the scalar curvature of g, and ∆ = d∗d = −∇·∇ is the
positive-spectrum Laplace-Beltrami operator associated with g. In other words,
λg can be expressed in terms of Raleigh quotients as
λg = inf
u
∫
M
[
sgu
2 + 4|∇u|2] dµ∫
M
u2dµ
where the inﬁmum is taken over all smooth, real-valued functions u on M .
One of Perelman’s remarkable observations is that the scale-invariant quan-
tity λgV
2/n
g is non-decreasing under the Ricci ﬂow, where Vg =
∫
M
dµg denotes
the total volume of (M, g). This led him to consider the diﬀerential-topological
invariant obtained by taking the supremum of this quantity over the space of all
Riemannian metrics [13, 6]:
∗Supported in part by NSF grant DMS-0604735.
72 K. Akutagawa, M. Ishida and C. LeBrun Arch. Math.
Deﬁnition 1. Let M be a smooth compact n-manifold, n ≥ 3. Perelman’s λ¯
invariant of M is deﬁned to be
λ¯(M) = sup
g
λgV
2/n
g
where the supremum is taken over all smooth metrics g on M .
The roˆle of the scalar curvature and Laplacian in deﬁning λg might immediately
make one wonder whether this invariant might somehow be related to the Yamabe
problem. Recall that, as was conjectured by Yamabe [21], and later proved by
Trudinger, Aubin, and Schoen [2, 3, 11, 18, 20], every conformal class on any
smooth compact manifold contains a metric of constant scalar curvature. If M is
a smooth compact manifold of dimension n ≥ 3, and if
γ = [g] = {vg | v : M → R+},
is the conformal class of an arbitrary metric, such a metric gˆ can in fact be con-
structed by minimizing the normalized total-scalar-curvature functional
gˆ →
∫
M
sgˆ dµgˆ(∫
M
dµgˆ
)n−2
n
,
among all metrics conformal to g. Indeed, by setting gˆ = u4/(n−2)g, this expression
can be rewritten as ∫
M
sgˆ dµgˆ(∫
M
dµgˆ
)n−2
n
=
∫
M
[
sgu
2 + 4n−1n−2 |∇u|2
]
dµg(∫
M
u2n/(n−2)dµg
)(n−2)/n ,
and the proof proceeds by showing that there is a smooth positive function u which
minimizes the right-hand expression. In particular, each conformal class γ has an
associated number Yγ , called its Yamabe constant, obtained by setting
Yγ = inf
g∈γ
∫
M
sg dµg(∫
M
dµg
)n−2
n
;
the content of the Trudinger-Aubin-Schoen theorem is exactly that this number
is actually realized as the constant scalar curvature of some unit-volume metric in
γ. A constant-scalar-curvature metric of this type is called a Yamabe minimizer.
It is not diﬃcult to show that any Riemannian metric with s = const ≤ 0 is a
Yamabe minimizer, and that, moreover, the Yamabe minimizer g ∈ γ is unique (up
to constant rescaling) whenever Yγ ≤ 0. The situation is much more complicated
when Yγ > 0, but it is still not diﬃcult to see that if g is a metric for which s has a
ﬁxed sign (positive, zero, or negative) everywhere on M , then this sign necessarily
agrees with that of the number Y[g].
Yamabe’s work was apparently motivated by the hope of constructing Einstein
metrics via a variational approach. This idea eventually led Kobayashi [7] and
Vol. 88 (2007) Perelman’s invariant, Ricci flow, and the Yamabe invariants 73
Schoen [19] to independently introduce the smooth manifold invariant
Y(M) = sup
γ
Yγ = sup
γ
inf
g∈γ
∫
M
sg dµg(∫
M
dµg
)n−2
n
.
By construction, this is a diﬀeomorphism invariant of M , and is now commonly
known as the Yamabe invariant of M ; note, however, that Schoen called Y(M) the
sigma constant, and that this terminology is still preferred by some authors. Notice
that Y(M) ≤ 0 iﬀ M does not admit metrics of positive scalar curvature, and that,
when this happens, Y(M) is simply the supremum of the scalar curvatures of unit-
volume constant-scalar-curvature metrics on M .
The fact that there is some fundamental relation between the Yamabe invariant
Y(M) and Perelman’s λ¯ invariant was probably ﬁrst pointed out by Anderson [1].
More recently, an e-print by Fang and Zhang [4] computed the Perelman invariant
for a large class of 4-manifolds where the Yamabe invariants had already been
computed by the present authors [9, 5] and others [14, 15], and, as was later
emphasized by Kotschick [8], their answers exactly agree with those previously
discovered in the Yamabe case. The point of this brief note is to observe that this
was no mere matter of coincidence:
Theorem A. Suppose that M is a smooth compact n-manifold, n ≥ 3. Then
λ¯(M) =
{
Y(M) if Y(M) ≤ 0,
+∞ if Y(M) > 0.
In fact, this will follow easily once we clearly understand the behavior of λV 2/n
on each individual conformal class.
Proposition 1. Suppose that γ is a conformal class on M which does not contain
a metric of positive scalar curvature. Then
Yγ = sup
g∈γ
λgV
2/n
g
Proof. Let g ∈ γ, and let gˆ = u4/(n−2)g be the Yamabe minimizer in γ. Then
0 ≥ Yγ =
∫
M
[
sgu
2 + 4n−1n−2 |∇u|2
]
dµg(∫
M
u2n/(n−2)dµg
)(n−2)/n .
74 K. Akutagawa, M. Ishida and C. LeBrun Arch. Math.
Thus
λg
∫
u2dµ ≤
∫ [
su2 + 4|∇u|2] dµ
≤
∫ [
su2 + 4
n − 1
n − 2 |∇u|
2
]
dµ
= Yγ
(∫
u2n/(n−2)dµ
)(n−2)/n
≤ YγV −2/n
∫
u2dµ
where, since Yγ ≤ 0, the last step is an the application of the Ho¨lder inequality∫
f1f2 dµ ≤
(∫
|f1|pdµ
)1/p (∫
|f2|qdµ
)1/q
,
1
p
+
1
q
= 1,
with f1 = 1, f2 = u2, p = n/2, and q = n/(n − 2). Moreover, equality holds
precisely when u is constant — which is to say, precisely when g has constant
scalar curvature.
Since this shows that
λgV
2/n ≤ Yγ
for every g ∈ γ, and since equality occurs if g is the Yamabe minimizer, it follows
that
sup
g∈γ
λgV
2/n
g = Yγ ,
exactly as claimed. 
We now need make only one more simple observation:
Lemma 2. If M carries a metric with s > 0, then λ¯(M) = +∞.
Proof. Given such a manifold M and any smooth non-constant function f : M →
R, Kobayashi [7] has shown that there exists a unit-volume metric on M with
s = f . In particular, given any real number L, there is a unit-volume metric gL on
M with s > L everywhere. But for such a metric, λ > L and V = 1. Thus, taking
L → ∞, λ¯(M) = supg λgV 2/ng = +∞. 
Theorem A is now follows immediately. Indeed, if Y(M) > 0, M admits a
metric with s > 0, and Lemma 2 therefore tells us that λ¯(M) = +∞. Otherwise,
no conformal class contains a metric of positive scalar curvature, and Proposition
1 therefore tells us that each constant-scalar-curvature metric maximizes λV 2/n in
its conformal class. Given an arbitrary maximizing sequence gˆj for λV 2/n, we may
thus, by conformal rescaling, construct a new maximizing sequence gj consisting
of unit-volume constant-scalar-curvature metrics. But for any such sequence, the
Vol. 88 (2007) Perelman’s invariant, Ricci flow, and the Yamabe invariants 75
numbers sgj may be viewed either as {λgjV 2/ngj } or as {Y[gj ]}. Thus the suprema
over the space of all Riemannian metrics of Y[g] and λgV
2/n
g must precisely coincide.
Now, there is a substantial literature [5, 9, 10, 14, 15, 16] concerning manifolds
of non-positive Yamabe invariant, and the exact value of the invariant is moreover
known for large numbers of such manifolds. By virtue of Theorem A, all of these
facts about Y(M) may therefore immediately be interpreted as instead pertaining
to λ¯(M).
On the other hand, we have also seen that the Perelman invariant jumps to pos-
itive inﬁnity whenever there is a metric of positive scalar curvature. By contrast,
the Yamabe invariant always remains ﬁnite; indeed, one of Aubin’s fundamental
contributions to the theory of the Yamabe problem is the fact that Y(M) ≤ Y(Sn)
for any smooth compact n-manifold M . This systematic discrepancy fundamen-
tally reﬂects the fact that constant-scalar-curvature metrics are generally not min-
imizers in the positive case [7, 18], and that, moreover, constant-scalar-curvature
metrics of arbitrarily high energy exist in profusion [17] in this setting. Thus, while
we have found a fascinating link between Perelman’s invariant and the Yamabe
problem, this observation actually teaches us nothing at all precisely in the regime
where the Yamabe problem is the most technically diﬃcult and, in some respects,
still the most poorly understood.
References
[1] M. T. Anderson, Canonical metrics on 3-manifolds and 4-manifolds. Asian J. Math. 10,
127–163 (2006).
[2] T. Aubin, E´quations diﬀe´rentielles non line´aires et proble`me de Yamabe concernant la
courbure scalaire. J. Math. Pures Appl. 55(9), 269–296 (1976).
[3] T. Aubin, Some Nonlinear Problems in Riemannian Geometry, Springer Monographs in
Mathematics. Springer-Verlag, Berlin, 1998.
[4] F. Fang and Y. Zhang, Perelman’s λ functional and the Seiberg-Witten equations. e-print
math.FA/0608439.
[5] M. Ishida and C. LeBrun, Curvature, connected sums, and Seiberg-Witten theory. Comm.
Anal. Geom. 11, 809–836 (2003).
[6] B. Kleiner and J. Lott, Notes on Perelman’s papers. e-print math.DG/0605667.
[7] O. Kobayashi, Scalar curvature of a metric of unit volume. Math. Ann. 279, 253–265 (1987).
[8] D. Kotschick, Monopole classes and Perelman’s invariant of four-manifolds. e-print
math.DG/0608504.
[9] C. LeBrun, Four-manifolds without Einstein metrics. Math. Res. Lett. 3, 133–147 (1996).
[10] C. LeBrun, Ricci curvature, minimal volumes, and Seiberg-Witten theory. Inv. Math. 145
279–316 (2001).
[11] J. Lee and T. Parker, The Yamabe problem. Bull. Am. Math. Soc. 17, 37–91 (1987).
[12] G. Perelman, The entropy formula for the Ricci ﬂow and its geometric applications. e-print
math.DG/0211159.
[13] G. Perelman, Ricci ﬂow with surgery on three-manifolds. e-print math.DG/0303109.
[14] J. Petean, Computations of the Yamabe invariant. Math. Res. Lett. 5, 703–709 (1998).
76 K. Akutagawa, M. Ishida and C. LeBrun Arch. Math.
[15] J. Petean, The Yamabe invariant of simply connected manifolds. J. Reine Angew. Math.,
523, 225–231 (2000).
[16] J. Petean and G. Yun, Surgery and the Yamabe invariant. Geom. Funct. Anal. 9,
1189–1199 (1999).
[17] D. Pollack, Nonuniqueness and high energy solutions for a conformally invariant scalar
equation. Comm. Anal. Geom. 1, 347–414 (1993).
[18] R. Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature.
J. Diﬀerential Geom. 20, 478–495 (1984).
[19] R. Schoen, Variational theory for the total scalar curvature functional for Riemannian
metrics and related topics. Lec. Notes Math. 1365, 120–154 (1987).
[20] N. Trudinger, Remarks concerning the conformal deformation of metrics to constant scalar
curvature. Ann. Scuola Norm. Sup. Pisa. 22, 265–274 (1968).
[21] H. Yamabe, On the deformation of Riemannian structures on compact manifolds. Osaka
Math. J. 12, 21–37 (1960).
Kazuo Akutagawa, Dept. Mathematics, Tokyo Univ. of Science, Noda 278-8510, Japan
e-mail: akutagawa kazuo@ma.noda.tus.ac.jp
Masashi Ishida, Department of Mathematics, SUNY, Stony Brook, NY 11794-3651,
USA
Permanent Address: Department of Mathematics, Sophia University, Tokyo 102-8554,
Japan
e-mail: ishida@mm.sophia.ac.jp
Claude LeBrun∗, Department of Mathematics, SUNY, Stony Brook, NY 11794-3651,
USA
e-mail: claude@math.sunysb.edu
Received: 6 October 2006
Revised: 17 October 2006


Perelman's Invariant, Ricci Flow, and the Yamabe Invariants of Smooth
  Manifolds

Perelman's Invariant, Ricci Flow, and the Yamabe Invariants of Smooth Manifolds

Abstract

Similar works

Full text

Available Versions

Springer

Springer - Publisher Connector

Springer - Publisher Connector