Classification of expanding and steady Ricci solitons with integral curvature decay

0.21 MB·English

by  Giovanni Catino

CLASSIFICATION OF EXPANDING AND STEADY RICCI SOLITONS
WITH INTEGRAL CURVATURE DECAY
G. Catino1, P. Mastrolia2 and D. D. Monticelli3
Abstract. Inthispaperweprovenewclassificationresultsfornonnegativelycurvedgradientexpand-
ingandsteadyRiccisolitonsindimensionthreeandabove,undersuitableintegralassumptionsonthe
scalarcurvatureoftheunderlyingRiemannianmanifold. Inparticularweshowthattheonlycomplete
expanding solitons with nonnegative sectional curvature and integrable scalar curvature are quotients
5 of the Gaussian soliton, while in the steady case we prove rigidity results under sharp integral scalar
1 curvature decay. As a corollary, we obtain that the only three dimensional steady solitons with less
0 thanquadraticvolumegrowtharequotients ofR×Σ2,whereΣ2 isHamilton’scigar.
2

n
1. Introduction and main results
a
J
AgradientRiccisolitonisasmoothn–dimensional,connected,RiemannianmanifoldMn withmetric
7
g satisfying

]
G (1.1) Ric+ 2f =λg
∇
D
for some smooth potential function f :Mn R and a real constant λ. The soliton is called expanding,
. →
h steady or shrinking if, respectively, λ<0, λ=0 or λ>0. When f is constant, a gradient Ricci soliton
t
a is an Einstein manifold. Ricci solitons generate self-similar solutions to the Ricci flow and often arise as
m
singularity models of the flow; therefore, it is crucial to study and classify them in order to understand
[
the geometry of singularities.

1 The two dimensional case is well understood and all complete Ricci solitons have been classified, see
v
for instance the very recent [2] and references therein. In particular, it is well known that the only
7
1 gradient steady Ricci soliton with positive curvature is Hamilton’s cigar Σ2, see [22].
5
In dimension three, due to the efforts of Ivey [24], Perelman[30], Ni and Wallach[29] and Cao, Chen
1
0 and Zhu [7], shrinking solitons have been completely classified: they are quotients of either the round
1. sphere S3, the round cylinder R S2 or the shrinking Gaussian soliton R3.
×
0 InthesteadythreedimensionalcasetheknownexamplesaregivenbyquotientsofR3,R Σ2 andthe
5 ×
1 rotationallysymmetricone constructedbyBryant[5]. Inthe seminalpaperbyBrendle [4], itwasshown
: that Bryant soliton is the only nonflat, k-noncollapsed, steady soliton, proving a famous conjecture by
v
i Perelman [30]. It is still an open problem to classify three dimensional steady solitons which do not
X
satisfy the k-noncollapsing condition; see Cao [8] for an interesting result in this direction.
r
a The case of expanding solitons is far less rigid; however, some properties and classification theorems
have been proved in the recent years by various authors, see for instance [31], [26], [?], [16], [33], [17],
[20] and references therein.
The aim of this paper is to prove new classification results of gradient expanding and steady solitons
in dimension three and above under integral assumptions on the scalar curvature. Note that similar
Date:January8,2015.
2010 Mathematics Subject Classification. 53C20, 53C25.
Key words and phrases. Riccisolitons,Weitzenb¨ock formula,weightedEinsteintensor,rigidityresults.
1Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, Milano, Italy, 20133. Email: gio-
[email protected]. SupportedbyGNAMPAproject“EquazionidievoluzionegeometricheestruttureditipoEinstein”.
2Dipartimento di Matematica, Universit`a degli Studi di Milano, Via Saldini 50, Milano, Italy, 20133. Email:
[email protected]. Supported byGNAMPAproject“AnalisiGlobaleedOperatoriDegeneri”.
3Dipartimento di Matematica, Universit`a degli Studi di Milano, Via Saldini 50, Milano, Italy, 20133. Email:
[email protected]. SupportedbyGNAMPAproject“AnalisiGlobaleedOperatoriDegeneri”.
The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilita` e le loro Applicazioni
(GNAMPA)oftheIstitutoNazionalediAltaMatematica(INdAM).
1
2 STEADY AND EXPANDING RICCI SOLITONS
conditions have been considered by Deruelle [19] in the steady case (although the author exploits a
completely different approach).
In particular we prove the following
Theorem 1.1. Let (Mn,g) be a complete gradient expanding Ricci soliton of dimension n 3 with
≥
nonnegative sectional curvature. If R L1(Mn), then Mn is isometric to a quotient of the Gaussian
∈
soliton Rn.
Theorem 1.2. Let (Mn,g) be a complete gradient steady Ricci soliton of dimension n 3 with non-
≥
negative sectional curvature. Suppose that
1
liminf R=0.
r→+∞ r Z
Br(o)
Then, Mn is isometric to a quotient of Rn or Rn−2 Σ2, where Σ2 is the cigar soliton.
×
In the three dimensional case we can prove the analogous results under weaker assumptions.
Theorem 1.3. Let M3,g be a three dimensional complete gradient expanding Ricci soliton with non-
negative Ricci curva(cid:0)ture. I(cid:1)f R L1(M3), then M3 is isometric to a quotient of the Gaussian soliton
∈
R3.
Theorem 1.4. Let M3,g be a three dimensional complete gradient steady Ricci soliton. Suppose that
(cid:0) (cid:1)
1
liminf R=0.
r→+∞ r Z
Br(o)
Then M3 is isometric to a quotient of R3 or R Σ2, where Σ2 is the cigar soliton.
×
Remark 1.5. As it will be clear from the proofs of Theorems 1.1 and 1.3, instead of R L1(Mn) we
∈
can assume that
liminf R=0.
r→+∞ ZB2r(o)\Br(o)
Remark 1.6. The quantity
1
(1.2) liminf R
r→+∞ r Z
Br(o)
thatappearsin Theorems1.2and1.4is independent ofthe choiceofthe centero Mn. Moreover,note
∈
that our assumptions in the steady case do not imply a priori that the scalar curvature goes to zero at
infinity, in contrast with the results in [19]. In fact, in [19] it is assumed that R L1(Mn). This, using
∈
the hypothesis that the steady Ricci soliton has nonnegative sectional curvature, implies that the scalar
curvature is nonnegative, bounded, and globally Lipschitz, and thus that R 0 at infinity.
→
Asaconsequenceoftheintegraldecayestimatein[19](seeLemma4.4),itfollowsthattheassumption
in Theorems 1.2 and 1.4 holds if g has less than quadratic volume growth, i.e. Vol(B (o)) = o(r2) as
r
r + . This immediately implies the following
→ ∞
Corollary 1.7. The only complete gradient steady Ricci solitons of dimension n 3 with nonnegative
≥
sectional curvature and less than quadratic volume growth are quotients of Rn−2 Σ2.
×
In particular, in dimension three the nonnegativity assumption on the curvature is automatically
satisfied (see [15]), implying
Corollary 1.8. The only three dimensional complete gradient steady Ricci solitons with less than qua-
dratic volume growth are quotients of R Σ2.
×
We note that the condition in Theorem 1.3 is sharp: in fact there exists a rotationally symmetric
exampleconstructedbyBryantin[5](seealsotheappendixin[17])whichhaspositivesectionalcurvature,
quadraticcurvaturedecayatinfinityandEuclideanvolumegrowth. Moreover,thesteadyBryantsoliton
STEADY AND EXPANDING RICCI SOLITONS 3
has positive sectional curvature, linear curvature decay and quadratic volume growth; this shows that
the assumptions in Theorem 1.4 and Corollary 1.8 are sharp as well.
We explicitly remark that Theorems 1.2 and 1.4 improve a result in [19], while the results in the
expanding case, to the best of our knowledge, are completely new and should be compared with [?,
Theorem 4] and [31, Theorem 4.5], where the required integral conditions involve the measure e−fdµ,
and the weight e−f, under mild assumptions on the curvature, has exponential growth (see e.g. [6,
Lemma 5.5]). We also note that, in dimension three, the condition
1
(1.3) liminf R k >0
r→+∞ r Z ≥
Br(o)
implies the k-noncollapsing of balls with sufficiently large radii, a priori nonuniformly with respect to
the center. It would be extremely interesting either to show that the only three dimensional gradient
steady Ricci soliton satisfying (1.3) is, up to scaling, the Bryant soliton, or to construct a (k-collapsed)
counterexample. The first case, together with Theorem 1.4, would complete the classification of steady
solitons in dimension three.
One of the main tool in our analysis is a geometric (0,2)-tensor that we call the weighted Einstein
tensor E, and which is defined as
b
(1.4) E = Ric 1Rg e−f,
−2
(cid:0) (cid:1)
where f is the soliton potential. The weigbhted Einstein tensor appeared for the first time in [11], where
the authors observedthat E is a Codazzitensor on every gradientthree dimensionalRicci soliton. Here
we prove, in Section 2, that on every gradient Ricci soliton E satisfies the Weitzenbo¨ck formula
b
1 1
(1.5) ∆E 2 = E 2 E 2, f b(n 2)λE 2+Q,
2 | | |∇ | − 2h∇| | ∇ i− − | |
whereQisacubiccurvaturetebrm. Quitbesurprisingbly,weareabletoshobwthatthisquantitysatisfiesnice
algebraicproperties(see Section2 andthe Appendix) undersuitable curvatureassumptions;namely, we
prove that Q 0 if the sectional curvature (or the Ricci curvature, in dimension three) is nonnegative,
≥
and we completely characterize the equality case. We highlight the fact that equation (1.5) is only
effective when λ 0; this feature allows us to exploit, in the expanding and steady case, a technique
≤
reminiscent of those used to prove earlier results concerning gradient shrinking Ricci solitons (see e.g.
[29], [32], [9], [28], [34]).
Exploitingtheaboveformulaandusingacarefulintegrationbypartsargument,inSection3weprove
Theorems 1.1 and 1.3 in the expanding case, while in Section 4 we prove Theorems 1.2 and 1.4 in the
steady case.
2. A Weitzenbo¨ck formula for the weighted Einstein tensor
Let (Mn,g) be a complete gradient Ricci soliton of dimension n 3, that is a Riemannian manifold
≥
satisfying the equation
(2.1) Ric+ 2f = λg
∇
for some smooth function f : M R and some constant λ R. For the weighted Einstein tensor E
→ ∈
defined in (1.4) we prove the following
b
Proposition 2.1. Let (Mn,g) be a complete gradient Ricci soliton of dimension n 3. Then
≥
1 1 n 2 1 2
(2.2) ∆E 2 = E 2 E 2, f (n 2)λE 2 2Rm(E,E) − R E 2 tr E ,
2 | | |∇ | −2h∇| | ∇ i− − | | − − 2 (cid:20)| | − n 2(cid:16) (cid:16) (cid:17)(cid:17) (cid:21)
−
b b b b b b b b
where Rm is the Riemann curvature tensor, Rm E,E =R E E and tr is the trace.
ijkl ik jl
(cid:16) (cid:17)
b b b b
4 STEADY AND EXPANDING RICCI SOLITONS
Proof. From (1.4) we have, on a local orthonormalframe,
1
efE =R Rδ ,
ij ij ij
− 2
which implies, taking the covariant derivativbe,
1
(2.3) ef f E +E =R R δ .
k ij ij,k ij,k k ij
(cid:16) (cid:17) − 2
Taking the divergence of the previousbequatbion we get
1
(2.4) ef f 2E +2f E +∆fE +E =R ∆Rδ .
ij k ij,k ij ij,kk ij,kk ij
(cid:16)|∇ | (cid:17) − 2
Now we recall that, for a gradbient Riccibsoliton, webhavebthe validity of the following equations (see e.g.
[21] or [27]):
R =f R +2λR 2R R ,
ij,kk k ij,k ij kt ikjt
−
∆R= f, R +2λR 2Ric2,
h∇ ∇ i − | |
R+∆f =nλ.
Inserting the previous relations in equation (2.4), using the definition of E (which implies that R =
ij
efE + 1Rδ and Ric2 =e2f E 2 (n−4)R2) and simplifying we deduce
ij 2 ij | | | | − 4 b
b b (n 2)
(2.5) ef E +f E +(n 2)λE = 2efE R − R2δ +e2f E 2δ .
ij,kk k ij,k ij kt ikjt ij ij
(cid:16) − (cid:17) − − 4 | |
Now we contracbt (2.5) withb E , observingbthat tr(E)=bE = (n−2)Re−f, and we obbtain
ij tt − 2
1 b b b (n 2) 1 2
(2.6) E E = E 2, f (n 2)λE 2 2Rm(E,E) − R E 2 tr(E) ,
ij ij,kk
−2h∇| | ∇ i− − | | − − 2 (cid:20)| | − n 2(cid:16) (cid:17) (cid:21)
−
b b b b b b b b
which easily implies (2.2) since 1∆E 2 = E 2+E E . (cid:3)
2 | | |∇ | ij ij,kk
b b b b
Corollary 2.2. Let (Mn,g) be a complete gradient Ricci soliton of dimension n 3. Then
≥
1 1
(2.7) ∆E 2 = E 2 E 2, f (n 2)λE 2+Q
2 | | |∇ | − 2h∇| | ∇ i− − | |
where b b b b
(n 2)3 (n 2)(n 4)
Q:=e−2f − R3 2R T T − − RT 2 ,
(cid:20) 4n2 − ikjt ij kt− 2n | | (cid:21)
where T is the traceless Ricci tensor.
Proof. We recall that, in a local orthonormalframe, the components T of T are
ij
R
T =R δ .
ij ij ij
− n
Using the definition of E we deduce that
2R b E E e2f = 2R T T + 2(n−2)RT 2 (n−2)2R3.
− ikjt ij kt − ikjt ij kt n | | − 2n2
Now the claim follows inserbtinbg the previous equation into (2.2). (cid:3)
In the particular case of dimension three we have
Corollary 2.3. Let (M3,g) be a three dimensional complete gradient Ricci soliton. Then
1 1
(2.8) ∆E 2 = E 2 E 2, f λE 2+Q,
2 | | |∇ | − 2h∇| | ∇ i− | |
with b b b b
7 3
Q=e−2f 4R R R RRic2+ R3 .
ij jk ki
(cid:20) − 2 | | 4 (cid:21)
STEADY AND EXPANDING RICCI SOLITONS 5
Proof. The proof of the corollary is a simple computation using the fact that, in dimension three, one
has
1
T 2 = Ric2 R2,
| | | | − 3
2 1
R T T =R R R RRic2+ R3
ijkt ij kt ijkt ij kt
− 3 | | 9
and
R
R =R δ R δ +R δ R δ (δ δ δ δ ).
ijkt ik jt it jk jt ik jk it ik jt it jk
− − − 2 −
(cid:3)
In the next proposition we prove the main integral estimate that will be used in the proof of our
results.
Proposition 2.4. Let (Mn,g) be a complete gradient Ricci soliton of dimension n 3. Then, either
≥
(Mn,g) is Ricci flat or, for every nonnegative cutoff function ϕ with compact support in Mn, we have
Q (n 2)λE 2 ϕ3ef
(2.9) h − − | | i 3 E , ϕ ϕ2ef
ZM E b ≤− ZMh∇| | ∇ i
| | b
Proof. For every ε 0 define b
≥
Ω := x Mn E(x) ε
ε
{ ∈ || |≥ }
and let b
E(x) if x Ω ,
ε
h (x):=| | ∈
ε
εb if x M Ωε.
∈ \
Let ϕ be a smooth nonnegative cutoff function with compact supportin M. We multiply equation(2.7)
by h−1ϕ3ef and integrate on Mn, deducing
ε
1 ∆E 2ϕ3ef E , h E ϕ3ef E , ϕ E ϕ2ef E , f ϕ3ef
ε
| | = h∇| | ∇ i| | 3 h∇| | ∇ i| | h∇| | ∇ i .
2Z h Z h2 − Z h −Z h
M b ε M b ε b M b ε b M b ε
Since h = E on Ω and h =0 on M Ω , we obtain
ε ε ε ε
| | ∇ \
b
1 ∆E 2ϕ3ef h 2ϕ3ef E , ϕ E ϕ2ef E , f ϕ3ef
ε
| | = |∇ | 3 h∇| | ∇ i| | h∇| | ∇ i .
2Z h Z h − Z h −Z h
M b ε M ε M b ε b M b ε
Equation (2.7) and Kato’s inequality yield
E 2ϕ3ef h 2ϕ3ef E , ϕ E ϕ2ef Q (n 2)λE 2 ϕ3ef
0= |∇ | |∇ ε| +3 h∇| | ∇ i| | + h − − | | i
ZM bhε −ZM hε ZM b hε b ZM hε b
E 2ϕ3ef h 2ϕ3ef E , ϕ E ϕ2ef Q (n 2)λE 2 ϕ3ef
|∇| || |∇ ε| +3 h∇| | ∇ i| | + h − − | | i
≥ZM bhε −ZM hε ZM b hε b ZM hε b
E 2ϕ3ef E , ϕ E ϕ2ef Q (n 2)λE 2 ϕ3ef
= |∇ | +3 h∇| | ∇ i| | + h − − | | i
ZM\Ωε bhε ZM b hε b ZM hε b
E , ϕ E ϕ2ef Q (n 2)λE 2 ϕ3ef
3 h∇| | ∇ i| | + h − − | | i .
≥ ZM b hε b ZM hε b
Now,since everycomplete Ricci solitonis realanalyticin suitable coordinates(see [1]and[13, Theorem
2.4]), by the unique continuation property one has that either E 0 or the zero set of E has zero
| | ≡ | |
measure. In the first case, Bianchi identity implies that g is Ricci flat while in the second case, taking
b b
the limit as ε 0, since E h−1 1 almost everywhere on Mn, inequality (2.9) follows. (cid:3)
→ | | ε →
b
6 STEADY AND EXPANDING RICCI SOLITONS
3. Expanding case: proof of Theorems 1.1 and 1.3
3.1. Then-dimensionalcase. Let(Mn,g)beacompletegradientexpandingRiccisolitonofdimension
n 3 with nonnegative sectional curvature and assume that R L1(Mn). From Proposition 2.4, we
≥ ∈
have that either the soliton is Ricci flat (hence flat, since g has nonnegative sectional curvature) or
Q (m 2)λE 2 ϕ3ef
(3.1) h − − | | i 3 E , ϕ ϕ2ef
ZM E b ≤− ZMh∇| | ∇ i
| | b
for every nonnegative smooth cutoff funcbtion ϕ with compact support in Mn. Since (Mn,g) has non-
negative sectional curvature, Rm αR, for some positive constant α (see e.g. [3]) Moreover, for
| | ≤
expanding solitons with nonnegative Ricci curvature we have that the scalar curvature R is bounded,
see for instance [20, Proposition 2.4]. Thus g has bounded curvature. We recall that for every gradient
Ricci soliton we have Hamilton’s identity
R+ f 2 2λf =c
|∇ | −
for some real constant c ([23]). Since R 0, we deduce that f 2 2λf +c. By [6, Lemma 5.5] there
≥ |∇ | ≤
exist positive constants c ,c ,c such that
1 2 3
λ λ
(3.2) (r(x) c )2 c f(x) (r(x)+c )2,
1 2 3
− 2 − − ≤− ≤−2
where r(x)=dist(x,o) for some fixed origin o Mn; in particular f is proper. We define, for t 1,
∈ ≫
Ω = x Mn : f(x) t .
t
{ ∈ − ≤ }
We choose ϕ(x) = ψ(f(x)), where ψ : [0,+ ) [0,+ ) is a smooth nonincreasing function with
∞ → ∞
support in [0,2t], such that ψ 1 in [0,t] and
≡
c c
ψ′(s) ψ3/4(s) and ψ′′(s) ψ1/2(s)
| |≤ s | |≤ s2
for some c>0. Since in Ω we have f c√f c√t, we deduce
t
|∇ |≤ ≤
c
(3.3) ϕ ψ′ f in Ω ;
t
|∇ |≤| ||∇ |≤ √t
moreover,since ∆f =nλ R nλ on M,
− ≤
c
(3.4) ∆ϕ ψ′∆f +ψ′′ f 2 in Ω .
t
| |≤(cid:12) |∇ | (cid:12)≤ t
(cid:12) (cid:12)
Then, integrating by parts we have (cid:12) (cid:12)
E , ϕ ϕ2ef = ∆ϕϕ2+2ϕ ϕ2+ f ϕϕ2 E ef.
(cid:12)(cid:12)ZMh∇| | ∇ i (cid:12)(cid:12) ZΩ2t\Ωt(cid:16)| | |∇ | |∇ ||∇ | (cid:17)| |
(cid:12) b (cid:12) b
(cid:12) (cid:12)
By the definition of E and the nonnegative curvature assumption one has E ef cR, and from the
(cid:12) (cid:12) ≤
previous estimates (3.b3) and (3.4) we get (cid:12)(cid:12)b(cid:12)(cid:12)
E , ϕ ϕ2ef c R.
(cid:12)Z h∇| | ∇ i (cid:12)≤ Z
(cid:12) M (cid:12) Ω2t\Ωt
(cid:12) b (cid:12)
By (3.2) and since R L1(Mn), th(cid:12)e left-hand side ten(cid:12)ds to zero as t + , and from (3.1) we obtain,
∈ → ∞
applying Fatou’s lemma,
Q (m 2)λE 2 ϕ3ef
h − − | | i 0.
ZM E b ≤
| |
Now,weusethefactthatunderourassumptionsQisnonnegative(seeProposition5.2intheAppendix),
b
and since λ is strictly negative we get E 0. By Bianchi identity we get R 0, and so g is flat by the
≡ ≡
nonnegative curvature assumption. This concludes the proof of Theorem 1.1.
b
STEADY AND EXPANDING RICCI SOLITONS 7
3.2. The 3-dimensional case. The proof of Theorem 1.3 in dimension three is formally the same
as the higher dimensional case, with some minor corrections. In fact, under the weak assumption of
nonnegativity of the Ricci curvature we still have that the full curvature tensor is controlled by the
scalar curvature, i.e. Rm αR for some constant α. Hence, following the proof in the previous
| | ≤
subsection, we obtain that either (M3,g) is flat, or
Q λE 2 ϕ3ef
h − | | i 0.
ZM Eb ≤
| |
Now,weusethefactthatunderourassumptionsQbisnonnegative(seeProposition5.4intheAppendix),
and since λ is strictly negative we get E 0. By Bianchi identity we get R 0, and so g is flat by the
≡ ≡
nonnegative curvature assumption. This concludes the proof of Theorem 1.3.
b
Remark 3.1. Theorems 1.1 and 1.3 can be proved also using a L1-Liouville property for the operator
∆ . In fact, since Q 0 it follows from equation (2.7) that ∆ E 0 in a distributional sense.
−f −f
≥ | | ≥
Moreover, the nonnegativity of the Ricci curvature implies that Ric = Ric 2f = 2Ric λg 0.
−fb −∇ − ≥
Therefore, since R L1(Mn) implies E L1 efdµ,Mn , we can apply [14, Proposition 4.1], which
∈ | | ∈
asserts that on a complete Riemannian manifol(cid:0)d (Mn,g)(cid:1)with Ric 0 every positive solution u of
b −f ≥
∆ u 0 with u L1 efdµ,Mn must be constant. Thus E is constant, and therefore zero from
−f
≥ ∈ | |
equation (2.7). (cid:0) (cid:1)
b
4. Steady case: proof of Theorems 1.2 and 1.4
4.1. The n-dimensional case. Let (Mn,g) be a complete gradient steady Ricci soliton of dimension
n 3 with nonnegative sectional curvature and assume that
≥
1
liminf R=0.
r→+∞ r Z
Br
In particular, there exists a sequence r , i N, of positive radii converging to + such that
i
{ } ∈ ∞
1
(4.1) lim R=0.
i→+∞ri ZBri
From Proposition 2.4, we have that either the soliton is Ricci flat (hence flat, since g has nonnegative
sectional curvature) or
Qϕ3ef
(4.2) 3 E , ϕ ϕ2ef 3 E ϕϕ2ef
ZM E ≤− ZMh∇| | ∇ i ≤ ZM|∇| |||∇ |
| | b b
for every nonnegative smootbh cutoff function ϕ with compact support in Mn. Since (Mn,g) has non-
negative sectional curvature, Rm αR, for some positive constant α. Hamilton’s identity
| |≤
R+ f 2 =c
|∇ |
implies that both the scalar curvature R and f 2 are bounded. Moreover, it follows e.g. from [12]
|∇ |
that either R > 0 or the soliton is Ricci flat, thus flat. So from now on we will assume that the scalar
curvature is strictly positive. Using Kato’s inequality and the fact that Ric R/√n, we get
|∇ |≥|∇ |
n 1
E E Ric + R + f Ric Rg e−f c Ric + f R e−f c Ric +R e−f
|∇| ||≤|∇ |≤ |∇ | 2|∇ | |∇ || −2 | ≤ |∇ | |∇ | ≤ |∇ |
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
for sbome positive constant c. Hence, the left-hand side of (4.2) can be estimate as
E ϕϕ2ef Ric ϕϕ2+ R ϕϕ2.
Z |∇| |||∇ | ≤Z |∇ ||∇ | Z |∇ |
M M M
b
Now we fix an index i and choose ϕ with support in B = B (o) for some origin o Mn and such
2ri 2ri ∈
that ϕ 1 in B , ϕ 2 on Mn. Then, by (4.1), the secondterm ofthe left-hand side tends to zero
≡ ri |∇ |≤ ri
8 STEADY AND EXPANDING RICCI SOLITONS
as i + . By Ho¨lder inequality and the fact that R>0, the remaining term can be estimate as
→ ∞
Ric 2ϕ2 1/2 1/2
(4.3) Ric ϕϕ2 |∇ | R ϕ2ϕ2
Z |∇ ||∇ | ≤(cid:18)Z R (cid:19) (cid:18)Z |∇ | (cid:19)
M M M
To conclude the estimate we need the following lemma.
Lemma 4.1. Let (Mn,g) be a complete, nonflat, gradient steady Ricci soliton of dimension n 3 with
≥
nonnegative sectional curvature. Then, for every nonnegative cutoff function ϕ with compact support in
Mn, there exists a positive constant c such that
Ric 2ϕ2
|∇ | c R+R ϕ2 ϕ2.
ZM R ≤ ZM(cid:16) |∇ | (cid:17)
Proof. First of all, in some local frame, we have
1
Ric 2 = R R 2+ R R .
k ij j ik k ij j ik
|∇ | 2|∇ −∇ | ∇ ∇
From the soliton equation and the commutation rule of covariant derivatives, one has
R R =R f.
k ij j ik kijl l
∇ −∇ ∇
Since Rm αR and f 2 c for some α,c>0, we obtain
| |≤ |∇ | ≤
(4.4) Ric 2 cR2+ R R .
k ij j ik
|∇ | ≤ ∇ ∇
Hence, to finish the proof we have to estimate the right-hand side. Integrating by parts, commuting
indices and using Young’s inequality we get
R R ϕ2 R R ϕ2 R R ϕϕ R R Rϕ2
k ij j ik ij k j ik ij j ik k ij j ik k
∇ ∇ = ∇ ∇ 2 ∇ ∇ + ∇ ∇
Z R −Z R − Z R Z R2
M M M M
R R ϕ2 (R R R +R R R )ϕ2
ij j k ik kjil ij kl ij ik jl
∇ ∇
≤−Z R −Z R
M M
Ric 2ϕ2
+ε |∇ | +c(ε) R ϕ2+ R2 ϕ2,
ZM R ZM(cid:16) |∇ | |∇ | (cid:17)
for every ε > 0 and some constant c(ε). Using Bianchi identity, the fact that Rm αR and the well
| | ≤
known soliton identity R=2Ric( f) (see e.g. [21] or [27]), we obtain
∇ ∇
R R ϕ2 1 R Rϕ2 Ric 2ϕ2
∇k ij∇j ik ij∇i∇j +ε |∇ | +c(ε) R+R ϕ2 ϕ2
ZM R ≤−2ZM R ZM R ZM(cid:16) |∇ | (cid:17)
1 R2ϕ2 Ric 2ϕ2
= |∇ | +ε |∇ | +c (ε) R+R ϕ2 ϕ2
1
4ZM R ZM R ZM(cid:16) |∇ | (cid:17)
Ric 2ϕ2
ε |∇ | +c (ε) R+R ϕ2 ϕ2,
2
≤ ZM R ZM(cid:16) |∇ | (cid:17)
for every ε>0 and some constant c (ε). Choosing ε 1, this estimate and (4.4) conclude the proof of
2
≪
the lemma. (cid:3)
Now we can return to the proof of Theorem 1.2. Using the previous lemma and (4.3), we obtain
1/2 1/2
c
Ric ϕϕ2 c R+R ϕ2 ϕ2 R ϕ2ϕ2 R
ZM|∇ ||∇ | ≤ (cid:18)ZM(cid:16) |∇ | (cid:17) (cid:19) (cid:18)ZM |∇ | (cid:19) ≤ ri ZB2ri
which, by (4.1), tends to zero as i + . Applying Fatou’s lemma, from (4.2), we get
→ ∞
Qϕ3ef
0.
ZM E ≤
| |
Hence, Proposition 5.2 implies that Q 0 on M.bThe equality case implies that the Ricci tensor at
≡
every point has at most two distinct eigenvalues Λ = 0 with multiplicity (n 2) and Υ = 1R with
− 2
multiplicity two. In order to conclude the proof we need the following general result (see Lemma 3.2
STEADY AND EXPANDING RICCI SOLITONS 9
in [32]) for constant rank, symmetric, nonnegative tensors. We recall the definition of the X-Laplacian
∆ =∆ g(X, ), for some smooth vector field X (see e.g. [12]).
X
− ·
Lemma 4.2. Let be a constant rank, symmetric, nonnegative tensor on some tensor bundle. If
T
(∆ )(V,V) 0 for V Ker and X is a vector field, then the kernel of is a parallel subbundle.
X
T ≤ ∈ T T
Now let e ,...,e be a local orthonormal frame such that Ric(e ,e ) = Ric(e ,e ) = 1R and
1 n 1 1 2 2 2
Ric(e , ) = 0 for every k = 3,...,n. In particular, the only nonzero sectional curvature is σ , where
k 12
·
σ is the sectional curvature defined by the two-plane spanned by e and e . Then, by the well known
ij i j
soliton identity (see e.g. [21] or [27])
∆ R =2λR 2R R = 2R R ,
∇f ij ij iljt lt iljt lt
− −
we see that, for every fixed k =3,...,n, e KerRic and
k
∈
(∆ Ric)(e ,e )= 2 Rm(e ,e ,e ,e )Ric(e ,e )= R(σ +σ )=0.
∇f k k k i k i i i 1k 2k
− −
Xi
Moreover, since g has nonnegative sectional curvature, 1Rg Ric is nonnegative, and a simple compu-
2 −
tation shows that
1 1
∆ Rg Ric (e ,e )= ∆ Rg Ric (e ,e )=0.
∇f 1 1 ∇f 2 2
(cid:18) (cid:18)2 − (cid:19)(cid:19) (cid:18) (cid:18)2 − (cid:19)(cid:19)
Now Lemma 4.2 applies, and by de Rham Decomposition Theorem (see for instance [25], Chapter 1,
Section 6) the metric splits and Theorem 1.2 follows since, as we said in the Introduction, the cigar
soliton Σ2 is the only complete two-dimensional steady soliton with positive curvature.
Remark 4.3. Note that, under our assumptions, f in generalis notproper,thus one cannotexploitthe
argument used in the expanding case involving a cutoff function depending on the potential f.
4.2. The 3-dimensional case. The proof of Theorem 1.4 in dimension three follows the lines of the
higher dimensional case. First of all, by Chen [15] we have that g must have nonnegative sectional
curvature, and since R+ f 2 = c, g has also bounded curvature. From Hamilton’s strong maximum
|∇ |
principle (see e.g. [18]) we deduce that M3,g is either flat, or it splits as a product R Σ2 (where
×
Σ2 is again the cigar steady soliton) or (cid:0)it has (cid:1)strictly positive sectional curvature. In the latter case,
following the proof of Theorem 1.2 we obtain
Qϕ3ef
0.
ZM E ≤
| |
Now,weusethefactthatunderourassumptionsQbisnonnegative(seeProposition5.4intheAppendix),
and we get Q 0. The equality case in Proposition 5.4 implies that the Ricci curvature has a zero
≡
eigenvalue, a contradiction. This concludes the proof of Theorem 1.3.
4.3. Proof of Corollary 1.7. Corollary 1.7 is a direct consequence of the following
Lemma 4.4 (Lemma 4.3 in [19]). Let (Mn,g) be a complete gradient steady Ricci soliton with nonneg-
ative Ricci curvature. Then, for every o Mn and every r >0,
∈
Vol(B (o))
R n√c r ,
Z ≤ r
Br(o)
where c is the constant in Hamilton’s identity R+ f 2 =c.
|∇ |
Proof. Integrating the equation R+∆f =0 one has
R= ∆f f CA(∂B (o)),
r
Z −Z ≤Z |∇ |≤
Br(o) Br(o) ∂Br(o)
where A(∂B (o)) is the (n 1)-dimensional volume of the geodesic sphere ∂B (o). Now, since (Mn,g)
r r
−
has nonnegativeRiccicurvature,by Bishop-Gromovtheorem(see for instance [18]) for everyo M and
∈
10 STEADY AND EXPANDING RICCI SOLITONS
every r >0 one has
rA(∂B (o))
r
n,
Vol(B (o)) ≤
r
which implies the thesis.
(cid:3)
5. Appendix
Weprovideheretheproofofsomenewalgebraiccurvatureestimatesneededforthe proofofthe main
theorems. First we recall the following lemma (see [10, Proposition 3.1])
Lemma 5.1. Let (Mn,g) be a Riemannian manifold of dimension n 3 with nonnegative sectional
≥
curvature. Then, the following estimate holds
n 2
R T T − RT 2,
ikjl ij kl
≤ 2n | |
with equality if and only if the Ricci tensor has at most two distinct eigenvalues, Λ of multiplicity (n 2)
−
and Υ of multiplicity two.
Proof. Let e , i = 1,...,n, be the eigenvectors of T and let λ be the corresponding eigenvalues.
i i
{ }
Moreover,let σ be the sectionalcurvature defined by the two-plane spanned by e and e . We want to
ij i j
prove that the quantity
n n
n 2 n 2
R T T − RT 2 = λ λ σ − R λ2
ikjl ij kl− 2n | | i j ij − 2n k
iX,j=1 kX=1
is nonpositive if σ 0 for all i,j =1,...,n. The scalar curvature can be written as
ij
≥
n
R = R = σ = 2 σ .
ijij ij ij
iX,j=1 Xi<j
Hence, one has the following
n n n
n 2 n 2
λ λ σ − R λ2 = 2 λ λ σ − σ λ2
i j ij − 2n k i j ij − n ij k
iX,j=1 Xk=1 Xi<j Xi<j kX=1
n
n 2
= 2λ λ − λ2 σ .
Xi<j(cid:16) i j − n Xk=1 k(cid:17) ij
On the other hand, one has
n
λ2 = λ2+λ2+ λ2.
k i j k
kX=1 kX6=i,j
Moreover,using the Cauchy-Schwarzinequality and the fact that n λ =0, we obtain
k=1 k
P
λ2 1 λ 2 = 1 λ +λ 2,
kX6=i,j k ≥ n−2(cid:16)kX6=i,j k(cid:17) n−2(cid:0) i j(cid:1)
with equality if and only if λk =λk′ for every k,k′ =i,j. Hence, the following estimate holds
6
n
n 1 2
λ2 − λ2+λ2 + λ λ .
k ≥ n 2 i j n 2 i j
Xk=1 − (cid:0) (cid:1) −
Using this, since σ 0, it follows that
ij
≥
n n
n 2 n 1
λ λ σ − R λ2 − 2λ λ λ2+λ2 σ
iX,j=1 i j ij − 2n Xk=1 k ≤ n Xi<j(cid:0) i j −(cid:0) i j(cid:1)(cid:17) ij
n 1
= − (λ λ )2σ 0.
i j ij
− n − ≤
Xi<j
This concludes the proof of the estimate.