DOI: 10.2478/auom-2013-0027
An. Şt. Univ. Ovidius Constanţa
Vol. 21(2),2013, 131–139
A necessary and sufficient condition for some
steady Ricci solitons to have positive
asymptotic volume ratio
Xiang Gao
Abstract
In this paper, we firstly establish a useful ODE relationship between
R1 (c) and V1 (c) on the steady Ricci soliton. Based on this, we obtain a
necessary and sufficient condition for some complete noncompact steady
gradient Ricci solitons to have positive asymptotic volume ratio.
1
Introduction and Main Results
Recall that a complete Riemannian manifold (M n , g) is called a steady gradient Ricci soliton if there exists a smooth function f : M n → R, called the
potential function such that
Rij + ∇i ∇j f = 0.
(1)
Moreover for the steady gradient Ricci soliton, we actually have
R + |∇f |2 = C
(2)
holds on M n , where C is a constant.
Key Words: steady Ricci soliton, asymptotic volume ratio, potential function.
2010 Mathematics Subject Classification: Primary 58C25; Secondary 35P05.
Received: March 2012.
Accepted: June 2012.
This work is supported by the Fundamental Research Funds for the Central Universities
and NSFC11101267.
131
132
XIANG GAO
In [2], it was proved by B.-L. Chen that the complete ancient solutions to
the Ricci flow, and in particular the steady Ricci soliton, must have nonnegative scalar curvature. As a consequence, the potential function f satisfies the
following estimate:
√
√
(3)
− Cr(x) + f (O) ≤ f (x) ≤ Cr(x) + f (O),
where r(x) denotes the distance function from x to a fixed point O in M n .
Moreover, if we assume that the Ricci curvature is positive and the scalar
curvature R approaches 0 towards spatial infinity, then by the following
Lemma 1.1 proved by H. X. Guo in [5], we can derive that there is one point
where R obtains its maximum, and the point of maximum is unique.
Lemma 1.1 (Guo). Let (M n , g) be a steady gradient Ricci soliton with positive
(or negative) Ricci curvature, then there is at most one critical point of R.
Thus we can denote O the unique point of maximum of R, called the origin,
and assume f (O) = 0 by adding a constant. Calculating the constant in (2)
at O we have
2
R + |∇f | = R (O) = R0 .
(4)
Based on these, H. X. Guo [5] also proved a more precise estimate for the
potential function of a complete steady gradient Ricci soliton as follows:
Theorem 1.2 (Guo). Assume (M n , g) is a complete steady gradient Ricci
soliton with positive Ricci curvature, and the scalar curvature approaches 0
towards infinity. Then for any ε > 0, there exists rε > 0 such that when
r(x) ≥ rε we have
p
p
R0 − ε r(x) ≤ −f (x) ≤ R0 r(x),
(5)
where r (x) = d (x, O) and R0 is the maximum of R.
Then define the functions
V : R → [0, ∞),
R : R → [0, ∞)
by
Z
V(c) =
Z
dµ,
{f <c}
R(c) =
R dµ.
{f <c}
In [1], the following ODE relating V(c) and R(c) was established for the shrinking Ricci soliton
n
0 ≤ V(c) − R(c) = c V0 (c) − R0 (c).
(6)
2
In this paper, to prove our main result, we establish a similar result to (6)
for the steady Ricci soliton as follows:
133
Steady RICCI Solitons to Have Positive Asymptotic Volume Ratio
Theorem 1.3. Define
D1 (c) = {x ∈ M n |−f (x) < c },
Z
R1 (c) =
Z
R dµ,
V1 (c) =
D1 (c)
dµ,
D1 (c)
then
R1 (c) + R01 (c) − R0 V10 (c) = 0.
(7)
Recall that the asymptotic volume ratio (AVR) of a complete noncompact
Riemannian manifold (N n , h) is defined by
Volh B(p, r)
r→∞
ωn rn
(8)
AVR(h) = lim
if the limit exists, where B(p, r) denotes the geodesic ball in N n with center
p and radius r and ωn is the volume of the unit Euclidean n-ball. It is easy
to check that the AVR(h) is independent of the choice of p. Moreover, if
(N n , h) has nonnegative Ricci curvature, then this limit (8) exists by the
Bishop-Gromov volume comparison theorem.
For the case of shrinking Ricci solitons, H.-D. Cao and D.-T. Zhou [1]
proved the following result aided by an observation of Munteanu [6].
Theorem 1.4 (Cao-Zhou). Any complete noncompact shrinking gradient
linebreak Ricci soliton must have at most Euclidean volume growth, i.e.,
lim sup
r→∞
Vol B(O, r)
< ∞.
ωn rn
(9)
For the case of steady Ricci solitons, by using Theorem 1.3, we can prove
the following estimate.
Theorem 1.5. For the steady gradient Ricci soliton we have
V1 (c) ≥
R1 (c) R0 V1 (c0 ) − R1 (c0 )
+
.
R0
R0
(10)
In particular, more recently, observing the results in [1], [2], [4] and [8], B.
Chow, P. Lu and B. Yang [3] derived a necessary and sufficient condition for
noncompact shrinking Ricci soliton to have positive AVR as follows:
134
XIANG GAO
Theorem 1.6 (Chow-Lu-Yang). Let (M n , g) be a complete noncompact
shrinking gradient Ricci soliton, then AVR(g) exists (and is finite). Moreover, AVR(g) > 0 if and only if
Z ∞
R(c)
dc < ∞.
(11)
c
n+2 V(c)
In this paper, for the case of steady Ricci solitons, we prove a similar
necessary and sufficient condition for some noncompact steady solitons to have
positive AVR:
Theorem 1.7. Let (M n , g) be a complete noncompact steady gradient Ricci
soliton such that the average scalar curvature
R
Rdµ
B(O,r)
< ∞,
(12)
0 < R (g) = lim
r→∞ Volg (B (O, r))
then AVR(g) exists (and is finite). Moreover, AVR(g) > 0 if and only if
!
Z ∞
n+1
R1 (c)
Rc
−
dc > −∞.
(13)
c
(s)ds
c0
0 R1
The paper is organized as follows. In section 2, we prove Theorem 1.3 by
calculating, and then obtain Theorem 1.5 applying Theorem 1.3. Based on
these, in section 3, we prove our main result Theorem 1.7.
2
Proof of Theorem 1.3 and 1.5
Proof of Theorem 1.3. Firstly, by Theorem 1.2, when r(x) is greater than
some constant rε , we have
p
p
R0 − ε r(x) ≤ −f (x) ≤ R0 r(x).
Denote by
n
D1 (c) = {x ∈ M |−f (x) < c }
Z
and
V1 (c) =
dV,
(14)
D(c)
then by the Co-Area formula (cf. [7]), we have
Z c Z
1
V1 (c) =
ds
dA.
|∇(−f
)|
0
∂D1 (s)
(15)
Steady RICCI Solitons to Have Positive Asymptotic Volume Ratio
135
Hence
Z
V10 (c) =
∂D1 (c)
1
dA.
|∇f |
(16)
Then taking the trace in
Rij + ∇i ∇j f = 0,
we have
R + ∆f = 0.
(17)
Thus by using the Divergence Theorem and (4)
Z
Z
−
Rdµ =
∆f dµ
D1 (c)
D1 (c)
Z
∇f ·
=
∂D1 (c)
−∇f
dA
|∇f |
Z
=−
|∇f |dA
∂D1 (c)
Z
R − R0
=
dA
∂D1 (c) |∇f |
Z
R
=
dA − R0 V10 (c)
|∇f
|
∂D1 (c)
Then by using the Co-Area formula again, we have
Z
Z
R1 (c) =
c
R dµ =
D1 (c)
Z
ds
0
∂D1 (s)
R
dA.
|∇f |
Hence
R01 (c) =
Z
∂D1 (c)
Therefore, we have
Z
Z
−R1 (c) = −
Rdµ =
D1 (c)
∂D1 (c)
R
dA.
|∇f |
(18)
R
dA − R0 V01 (c) = R01 (c) − R0 V01 (c).
|∇f |
Now we turn to prove Theorem 1.5 by using Theorem 1.3.
136
XIANG GAO
Proof of Theorem 1.5. Integrate the identity (7) from c0 to c we get
Z c
R0 (V1 (c) − V1 (c0 )) =
R0 V0 1 (s)ds
c
Z 0c
(R0 1 (s) + R1 (s)) ds
=
c0
Z c
= R1 (c) − R1 (c0 ) +
R1 (s)ds
c0
Therefore, (10) follows from the observation that R1 (c) is nonnegative, because
the scalar curvature R ≥ 0.
3
Proof of Theorem 1.7
In this section, we prove Theorem 1.7 by using Theorem 1.2 and 1.3.
Proof of Theorem 1.7. Let
P(c)=
R1 (c) − R0 V1 (c)
cn+1
then
N (c)
=
N (c) − 1
R1 (c)
R0 V1 (c)
R1 (c)
R0 V1 (c) −
=
1
and
N (c) =
R1 (c)
,
R0 V1 (c)
R1 (c)
R1 (c)
= n+1
.
c
R1 (c) − R0 V1 (c)
P(c)
(19)
(20)
R1 (c)
Note that V
is the average scalar curvature over the set D(c), and the
1 (c)
ODE (7) implies
R01 (c) cn+1 − (n + 1) R1 (c)cn − R0 V10 (c) cn+1 + (n + 1) R0 V1 (c) cn
c2n+2
0
0
n+1
(R1 (c) − R0 V1 (c)) c
− (n + 1) cn (R1 (c) − R0 V1 (c))
=
c2n+2
n+1
2n+1
−R1 (c)c
− (n + 1) c
P(c)
=
c2n+2
(c)
2n+2
− NN
P(c) − (n + 1) c2n+1 P(c)
(c)−1 c
=
c2n+2 (c)
n
+1
N
=−
+
P(c).
c
N (c) − 1
P0 (c) =
Then we choose c0 such that P(c0 ) 6= 0, and integrate
n+1
N (c)
0
P (c) = −
+
P(c).
c
N (c) − 1
(21)
Steady RICCI Solitons to Have Positive Asymptotic Volume Ratio
137
from c0 to c we have
P(c) = P(c0 )e
−
Rc
c0
N(c)
+ n+1
( N(c)−1
c )dc
.
(22)
From ODE (7) it is easy to see that
Z
R1 (c) − R0 V1 (c) = −
c
R1 (s)ds,
0
which implies
R1 (c)
n+1
c
(c)
P
=
R1 (c)
N (c)
R1 (c)
.
=
= −R c
R1 (s)ds
N (c) − 1
R1 (c) − R0 V1 (c)
0
(23)
Note that (23) implies P(c) ≤ 0. Furthermore, by the following bounds
p
p
R0 − ε r(x) ≤ −f (x) ≤ R0 r(x)
we have
R1 (c) − R0 V1 (c)
cn+1
0
(c)
−
R0 V01 (c)
R
= lim 1
c→∞
(n + 1) cn
R1 (c)
= − lim
c→∞ (n + 1) cn
1
R1 (c)
V1 (c)
lim
lim
=−
c→∞
c→∞
n+1
cn
V1 (c)
√c
Vol(B
O,
)
1
R0
=−
R (g) lim
c→∞
n+1
cn
ωn
=−
R (g) AVR(g),
n/2
(n + 1) R0
lim P(c) = lim
c→∞
that is to say
−
c→∞
ωn
n/2
(n + 1) R0
R (g) AVR(g) = lim P(c),
c→∞
which exists by (21). Since the average scalar curvature
R
Rdµ
B(O,r)
> 0,
R (g) = lim
r→∞ Volg (B (O, r))
we have AVR(g) exists (and is finite).
(24)
138
XIANG GAO
Moreover, by using (22) and (23) we have
n/2
AVR(g) = −
N(c)
(n + 1) R0 P(c0 ) − Rc∞ ( N(c)−1
+ n+1
c )dc
e 0
ωn R (g)
n/2
(n + 1) R0 P(c0 )
=−
e
ωn R (g)
R∞
c0
R (c)
Rc 1
− n+1
c
0 R1 (s)ds
dc
.
Note that P(c0 ) < 0, and by using (12) we obtain that AVR(g) > 0 if and
only if
!
Z ∞
n+1
R1 (c)
Rc
−
dc > −∞.
c
(s)ds
c0
0 R1
4
Acknowledgment
I would especially like to thank the referee for meaningful suggestions that led
to improvements of the article.
References
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Steady RICCI Solitons to Have Positive Asymptotic Volume Ratio
139
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Xiang Gao,
School of Mathematical Sciences,
Ocean University of China,
Lane 238, Songling Road, Laoshan District, Qingdao City, Shandong
Province, 266100, People’s Republic of China.
Email: gaoxiangshuli@126.com
140
XIANG GAO