MAXIMUM SOLUTIONS OF NORMALIZED RICCI FLOWS ON
4-MANIFOLDS
FUQUAN FANG, YUGUANG ZHANG, AND ZHENLEI ZHANG
Abstract. We consider maximum solution g(t), t ∈ [0,+∞), to the normalized Ricci
flow. Among other things, we prove that, if (M,ω) is a smooth compact symplectic
4-manifold such that b+
(M) > 1 and let g(t), t ∈ [0,∞), be a solution to (1.3) on M
whose Ricci curvature satisfies that |Ric(g(t))| ≤ 3 and additionally χ(M) = 3τ(M) >
0, then there exists an m ∈ N, and a sequence of points {xj,k ∈ M}, j = 1, · · · ,m,
satisfying that, by passing to a subsequence,
(M, g(tk + t), x1,k, · · · , xm,k)
dGH−→ (
Nj, g∞, x1,∞, · · · , , xm,∞),
t ∈ [0,∞), in the m-pointed Gromov-Hausdorff sense for any sequence tk −→ ∞,
where (Nj , g∞), j = 1, · · · ,m, are complete complex hyperbolic orbifolds of complex
dimension 2 with at most finitely many isolated orbifold points. Moreover, the con-
vergence is C∞ in the non-singular part of
Nj and Volg0(M) =
Volg∞(Nj),
where χ(M) (resp. τ(M)) is the Euler characteristic (resp. signature) of M .
1. Introduction
Let (M, g) be a compact Riemannian manifold. The Perelman λ-functional
(1.1) λM(g) = inf
f∈C∞(M)
{F(g, f) :
e−fdvolg = 1}
where F(g, f) =
(Rg + |∇f |2)e−fdvolg and Rg is the scalar curvature of g. Note
that λM(g) is the lowest eigenvalue of the operator −4△ + Rg. By [Pe1] the gradient
flow of the Perelman λ-functional is the Hamilton’s the Ricci-flow evolution equation
(1.2)
g(t) = −2Ric(g(t))
The normalized Ricci flow equation on an n-manifold M reads
(1.3)
g(t) = −2Ric(g(t)) + 2R
where Ric (resp. R) denotes the Ricci tensor (resp. the average scalar curvature
). Note that (1.2) and (1.3) differ only by a change of scale in space and time,
and the volume Vol(g(t)) is constant in t. If dimM = n, λM(g) = λM(g)Volg(M)
is invariant up to rescaling the metric. Perelman [Pe1] has proved that λM(g(t)) is
non-decreasing along the Ricci flow g(t) whenever λM(g(t)) ≤ 0. This leads to the
The first author was supported by NSF Grant 19925104 of China, 973 project of Foundation Science
of China, and the Capital Normal University.
http://arxiv.org/abs/0704.0714v1
2 F. FANG, Y. ZHANG, AND Z. ZHANG
Perelman invariant λM by taking supremum of λM(g) in the set of all Riemannian
metrics on M .
By [AIL] the Perelman invariant λM is equal to the Yamabe invariant whenever
λM ≤ 0, after the earlier estimations (cf. [An5] [Pe2] [Le4] [FZ] and [Kot]). In
particular, if (M, g) is a smooth compact oriented 4-manifold with a Spinc-structure
c which is a monopole class (i.e., the associated Seiberg-Witten equation possesses
an irreducible solution) so that that c21(c)[M ] > 0, by [FZ] λM ≤ −
32π2c21(c)[M ].
Moreover, g is a Kähler-Einstein metric of negative scalar curvature if and only if
λM(g) = −
32π2c21(c)[M ]. However, there are plenty of 4-manifolds where the Perel-
man invariant λM = −
32π2c21(c)[M ] but do not admit any Kähler Einstein metric.
It is natural to study 4-manifolds with these extremal property. For such a 4-manifold
M , to seek for an ”optimal” Riemannian metric on M with respect to the Perelman
functional λM : M → R, we want to consider a maximal solution g(t) which is a solu-
tion of the Ricci flow (1.3). We call a longtime solution g(t), t ∈ [0,+∞), to the Ricci
flow (1.3) a maximum solution if lim
λM(g(t)) = λM . For a compact 3-manifold, by
Perelman [Pe2] all solutions of the Ricci flow (1.2) with surgery exist for longtime and
are maximum solutions, provided λM ≤ 0. In the paper [FZZ] obstructions are found
for the longtime solutions with bounded curvature to (1.3).
In this paper we are going to study the maximum solutions of (1.3) with bounded
Ricci curvatures instead. To avoid technique terminology we only state our results
for symplectic 4-manifolds by using the celebrated work of Taubes [Ta]: if (M,ω) is
a compact symplectic manifold with b+2 (M) > 1 (the dimension of self-dual harmonic
2-forms of M), the spinc-structure induced by ω is a monopole class. Moreover, in
this situation c21(c)[M ] = 2χ(M) + 3τ(M), where χ(M) (resp. τ(M)) is the Euler
characteristic (resp. signature) of M .
Theorem 1.1. Let (M,ω) be a smooth compact symplectic 4-manifold satisfying that
b+2 (M) > 1 and 2χ(M) + 3τ(M) > 0. If g(t), t ∈ [0,∞), is a solution to (1.3) such
that |Ric(g(t))| ≤ 3, and
λM(g(t)) = −
32π2(2χ(M) + 3τ(M)),
then there exists an m ∈ N, and sequences of points {xj,k ∈ M}, j = 1, · · · , m,
satisfying that, by passing to a subsequence,
(M, g(tk + t), x1,k, · · · , xm,k)
dGH−→ (
Nj, g∞, x1,∞, · · · , , xm,∞),
t ∈ [0,∞), in the m-pointed Gromov-Hausdorff sense for any sequence tk −→ ∞, where
(Nj , g∞), j = 1, · · · , m, are complete Kähler-Einstein orbifolds of complex dimension 2
with at most finitely many isolated orbifold points. The scalar curvature (resp. volume)
of g∞ is
−Volg0(M)−
32π2(2χ(M) + 3τ(M)) (resp. Volg0(M) =
Volg∞(Nj))
Moreover, the convergence is C∞ in the non-singular part of
1 Nj.
MAXIMUM SOLUTIONS OF NORMALIZED RICCI FLOWS ON 4-MANIFOLDS 3
We first remark that, if the diameters diamg(tk)(M) possess a uniform upper bound,
then m = 1, and N1 is a compact Kähler-Einstein orbifold. Secondly, if the Ricci curva-
ture bound in the above theorem is replaced by a uniform bound of sectional curvature,
then every (Nj, g∞), j = 1, · · · , m are complete Kähler-Einstein manifolds. By the
same arguments as in [An5][An6],
j=1Nj can weakly embed in M ,
j=1Nj ⊂⊂ M ,
i.e. for any compact subset K ⊂
j=1Nj , there is a smooth embedding FK : K −→ M .
Furthermore, there exists a sufficiently large compact subset K ⊂
j=1Nj such that
M\K admits an F-structure of positive rank. This type geometric decomposition seems
very useful to understand the diffeomorphism type of 4-manifolds.
Theorem 1.2. Let (M,ω) be a smooth compact symplectic 4-manifold such that b+2 (M) >
1 and let g(t), t ∈ [0,∞), be a solution to (1.3) such that |R(g(t))| ≤ 12. If in addition
χ(M) = 3τ(M) > 0, then
λM(g(t)) = −
32π2(2χ(M) + 3τ(M))
Moreover, if |Ric(g(t))| ≤ 3, the Kähler-Einstein metric g∞ in Theorem 1.1 is complex
hyperbolic.
To conclude the section we point out that the main result in Theorem 1.1 (resp.
Corollary 1.2) holds if the manifold is not symplectic but a compact oriented 4-manifold
with a monopole class c1 (i.e. with a spin
c-structure with non-vanishing Seiberg-Witten
invariant) so that c21 = 2χ(M) + 3τ(M) > 0.
2. Preliminaries
2.1. Monopole class. Let (M, g) be a compact oriented Riemannian 4-manifold with
a Spinc structure c. Let b+2 (M) denote the dimension of the space of self-dual harmonic
2-forms in M . Let S±
denote the Spinc-bundles associated to c, and let L be the
determinant line bundle of c. There is a well-defined Dirac operator
DA : Γ(S+c ) −→ Γ(S−c )
Let c : ∧∗T ∗M −→ End(S+
) denote the Clifford multiplication on the Spinc-
bundles, and, for any φ ∈ Γ(S±
), let
q(φ) = φ⊗ φ− 1
|φ|2id.
The Seiberg-Witten equations read
(2.1)
DAφ = 0
c(F+A ) = q(φ)
where A is an Hermitian connection on L, and F+A is the self-dual part of the curvature
of A.
A solution of (2.1) is called reducible if φ ≡ 0; otherwise, it is called irreducible. If
(φ,A) is a resolution of (2.1), one calculates easily that
(2.2) |F+A | =
|φ|2,
4 F. FANG, Y. ZHANG, AND Z. ZHANG
The Bochner formula reads
(2.3) 0 = −2△|φ|2 + 4|∇Aφ|2 +Rg|φ|2 + |φ|4,
where Rg is the scalar curvature of g.
The Seiberg-Witten invariant can be defined by counting the irreducible solutions
of the Seiberg-Witten equations (cf. [Le2]).
Definition 2.2. ([K1]) Let M be a smooth compact oriented 4-manifold. An element
α ∈ H2(M,Z)/torsion is called a monopole class ofM if and only if there exists a Spinc-
structure c on M with first Chern class c1 ≡ α(mod torsion), so that the Seiberg-Witten
equations have a solution for every Riemannian metric g on M .
By the celebrated work of Taubes [Ta], if (M,ω) is a compact symplectic 4-manifold
with b+2 (M) > 1, the canonical class of (M,ω) is a monopole class.
2.3. Kato’s inequality. Let (M, g) be a Riemannian Spinc-manifold of dimension n,
the following Kato inequality is useful.
Proposition 2.4. (Proposition 2.2 in [BD]) Let φ be a harmonic Spinc-spinor on
(M, g), i.e. DAφ = 0, where DA is the Dirac operator and A is an Hermitian connection
on the determinant line bundle. Then
(2.4) |∇|φ||2 ≤ n− 1
|∇Aφ|2 ≤ |∇Aφ|2
at all points where φ is non-zero. Moreover, |∇|φ||2 = |∇Aφ|2 occurs only if ∇Aφ ≡ 0.
Note that the arguments in the proof of Proposition 2.2 in [BD] can be used to
prove this proposition without any change, where the same conclusion was derived for
Spin-spinor φ. For any ǫ > 0, let |φ|2ǫ = |φ|2+ǫ2. If φ is harmonic, by above proposition,
(2.5) |∇|φ|ǫ|2 ≤
|∇|φ||2 ≤ n− 1
|∇Aφ|2 ≤ |∇Aφ|2
at points where φ(p) 6= 0. Since {p ∈ M : φ(p) 6= 0} is dense in M for harmonic φ, we
conclude that (2.5) holds everywhere in M .
2.5. Chern-Gauss-Bonnet formula and Hirzebruch signature formula. Let
(M, g) be a compact closed oriented Riemannian 4-manifold, χ(M) and τ(M) are the
Euler number and the signature of M respectively. The Chern-Gauss-Bonnet formula
and the Hirzebruch signature theorem say that
(2.6) χ(M) =
+ |Wg|2 −
|Rico|2)dvg, and
(2.7) τ(M) =
(|W+g |2 − |W−g |2)dvg,
where Rico = Ric(g)− Rg
g is the Einstein tensor, W+g and W
g are the self-dual and
anti-self-dual Weyl tensors respectively (cf. [B]). If g is a Kähler-Einstein metric, then
(2.8) R2g = 24|W+g |2,
MAXIMUM SOLUTIONS OF NORMALIZED RICCI FLOWS ON 4-MANIFOLDS 5
(cf. [Le3]) which will be used in the proof of Theorem 1.1.
By Chern-Gauss-Bonnet formula, one has an L2-bound of the curvature operator
Rm(g) by the bounds of Ricci curvature, i.e. if |Ric(g)| < C, then
(2.9)
|Rm(g)|2dvg ≤ 8π2χ(M) + C1V olg(M),
where C and C1 are constants independent of (M, g).
Let (N, g) be a complete Ricci-flat Einstein 4-manifold. Assume that
(2.10)
|Rm(g)|2dvg < ∞, and Volg(Bg(x, r)) ≥ Cr4,
for all r > 0, a point x ∈ N , and a positive constant C. By Theorem 2.11 of [N], (N, g)
is ALE. (i.e, Asymptotically Locally Euclidean space) of order 4. It is well-known that
N is asymptotic to the cone on the spherical space form S3/Γ, where Γ ⊂ SO(4) is a
finite group. The Chern-Gauss-Bonnet formula implies that
(2.11) χ(N) =
|Rm(g)|2dvg +
(cf. [N] and [An1]).
2.6. Curvature estimates for 4-manifolds. Now let’s recall a result of [CT], which
is important to the proof of Theorem 1.1. Let (M, g) be a complete Riemannian 4-
manifold. A subset U ⊂ M such that for all p ∈ U , sup
Bg(p,1)
Ric(g) ≥ −3, is called
̺-collapsed if for all p ∈ U ,
V olg(Bg(p, 1)) ≤ ̺.
By Theorem 0.1 in [CG], there is a constant ε4 such that if U is ̺-collapsed with
sectional curvature |Kg| ≤ 1 and ̺ ≤ ε4, then U carries an F-structure of positive
rank.
Theorem 2.7. (Remark 5.11 and Theorem 1.26 in [CT]) There exist constants δ > 0,
c > 0 such that: if (M, g) is a complete Riemannian 4-manifold with |Ric(g)| ≤ 3 and
|Rm(g)|2dvg ≤ C,
and if E ⊂ M is a bounded open subset such that T1(E) = {x ∈ M : dist(x, E) ≤ 1} is
ε4-collapsed with
Bg(x,1)
|Rm(g)|2dvg ≤ δ (for all T1(E)),
then ∫
|Rm(g)|2dvg ≤ cV olg(A0,1(E)),
where A0,1(E) = T1(E)\E.
6 F. FANG, Y. ZHANG, AND Z. ZHANG
3. The limiting behavior of Ricci flow
In this section we study the limiting behavior of Ricci-flow with bounded Ricci
curvatures on 4-manifolds. We will assume in this section that M is a smooth closed
oriented 4-manifold with λM < 0, and g(t), t ∈ [0,+∞), is a longtime solution of the
normalized Ricci flow (1.3) with bounded Ricci-curvature. By normalization we may
assume that |Ric(g(t))| ≤ 3. By (2.9) there is a constant C independent of t such that
|Rm(g(t))|2dvg(t) ≤ C.
Let us denoted by V the volume Volg(0)(M) = Volg(t)(M), and R̆(g(t)) = min
R(g(t))(x)
the minimum of the scalar curvature of g(t). It is easy to see that R̆(g(t)) ≤ λMV −
Lemma 3.1. (3.1.1) lim
λM(g(t)) = lim
R(g(t)) = lim
R̆(g(t)) = R∞
(3.1.2) lim
|R(g(t))−R(g(t))|dvg(t) = 0,
(3.1.3) lim
|Rico(g(t))|2dvg(t) = 0.
Proof. By Perelman [Pe1] λM(g(t)) is a non-decreasing function on t, therefore the limit
λM(g(t)) exists since λM < 0. Now let us denote by R∞ the limit lim
λM(g(t)).
Note that R∞ ≤ λMV −
2 < 0. To prove (3.1.1), we first prove that both lim
R(g(t))
and lim
R̆(g(t)) exist and take values R∞. By the same arguments as in the proof of
Proposition 2.6 and Lemma 2.7 of [FZZ] we get that
R(g(t))− R̆(g(t)) = 0.
Observe that R(g(t)) ≥ λM(g(t)) ≥ R̆(g(t)) (cf. [KL] (92.3)). Therefore lim
R(g(t)) =
R∞ = lim
R̆(g(t)). This proves (3.1.1).
Note that∫
|R(g(t))− R(g(t))|dvg(t) ≤
(R(g(t))− R̆(g(t)))dvg(t) +
(R(g(t))− R̆(g(t)))dvg(t)
= 2(R(g(t))− R̆(g(t)))V
(3.1.2) follows from (3.1.1).
By Lemma 3.1 in [FZZ],
|Rico(g(t))|2dvg(t)dt < ∞,
and, by Lemma 1 in [Ye], we have
|Rico(g(t))|2dvg(t) ≤ −2
|∇Rico(g(t))|2dvg(t)+4
|Rm||Rico(g(t))|2dvg(t) < D,
where D is a constant independent of t. By the same argument as in the proof of
Proposition 2.6 in [FZZ] (3.1.3) follows. �
MAXIMUM SOLUTIONS OF NORMALIZED RICCI FLOWS ON 4-MANIFOLDS 7
The following is the main result of this section, which is an analogy of Theorem 10.5
in [CT], where the same conclusion was derived for closed oriented Einstein 4-manifolds
with the same negative Einstein constant. The key point in our case is to use Lemma
3.1 to get non-collapsing balls and to prove the limiting metric is an Einstein metric
(cf. Lemma 3.3 and Lemma 3.4 below).
Proposition 3.2. Let M be a smooth closed oriented 4-manifold with λM < 0. If
g(t), t ∈ [0,∞) is a solution to (1.3) such that |Ric(g(t))| ≤ 3, and {tk} is a sequence
of times tends to infinity such that
diamgk(M) −→ ∞,
when k −→ ∞, where gk = g(tk), then there exists an m ∈ N, and sequences of points
{xj,k ∈ M}, j = 1, · · · , m, satisfying that, by passing to a subsequence,
(M, gk, x1,k, · · · , xm,k)
dGH−→ (
Nj , g∞, x1,∞, · · · , , xm,∞)
in the m-pointed Gromov-Hausdorff sense for k → ∞, where (Nj , g∞) j = 1, · · · , m are
complete Einstein 4-orbifolds with at most finitely many isolated orbifold points {qi}.
The scalar curvature (resp. volume) of g∞ is
R∞ = lim
λM(g(t)), (resp. V = Volg0(M) =
Volg∞(Nj)).
Furthermore, in the regular part of Nj, {gk} converges to g∞ in both L2,p (resp. C1,α)
sense for all p < ∞ (resp. α < 1).
We divide the proof of Proposition 3.2 into several useful lemmas.
A key result in the paper [CT] shows that, for any compact oriented Einstein 4-
manifold (X, g) with Einstein constant −3, there exists a constant C depending only
on the Euler number of X , and a point x ∈ X such that Volg(Bg(x, 1)) ≥ CVolg(X) (cf.
Theorem 0.14 [CT]). Cheeger-Tian remarked that the same result continues to hold
for 4-manifolds which are sufficiently negatively Ricci pinched. The following lemmas
is an analogy of the result for the metric gk in Proposition 3.2.
Lemma 3.3. There exists a constant v > 0, and a sequence {xk} ⊂ M such that
Volgk(Bgk(xk, 1)) ≥ v.
Proof. Let ε4 > 0 be the critical constant of Cheeger-Tian (cf. §1 [CT]), i.e., if X is a
Riemannian 4-manifold which is ε4-collapsed with locally bounded curvature, then X
carries an F-structure of positive rank. We may assume that, for all x ∈ M and gk,
Volgk(Bgk(x, 1)) < ε4. By a standard covering argument, for any k, there exist finitely
many points q1, · · · , ql such that E = M\
i=1Bgk(qi, 1) satisfies the hypothesis of
Theorem 2.7. Moreover, l ≤ Cδ−1 where C and δ are the constants in Theorem 2.7.
Therefore, by Theorem 2.7 we conclude that, there is a constant C1 independent of k
such that
(3.1)
|R(gk)|2dvk ≤ 6
|Rm(gk)|2dvk ≤ C1
Volgk(Bgk(qi, 1)).
8 F. FANG, Y. ZHANG, AND Z. ZHANG
On the other hand, by Lemma (3.1.2)
(R(gk)
2 − R(gk)2)dvk| ≤ 24
|R(gk)− R(gk)|dvk
k→∞−→ 0.
Therefore
(3.2)
∞Volgk(E)−
R(gk)
2dvk ≤ R(gk)2Volgk(E)−
R(gk)
(R(gk)
2 − R(gk)2)dvk
for sufficiently large k since R∞ ≤ λMV −
2 < 0. By inserting (3.1) we get that
∞(V −
V olgk(Bgk(qi, 1))−
∞Volgk(E)−
Volgk(Bgk(qi, 1)),
V ≤ C2
Volgk(Bgk(qi, 1)),
where C2 is a constant independent of k. Therefore, there is at least a ball among the
l balls whose volume is at least V
. The desired result follows. �
Assuming that diamgk(M) → ∞ for k → ∞, by using the technique developed in
[An3], the analogue of Theorem 3.3 in [An2] holds (cf. Theorem 2.3 in [An4]), i.e.
there exist a sequence of points {xk} ⊂ M such that, by passing to a subsequence,
{(M, gk, xk)}
dGH−→ (N∞, g∞, x∞)
where N∞ is a 4-orbifold with only isolated orbifold points {qi}, g∞ is a complete C0
orbifold metric, and g∞ is a C
1,α ∩L2,p Riemannian metric on the regular part of N∞,
for all p < ∞ and α < 1. Furthermore, {gk} converges to g∞ in the L2,p (resp. C1,α)
sense on the regular part of N∞, i.e. for any r ≫ 1 and k, there is a smooth embedding
Fk,r : Bg∞(x∞, r)\
i Bg∞(qi, r
−1) ⊂ N∞ → M such that, by passing to a subsequence,
F ∗k,rgk converge to g∞ in both L
2,p and C1,α senses.
Lemma 3.4. g∞ is an Einstein orbifold metric with scalar curvature R∞.
Proof. We first prove that g∞ is an Einstein metric with scalar curvature R∞ on the
regular part of N∞. Since F
k,rgk converge to g∞ in the L
2,p(resp. C1,α) sense on
Bg∞(p∞, r)\
i Bg∞(qi, r
−1), for any r, by Lemma 3.1, we obtain that
Bg∞ (p∞,r)\
i Bg∞(qi,r
|Rico(g∞)|2dv∞ ≤ lim
|Rico(gk)|2dvk = 0,
MAXIMUM SOLUTIONS OF NORMALIZED RICCI FLOWS ON 4-MANIFOLDS 9
Bg∞ (p∞,r)\
i Bg∞ (qi,r
|R(g∞)− R∞|dv∞ ≤ lim
|R(gk)− R(gk)|dvk = 0.
Therefore g∞ is a C
1,α Riemannian metric on Bg∞(p∞, r)\
i Bg∞(qi, r
−1) which sat-
isfies the Einstein equation in the weak sense. By elliptic regularity theory, g∞ is a
smooth Einstein metric with scalar curvature R∞.
Since g∞ is a C
0-orbifold metric, i.e. for any orbifold point qi ∈ N∞, there is
a neighborhood Ui ∼= B(0, r)/Γ of qi such that g̃∞ is a C0-Riemannian metric on
B(0, r) ⊂ R4 where Γ ⊂ SO(4) is a finite subgroup acting freely on S3, and g̃∞|B(0,r)\{0}
is the pull-back metric of g∞. Note that g̃∞ is a smooth Einstein metric on B(0, r)\{0}
satisfying that
B(0,r)
|Rm(g̃∞)|2dveg∞ < C < ∞. By the arguments as in [An1] and
[Ti], g̃∞ is a C
∞ Einstein metric on B(0, r) (cf. the proof of Theorem C in [An1], and
Section 4 in [Ti]). Hence g∞ is an Einstein orbifold metric. �
By the discussion before Lemma 3.4 we may choose ℓ sequences of points {xj,k} ⊂
M , j = 1, · · · , ℓ, such that distgk(xi,k, xj,k)
k→∞−→ ∞ for any i 6= j, and
(3.3) {(M, gk, x1,k, · · · , xℓ,k)}
dGH−→ (
Nj , g∞, x1,∞, · · · , xℓ,∞)
where (Nj , g∞, xj,∞), j = 1, · · · , ℓ are complete Einstein 4-orbifolds with only isolated
singular points and scalar curvatures R∞. Furthermore, {gk} converges to g∞ in both
L2,p (resp. C1,α) sense on the regular parts of Nj , j = 1, · · · , ℓ. Note that
(3.4) V ≥
Volg∞(Nj).
Lemma 3.5. The number of orbifold points of
Nj is less than a constant depending
only on the Euler characteristic χ(M).
Proof. For each orbifold point q ∈ Nj , there exist a sequence {qk} ⊂ M , and two
constants r ≫ r1 > 0 such that:
(3.5.1) q ∈ Bg∞(xj,∞, r);
(3.5.2) Bg∞(q, r1)\Bg∞(q, σ) lies in the regular part of Bg∞(xj,∞, r) for any σ < r1;
(3.5.3) (Bgk(qk, r1)\Bgk(qk, σ), gk)
C1,α−→ (Bg∞(q, r1)\Bg∞(q, σ), g∞).
By the definition of harmonic radius (cf. [An3]), the harmonic radii of all points in
Bgk(qk, r1)\Bgk(qk, σ) have a uniform lower bound, saying µ > 0, a constant depending
on σ but independent of k.
Clearly, there is a positive constant v0 (e.g.,
Volg∞(Bg∞(xj,∞, r))) such that
Volgk(Bgk(xj,k, r)) ≥ v0. Note that the Sobolev constants CS,k of Bgk(xj,k, r) are
bounded from below by a constant depending only on v0, r (cf. [An2] and [Cr]). There-
fore, by [An2] again we get that Volgk(Bgk(qk, s)) ≥ Cs4 for any s ≪ 1, where C is
independent of k.
10 F. FANG, Y. ZHANG, AND Z. ZHANG
Let us denote by rh,k the infimum of the harmonic radii of gk in the ball Bgk(qk, r1).
Note that rh,k
k→∞−→ 0 since q is a orbifold point (cf. [An3]). Therefore, there is a point
q̄k ∈ Bgk(qk, σ) so that rh(q̄k) = rh,k for sufficiently large k.
Consider the normalized balls (Bgk(qk, r1), r
h,kgk), which have harmonic radii at
least 1. By passing to a subsequence if necessary,
(Bgk(qk, r1), r
h,kgk, q̄k)
C1,α−→ (W, ḡ∞, q̄)
where (W, ḡ∞) is a complete Ricci-flat 4-manifold satisfying that
(3.5) Volḡ∞(Bḡ∞(q̄, r)) ≥ Cr4
for any r > 0. It is obvious that
|Rm(ḡ∞)|2dvḡ∞ ≤ lim inf
|Rm(gk)|2dvk ≤ C.
Therefore (W, ḡ∞) is an Asymptotically Locally Euclidean space (cf. Theorem 2.11 in
[N] or [An1]), which is asymptotic to a cone of S3/Γ where Γ ⊂ SO(4) is a finite group
acting freely on S3. By the Chern-Gauss-Bonnet formula
(3.6) χ(W ) =
|Rm(ḡ∞)|2dvḡ∞ +
|Γ| .
By [An1] W is isometric to R4, provided |Γ| = 1. Since the harmonic radius of ḡ∞ at
q̄ is 1, hence ḡ∞ can not be the Euclidean metric. Hence |Γ| ≥ 2. It is easy to verify
that χ(W ) ≥ 1. By (3.6) we get that
|Rm(ḡ∞)|2dvḡ∞ ≥ 4π2.
This proves that every orbifold point contributes to lim inf
|Rm(gk)|2dvk at least
4π2. By the rescaling invariance of the integral we conclude that the number of orbifold
points β ≤ C
The following lemma is an analogue of a result in Cheeger-Tian [CT].
Lemma 3.6. ℓ < χ(M)+β+1, where β := #{number of orbifold points in Lemma 3.5}.
Proof. Suppose not, i.e, ℓ ≥ χ(M) + β + 1, by definition there are at least χ(M) + 1
components of
1Nj which are smooth complete non-compact Einstein 4-manifolds of
finite volume, for simplicity saying N1, · · · , Ns, where s ≥ χ(M) + 1. By Theorem 4.5
in [CT], for each 1 ≤ j ≤ s,
|Rm(g∞)|2dvg∞ ≥ 8π2.
Since (M, gk, xk,j)
L2,p−→ (Nj , g∞, x∞,j), by Chern-Gauss-Bonnet formula and (3.1.3) in
Lemma 3.1 we get that
8π2χ(M) = lim
|Rm(gk)|2dvgk ≥
|Rm(g∞)|2dvg∞ ≥ 8π2(χ(M) + 1).
A contradiction. �
MAXIMUM SOLUTIONS OF NORMALIZED RICCI FLOWS ON 4-MANIFOLDS 11
Let m denote the maximal value of all possible choice of the base point sequences
in (3.3), which has a upper bound by Lemma 3.6.
Lemma 3.7. Let Mk,r = M\
Bgk(xj,k, r). For sufficiently large r, there is a
constant C independent of r such that
(3.7) lim
Volgk(Mk,r) ≤ C
Volg∞(Nj\Bg∞(xj,∞,
(3.8)
Volg∞(Nj) = V.
Proof. We may choose r ≫ 1 such that, for any y ∈
j=1(Nj\Bg∞(xj,∞, r − 1)),
Volg∞(Bg∞(y, 1)) ≤ 12ε4, where ε4 > 0 is the critical constant of Cheeger-Tian (cf.
proof of Lemma 3.3 or §1 [CT] ).
Now we claim that there is a constant k0 ≫ 1 such that, for any k > k0 and any
x ∈ Mk,r, Volgk(Bgk(x, 1)) ≤ ε4.
If it is false, without loss of generality we may assume a sequence of points {yk} ⊂
Mk,r such that
(3.9) Volgk(Bgk(yk, 1)) > ε4
Observe that the distance distgk(yk, xj,k) → ∞ as k → ∞ for all 1 ≤ j ≤ m. Otherwise,
assuming distgk(yk, xj,k) < ρ for some j and ρ > 0, we get that F
j,k,ρ(yk) → y∞ ∈
Bg∞(xj,∞, ρ)\Bg∞(xj,∞, r − 1), and so
(3.10) Volgk(Bgk(yk, 1)) → Volg∞(Bg∞(y∞, 1)) ≤
when k → ∞, since F ∗j,k,ρgk C1,α-converges to gj,∞, where
(3.11) Fj,k,ρ : Bg∞(xj,∞, ρ)\
Bg∞(qi, ρ
−1) ⊂ N∞ → M
is a smooth embedding so that F ∗j,k,ρgk converges to g∞ in the C
1,α-sense (cf. the
discussion before Lemma 3.4). A contradiction to (3.9).
Note that (M, gk, yk)
dGH−→ (N∞, g∞, y∞) where N∞ is a complete 4-orbifold different
from each of Nj, 1 ≤ j ≤ m. This violates the choice of maximality of m. Hence we
have proved the claim.
By a standard covering argument, for any k, there exist finitely many points
z1,k, · · · , zI,k such that Ek,r = Mk,r\
i=1Bzi,k(1) satisfies the hypothesis of Theorem
2.7, where I is independent of k. By Theorem 2.7, there is a constant C independent
of k such that
|R(gk)|2dvk ≤ 6
|Rm(gk)|2dvk ≤ C(
Volgk(Bgk(zi,k, 1))+Volgk(A0,1(Mk,r))).
By Lemma 3.1, for k ≫ 1, we have
(3.12)
|R(gk)− R(gk)|dvk <
|R(gk)− R(gk)|dvk −→ 0.
12 F. FANG, Y. ZHANG, AND Z. ZHANG
By (3.2) we get
∞Volgk(Ek,r)−
R(gk)
2dvk ≤
(R(gk)
2 − R(gk)2)dvk
|R(gk)− R(gk)|dvk,
Since Volgk(Ek,r) ≥ Volgk(Mk,r) −
Volgk(Bgk(zi,k, 1)), by the above together we get
immediately that
(3.13)
Volgk(Mk,r) ≤ C(
Volgk(Bgk(zi,k, 1))+Volgk(A0,1(Mk,r)))+24
|R(gk)−R(gk)|dvk.
If distgk(zi,k, xj,k) → ∞ for all 1 ≤ j ≤ m, by the same argument as above we get that
Volgk(Bgk(zi,k, 1)) → 0
when k → ∞. Otherwise, there exists a subsequence ks → ∞ and an index j such that
distgks (zi,ks, xj,ks) < ρ
for some constant ρ. In both cases, we obtain
lim sup
Volgk(Bgk(zi,ks, 1)) ≤
Volg∞(Nj\Bg∞(xj,∞,
for r ≫ ρ. Therefore, by (3.12) and (3.13) we conclude immediately (3.7).
By (3.7) it follows that lim
k,r→∞
Volgk(Mk,r) → 0. Hence (3.8) follows. �
By now Proposition 3.2 follows by the above lemmas.
4. Smooth convergence on the regular part
The main result of this section is the following:
Proposition 4.1. Let M be a closed 4-manifold satisfying that λ̄M < 0 and let
g(t), t ∈ [0,∞), be a solution to the normalized Ricci flow equation (1.3) on M with
uniformly bounded Ricci curvature. If (M, g(tk), pk)
dGH−→ (N∞, g∞, p∞), where tk → ∞
and N∞ is a 4-dimensional orbifold, and g(tk)
C1,α−→ g∞ on the regular part R of N∞ (the
compliment of the orbifold points), then, by passing to a subsequence, for all t ∈ [0,∞),
(M, g(tk + t), pk)
dGH−→ (N∞, g∞(t), p∞), where g∞(t) is a family of smooth metrics on
R solving the normalized Ricci flow equation on R with g∞(0) = g∞. Moreover, the
convergence is smooth on R× [0,∞).
In [Se] the convergence of Kähler-Ricci flow on compact Kähler manifolds with
bounded Ricci curvature was studied. It seems that the arguments in [Se] could be
applied to prove Proposition 4.1, but the authors can not follow completely her line.
Therefore, we give a quite different approach, where we first give a curvature estimate
of the Ricci flow similar to Perelman’s pseudolocality theorem. Using this curvature
MAXIMUM SOLUTIONS OF NORMALIZED RICCI FLOWS ON 4-MANIFOLDS 13
estimation we prove the limit Ricci flow exists on R × [0,∞). Finally, we prove that
R is exactly the regular part of every subsequence limit of (M, g(tk + t), pk), for all
t ∈ [0,∞). It deserves to point out that our approach works only in dimension 4.
We now give a curvature estimate for the Ricci flow which is an analogy of Perel-
man’s pseudolocality theorem (cf. [Pe1] Thm. 10.1). The difference is that here we
use the hypothesis of local almost Euclidean volume growth, instead of the almost
Euclidean isoperimetric estimate. The proof is much easier than that of Perelman’s
pseudolocality theorem.
Theorem 4.2. There exist universal constants δ0, ǫ0 > 0 with the following property.
Let g(t), t ∈ [0, (ǫP r0)2], be a solution to the Ricci flow equation (1.2) on a closed
n-manifold M and x0 ∈ M be a point. If the scalar curvature
R(x, t) ≥ −r−20 whenever distg(t)(x0, x) ≤ r0,
and the volume
Volg(t)(Bg(t)(x, r)) ≥ (1− δ0)Vol(B(r)) for all Bg(t)(x, r) ⊂ Bg(t)(x0, r0),
where B(r) denotes a ball of radius r in the n-Euclidean space and Vol(B(r)) denotes
its Euclidean volume, then the Riemannian curvature tensor satisfies
|Rm|g(t)(x, t) ≤ t−1, whenever distg(t)(x0, x) < ǫ0r0, and 0 < t ≤ (ǫ0r0)2.
In particular, |Rm|g(t)(x0, t) ≤ t−1 for all time t ∈ (0, (ǫ0r0)2].
Proof. We use Claim 1 and Claim 2 of Theorem 10.1 in [Pe1] and adopt a contradiction
argument. For any given small constants ǫ, δ > 0, set ǫ0 = ǫ, δ0 = δ, then there is a
solution to the Ricci flow equation (1.2), say (M, g(t)), not satisfying the conclusion
of the theorem. After a rescaling, we may assume that r0 = 1. Denote by M̄ the
non-empty set of pairs (x, t) such that |Rm|g(t)(x, t) > t−1, then as in Claim 1 and
Claim 2 of Theorem 10.1 in [Pe1], we can choose another space time point (x̄, t̄) ∈ M̄
with 0 < t̄ ≤ ǫ2, distg(t̄)(x0, x̄) < 110 , such that |Rm|g(t)(x, t) ≤ 4Q whenever
t̄− 1
Q−1 ≤ t ≤ t̄, distg(t̄)(x̄, x) ≤
(100nǫ)−1Q−1/2,
where Q = |Rm|g(t̄)(x̄, t̄). It is remarkable that from the proof of Claim 2 of Theorem
10.1 in [Pe1], each such a space time point (x, t) satisfies
distg(t)(x, x0) < distg(t̄)(x0, x̄) + (100nǫ)
−1Q−1/2 <
+ (100n)−1 <
Now choosing sequences of positive numbers ǫk → 0 and δk → 0, we obtain a
sequence of solutions (Mk, gk(t)), t ∈ [0, ǫ2k] and a sequence of points x0,k, x̄k ∈ Mk
and times t̄k, with each satisfying the assumptions of the theorem and the properties
described above. In particular, we have that Qk = |Rmk|gk(t̄k)(x̄k, t̄k) → ∞. Consider
the sequence of pointed Ricci flow solutions
(Bgk(t̄k)(x̄k,
(100nǫk)
k ), Qkgk(Q
k t+ t̄k), x̄k), t ∈ [−
, 0].
Using Hamilton’s compactness theorem for solutions to the Ricci flow, we can extract
a subsequence which converge to a complete Ricci flow solution (M∞, g∞(t), x̄∞), t ∈
, 0], with |Rm∞|g∞(0)(x̄∞, 0) = 1.
14 F. FANG, Y. ZHANG, AND Z. ZHANG
By assumption, the balls
Bgk(t̄k)(x̄k,
(100nǫk)
k ) ⊂ Bgk(t)(x0,k,
for any t ∈ [t̄− 1
Q−1, t̄], so the scalar curvature Rk(x, t) ≥ −1 for t ∈ [t̄− 12nQ
−1, t̄] and
x ∈ Bgk(t̄k)(x̄k, 110(100nǫk)
k ) and Volgk(t)(Bgk(t)(x, r)) ≥ (1−δk)Vol(B(r)) for any
metric ball Bgk(t)(x, r) ⊂ Bgk(t̄k)(x̄k, 110(100nǫk)
k ), t ∈ [t̄− 12nQ
−1, t̄]. Passing to
the limit, we see that g∞(t) has scalar curvature R∞ ≥ 0 everywhere and local volume
Volg∞(t)(Bg∞(t)(z, r)) ≥ Vol(B(r)) for any balls Bg∞(t)(z, r) at time t ∈ (− 12n , 0]. Then
the local variation formula of volume implies that R∞ ≡ 0 on M∞×(− 12n , 0], see [STW]
for details. By the evolution of the scalar curvature ∂
R∞ = △R∞ + 2|Ric∞|2, we get
that Ric∞ ≡ 0 over M∞ × (− 12n , 0]. Then the Bishop-Gromov volume comparison
theorem implies that g∞(t) are flat solutions to the Ricci flow, which contradicts the
fact that |Rm∞|(x̄∞, 0) = 1. This ends the proof of the theorem. �
The next lemma provides a comparison of the curvature of the normalized and un-
normalized Ricci flow. By assumption, there is C̄ < ∞ such that |Ric| ≤ C̄ everywhere
along the flow (M, g(t)). Note that by Lemma 3.1, there is some time T < ∞ such that
2R∞ ≤ R(g(t)) ≤ 12R∞ < 0 whenever t > T . Fix any such a time t̄ > T and let h(t)
and h̃(t̃) be the solutions to the normalized and unnormalized Ricci flow with initial
metric h(0) = h̃(0) = g(t̄) respectively, where t̃ = t̃(t) is the corresponding rescaled
time for t. Denote by Rmt̄, Rict̄, Rt̄ and R̃mt̄, R̃ict̄, R̃t̄ the corresponding Riemann-
ian curvature, Ricci curvature and scalar curvature of them, where |Rict̄| ≤ C̄ since
h(t) = g(t̄+ t). Then we have
Lemma 4.3. The solution h̃(t̃) exists for all time t̃ ∈ [0,∞). Furthermore, there exist
constants C and τ depending on λ̄M and C̄, such that
t ≤ t̃ ≤ Ct, |R̃mt̄|(x, t̃) ≤ |Rmt̄|(x, t) ≤ C|R̃mt̄|(x, t̃), whenever t ≤ τ.
Proof. The solution h(t) has average scalar curvature R(t̄ + t) ≤ 1
R∞ < 0, so h̃(t̃)
also has average scalar curvature R̃ < 0. From the evolution d
lnVol(h̃(t̃)) = −R̃, the
volume Vol(h̃(t̃)) increases strictly in t̃, so to normalize it, we need to compress the
space and time. Thus t̃ ≥ t and |R̃mt̄|(x, t̃) ≤ |Rmt̄|(x, t) for all (x, t). So h̃(t̃) exists
for all time.
The last assertion means that the scaling factor from normalized Ricci flow to the
unnormalized one is less than C on the time interval [0, τ ]. Consider the evolution of
average scalar curvature R̃(t̃):
(2|R̃ict̄|2 − R̃2t̄ )dvk
V olh̃( t̃)(M)
for some constant Λ = Λ(C̄), since |R̃ict̄| ≤ |Rict̄| ≤ C̄, |R̃t̄| ≤ |Rt̄| ≤ C̄, |R̃| ≤ |R| ≤ C̄.
Note that the initial value R̃(0) = R(g(t̄)) ≤ 1
R∞, so there is some constant τ̃ = τ̃(Λ)
such that R̃(t̃) ≤ 1
R∞ for t̃ ∈ [0, τ̃ ]. Thus the scaling factor from normalized Ricci
MAXIMUM SOLUTIONS OF NORMALIZED RICCI FLOWS ON 4-MANIFOLDS 15
flow to the unnormalized one, which equals
R(h(t))
R̃(t̃)
, is less than 8 on the time interval
t̃ ∈ [0, τ̃ ]. Now the result follows, by setting τ = τ̃
and C = 8. �
The following lemma gives the estimation of the local volume along the Ricci flow.
As in [Se], the proof uses Theorem A 1.5 of [CC]. By assumption, we have a solution
(M, g(t)) to the normalized Ricci flow (1.3) and a sequence of times tk → ∞ and points
pk such that (M, g(tk), pk)
dGH−→ (N∞, g∞, p∞) with g(tk)
C1,α−→ g∞ on the regular part R
of the orbifold N∞. For the space M or N∞, let Rǫ,ρ be the set of points x such that
dGH(B(x, r), B(r)) < ǫr for any r ≤ u, where u ≥ ρ is some constant depending on x.
Here and after, B(r) denotes a ball of radius r in 4-Euclidean space and B(x, r) the
metric ball of radius r with center x in a metric space. A weak version is WRǫ,ρ, the
set of points x such that there is u ≥ ρ with dGH(B(x, u), B(u)) < ǫu.
Lemma 4.4. For each q ∈ R, choose a sequence qk ∈ M that converge to q. Then for
any ǫ > 0, there exist k0, η, ρ > 0 such that
Vol(Bg(tk+t)(q
k, r)) ≥ (1− ǫ)Vol(B(r)), ∀r < ρ, k0 < k,
whenever Bg(tk+t)(q
k, r) ⊂ Bg(tk)(qk, ρ) and t ∈ [−η, η].
Proof. By the boundedness of Ricci tensor, there is a universal constant Λ = Λ(C̄) > 1
such that Bg(t)(p,Λ
−1r) ⊂ Bg(s)(p, r) ⊂ Bg(t)(p,Λr) for all t, s ∈ [tk − 1, tk + 1], p ∈ M
and r > 0. By Theorem A.1.5 of [CC], for fixed ǫ > 0, there are δ = δ(ǫ, n), ρ =
ρ(ǫ, n) > 0 such that x ∈ WRδ,r implies Vol(Bg(t)(x, r)) ≥ (1 − ǫ)Vol(B(r)) for each
r ≤ ρ and x ∈ M . So by definition, it suffice to show q′k ∈ Rδ,ρ with respect to each
metric g(t), t ∈ [tk − η, tk + η], whenever q
k ∈ Bg(tk)(qk,Λρ), for some constant η > 0.
The constant ρ may be modified by a smaller one if necessary.
Using Theorem A.1.5 of [CC] again, for fixed δ as above, there is δ1 = δ1(δ, n) > 0
such that qk ∈ WR
(Λ2+1)ρ
implies q
k ∈ Rδ,ρ for any q
k ∈ Bg(t)(qk,Λ2ρ). So it re-
duces to show qk ∈ WR
(Λ2+1)ρ
with respect to each time t ∈ [tk − η, tk + η] for some
η > 0 small enough. In fact, as showed in [Se], dGH(Bg(tk)(qk, ρ1), B(ρ1)) <
for some small number ρ1 and all k large enough. By the boundedness of Ricci ten-
sor again, there is a constant η ≤ 1 such that for each time t ∈ [−η, η], we have
dGH(Bg(tk+t)(qk, ρ1), Bg(tk)(qk, ρ1)) <
δ1ρ1 for all k. Thus dGH(Bg(tk+t)(qk, ρ1), B(ρ1)) <
δ1ρ1 for each t ∈ [−η, η]. Now the result follows by setting ρ = (1−δ)ρ1Λ2+1 . �
Note that in the proof, the constant δ1 = δ1(ǫ, n), so the constant η depends only on
ǫ, n and C̄. By assumption, there is a compact exhaustion {Ki}∞i=1 of R and a sequence
of smooth embeddings Fi : Ki → M such that Fi(p∞) = pi and F ∗i g(ti) converges to
g∞ in the local C
1,α sense. Following the lines described in [Se], we can prove
Lemma 4.5. Denote by Ki,k = Fk(Ki), then for any ǫ > 0 and i, there are k0, η, ρ > 0
such that
Vol(Bg(tk+t)(q
k, r)) ≥ (1− ǫ)Vol(B(r)), ∀q
k ∈ Ki,k, k0 < k, t ∈ [−η, η] and r < ρ.
Now we are ready to prove the Proposition 4.1.
16 F. FANG, Y. ZHANG, AND Z. ZHANG
Proof of Proposition 4.1. Assume that p∞ ∈ Ki for each i. Set ǫ = δ0 in the the
previous lemma, where δ0 is just the constant in Theorem 4.2, then for one fixed Ki,
there exist k0, η, ρ > 0 such that Vol(Bg(tk+t)(q, r)) ≥ (1 − δ0)Vol(B(r)) whenever
q ∈ Ki,k, k0 < k, t ∈ [−η, η] and r < ρ. Modifying ρ and η by smaller constants, we
assume (ǫ0ρ)
2 ≤ 2η < τ , where τ and ǫ0 are constants in Lemma 4.3 and Theorem 4.2
respectively.
Let hk(t̃) be the corresponding solutions to the unnormalized Ricci flow equation
with initial value hk(0) = g(tk−η), then Vol(Bhk(t̃)(q, r)) ≥ (1−δ0)Vol(B(r)) whenever
q ∈ Ki,k, r < ρ, k0 < k and t̃ satisfying t(t̃) ∈ [0, 2η], since the inequality Vol(B(q, r)) ≥
(1− δ0)Vol(B(r)) is scale invariant and Bhk(t̃) ⊂ Bg(tk+t(t̃))(q, r) for k large enough such
that tk ≥ T + η for T chosen as above. Denote by R̃mk the Riemannian curvature
tensor of hk, then by Theorem 4.2 and Lemma 4.3, we have
|Rm|(q, tk + t) ≤ C|R̃mk|(q, t̃) ≤ C(t̃)−1 ≤ C(t− tk + η)−1,
for all q ∈ Ki,k. Hence |Rm|(q, t) is uniformly bounded on Ki,k × [tk − η2 , tk +
By Hamilton’s compactness theorem of Ricci flow solution, {(Ki,k, g(tk+ t), pk)}∞k=1
converge along a subsequence to a solution to the normalized Ricci flow (Ki,∞, gi,∞(t), pi,∞), t ∈
), in the local C∞ sense. When we consider the time t = 0, then using a diag-
onalization argument, a subsequence of {(Ki,k, g(tk), pk)}i,k will converge in the local
C∞ sense to a smooth Riemannian manifold (K∞, g∞, p∞), which is just (R, g∞), by
the uniqueness of the limit space.
For fixed i, there is a family of metrics gi,∞(t), t ∈ (−η2 ,
), on Ki. As showed
in [Se], we translate the time by η
, say considering the sequence {(Ki,k, g(tk + η4 +
t), pk)}k, and repeat the above argument, then obtain that {(Ki,k, g(tk + t), pk)}k
loc−→
(Ki,∞, gi,∞(t), pi,∞) along another subsequence, on the time interval t ∈ (−η2 ,
The essential point is that the estimate dGH(Bg(tk)(qk, ρ1), B(ρ1)) <
δ1ρ1 in the proof
of Lemma 4.4 holds for some constant ρ1, simultaneously the time tk is replaced by
, but the constant η in Lemma 4.5 is fixed in this procedure. Iterating this process
infinite times we obtain the convergence on Ki for all t ∈ [0,∞). Then do the same
thing for each Ki, i = 1, 2, · · · , and after a diagonalization argument, we get that a
subsequence of {(Ki,k, g(tk + t), pk)}k, say (Ki,ki, g(tki + t), pki)
loc−→ (R, g∞(t), p∞) for
all t ∈ [0,∞), with g∞(0) = g∞.
We finally show that the completion of R with respect to the metric g∞(t), say R̄t,
is just N∞, for each time t ∈ [0,∞). Denote by S = N∞\R the set of singular points
of (N∞, g∞(0)), then it suffice to show that R̄t = R ∪ S for fixed time t. Assume
S = {ql}Ql=1, where Q ≤ β for β = β(M) by Lemma 3.5, and let ε > 0 be any
small constant such that Bg∞(0)(qi, ε) ∩ Bg∞(0)(qj , ε) = ∅ whenever i 6= j. Denote by
Kε = R\
Bg∞(0)(pl, ε), then using |Ric∞| ≤ C̄ on R× [0,∞) and by the evolution
of the distance function, we obtain dGH((R\Kε, g∞(t)),S) ≤ e2C̄tε and consequently
R̄t = R ∪ S, by letting ε → 0. �
5. Proofs of Theorems 1.1 and 1.2
The main result of this section is the following
MAXIMUM SOLUTIONS OF NORMALIZED RICCI FLOWS ON 4-MANIFOLDS 17
Theorem 5.1. Let (M, c) be a smooth oriented closed 4-manifold with a Spinc-structure
c. Assume that the first Chern class c1(c) of c is a monopole class of M satisfying that
(5.1) c21(c)[M ] ≥ 2χ(M) + 3τ(M) > 0.
Let g(t), t ∈ [0,∞), be a solution to (1.3) so that |Ric(g(t))| ≤ 3, and
(5.2) lim
λM(g(t)) = −
32π2c21(c)[M ].
Then there exists an m ∈ N, and sequences of points {xj,k ∈ M}, j = 1, · · · , m,
satisfying that, by passing to a subsequence,
(M, g(tk + t), x1,k, · · · , xm,k)
dGH−→ (
Nj, g∞, x1,∞, · · · , , xm,∞),
t ∈ [0,∞), in the m-pointed Gromov-Hausdorff sense for any tk → ∞, where (Nj , g∞)
j = 1, · · · , m are complete Kähler-Einstein orbifolds of complex dimension 2 with at
most finitely many isolated orbifold points {qi}. The scalar curvature (resp. volume)
of g∞ is
−Volg0(M)−
32π2c21(c)[M ] (resp. V = Volg0(M) =
Volg∞(Nj)).
Furthermore, in the regular part of Nj, {g(tk + t)} converges to g∞ in C∞-sense.
Comparing with Proposition 3.2, Theorem 5.1 shows that the Einstein orbifolds are
actually Kähler Einstein orbifolds under the additional assumptions. The key point in
the proof is that the sequence of the self-dual parts of the curvatures of the connections
on the determinant line bundles given by the irreducible solutions in the Seiberg-Witten
equations converges to a non-trivial parallel self-dual 2-form on every component Nj ,
which is a candidate of the Kähler form.
Let (M, c) and g(t) be the same as in Thoerem 5.1, and let V , m, tk, xj,k, R̆(g(t)),
gk, g∞, Nj and Fj,k,r be the same as in Section 3. Assume that, for each k, (φk, Ak) is
an irreducible solution to the Seiberg-Witten equations (2.1). Let | · |k denote the norm
with respect to the metric gk = g(tk). The following lemma shows that the L
2-norms
of the self-dual parts F+Ak tends to zero.
Lemma 5.2.
|∇kF+Ak |
kdvk = 0,
where ∇k is the connection on Λ2T ∗(M) induced by Levi-civita connection.
Proof. The Bochner formula implies that
0 = −1
∆k|φk|2k + |∇Akφk|2k +
R(gk)
|φk|2k +
|φk|4k,
By taking integration we get that
(5.3)
(|∇Akφk|2k +
R(gk)
|φk|2k)dvk = −
|φk|4kdvk.
18 F. FANG, Y. ZHANG, AND Z. ZHANG
Since λM(gk) is the lowest eigenvalue of the operator −4△k+R(gk), for any 1 ≫ ǫ > 0,
by definition
(5.4) λM(gk)
|φk|2k,ǫdvk ≤
(4|∇|φk|k,ǫ|2 +R(gk)|φk|2k,ǫ)dvk,
where | · |2k,ǫ = | · |2k + ǫ2. By Kato’s inequality (cf. (2.5)) and letting ǫ → 0,
λM(gk)
|φk|2kdvk ≤
(4|∇Akφk|2k +R(gk)|φk|2k)dvk = −
|φk|4kdvk ≤ 0.
As λM(gk) ≤ 0, by Schwarz inequality,
λM(gk)(
|φk|4k,ǫdvk)
2 = λM(gk)Volgk(M)
|φk|4k,ǫdvk)
2 ≤ λM(gk)
|φk|2k,ǫdvk.
Therefore
λM(gk)(
|φk|4k,ǫdvk)
(4|∇|φk|k,ǫ|2 +R(gk)|φ|2k,ǫ)dvk.
(5.5) 4
(|∇Akφk|2k − |∇|φk|k,ǫ|2)dvk ≤ −
|φk|4kdvk − λM(gk)(
|φk|4k,ǫdvk)
From (2.5), |∇|φk|k,ǫ|2 ≤ 34 |∇
Akφk|2k. Hence, by letting ǫ −→ 0, we have
(5.6)
|∇Akφk|2kdvk ≤ −((
|φk|4kdvk)
2 + λM(gk))(
|φk|4kdvk)
If c+1,k denotes the self-dual part of the harmonic form representing the first Chern class
c1(c) of c, by the Seiberg-Witten equation we get that
(5.7)
|φk|4kdvk = 8
|F+Ak |
2dvk ≥ 32π2[c+1,k]2[M ] ≥ 32π2c21(c)[M ].
Note that, by the standard estimates for Seiberg-Witten equations,
−R̆(gk) ≥ |φk|2k
and, by Theorem 1.1 in [FZ],
32π2c21(c)[M ] + λM(gk) is non-positive. Hence
(5.8)
|∇Akφk|2kdvk ≤ −(
32π2c21(c)[M ] + λM(gk))(
|φk|4kdvk)
≤ R̆(gk)V
32π2c21(c)[M ] + λM(gk)) −→ 0,
when k −→ ∞, by (5.2) and Lemma 3.1.
By the second one of the Seiberg-Witten equations again (cf. [Le2]),
(5.9) |∇kF+Ak |
|φk|2k|∇Akφk|2k,
where ∇Ak is the connection on Γ(S
) induced by the Levi-civita connection. Hence
|∇kF+Ak |
kdvk ≤
|R̆(g(tk))|
|∇Akφk|2kdvk −→ 0,
when k −→ ∞. �
MAXIMUM SOLUTIONS OF NORMALIZED RICCI FLOWS ON 4-MANIFOLDS 19
Regard F+Ak as self-dual 2-forms of g
k on Uj,r = Bg∞(xj,∞, r)\
i Bg∞(qi,j, r
where g′k = F
j,k,r+1gk, and qi,j are the orbifold points of Nj. Since
(5.10) |F+Ak |
|φk|4k ≤
R̆(gk)
2 ≤ C,
where C is a constant independent of k, F+Ak ∈ L
1,2(g′k), and
‖F+Ak‖L1,2(g′k) ≤ C
where C ′ is a constant independent of k. Note that ‖ · ‖L1,2(g∞) ≤ 2‖ · ‖L1,2(g′k) for k ≫ 1
since g′k
C1,α−→ g∞ on Uj,r. Thus, by passing to a subsequence, F+Ak
L1,2−→ Ωj ∈ L1,2(g∞), a
self-dual 2-form with respect to g∞.
Lemma 5.3. For any j, Ωj is a smooth self-dual 2-form on Uj,r\∂Uj,r such that
∇∞Ωj ≡ 0, and |Ωj |∞ ≡ cont. 6= 0, where ∇∞ is the connection induced by the
Levi-civita connection of g∞. Hence, g∞ is a Kähler metric with Kähler form
|Ωj |
on Uj,r.
Proof. By Lemma 5.2
|∇∞Ωj |2∞dv∞ = lim
|∇∞F+Ak |
∞dv∞ ≤ 2 lim
|∇kF+Ak |
kdvk = 0.
It is easy to see that Ωj is a weak solution of the elliptic equation ∇∞Ωj = 0 on Uj,r.
By elliptic equation theory, Ωj is a smooth self-dual 2-form on Uj,r\∂Uj,r, ∇∞Ωj ≡ 0,
and |Ωj|∞ ≡ cont..
Now we claim that, for any j and r ≫ 1,
|Ωj |2∞dv∞ 6= 0. If not, there exist
js, s = 1, · · · , m0, m0 ≤ m, such that
Ujs,r
|Ωjs|2∞dv∞ ≡ 0. By Lemma 3.1, R∞ =
R(gk) = lim
R̆(gk) = λMV
2 , which is the scalar curvature of g∞, i.e. R∞ =
R(g∞). Note that, by (5.10) and Lemma 3.7,
|Ωj |2∞dv∞ = lim
|F+Ak |
R̆(gk)
2Volg′
(Uj,r)
∞Volg∞(Uj,r),
|F+Ak |
kdvk −
|F+Ak |
kdvk| ≤
R̆(gk)
2Volgk(M\
Fk,j,r(Uj,r))
Volg∞(Nj\Uj, r2 ),
and, by Lemma 3.1,
(R(gk)
2 − R2∞)dvk| ≤ 24 lim
(|R(gk)−R(gk)|+ |R∞ − R(gk)|)dvk = 0,
20 F. FANG, Y. ZHANG, AND Z. ZHANG
where C is a constant in-dependent of k. Hence, we obtain
j 6=j1,··· ,jm0
Volg∞(Uj,r) ≥
8|Ωj|2∞dv∞ = lim
8|F+Ak|
≥ lim
8|F+Ak |
kdvk − CR
Volg∞(Nj\Uj, r2 )
≥ 32π2c21(c)[M ]− CR
Volg∞(Nj\Uj, r2 ).
The last inequality is obtained by (5.7). Thus, by (5.1),
j 6=j1,··· ,jm0
Volg∞(Uj,r) ≥ 32π2(2χ(M) + 3τ(M))− CR
Volg∞(Nj\Uj, r2 ).
By the Chern-Gauss-Bonnet formula and the Hirzebruch signature theorem,
2χ(M) + 3τ(M) ≥ 1
R(gk)
2 + 2|W+(gk)|2k)dvk −
|Rico(gk)|2dvk.
By Lemma 3.1, and the fact that g′k
L2,p−→ g∞ on Uj,r, we obtain that
j 6=j1,··· ,jm0
Volg∞(Uj,r) ≥
+ 2|W+(g∞)|2∞)dv∞
−CR2∞
Volg∞(Nj\Uj, r2 ).
Note that, on any Uj,r, j 6= j1, · · · , jm0 , ∇∞Ωj ≡ 0, |Ωj|∞ ≡ cont. 6= 0, and Ωj
is a self-dual 2-form. Thus g∞ is a Kähler metric with Kähler form
|Ωj |
on Uj,r,
j 6= j1, · · · , jm0 . It is well known that R
∞ = 24|W+(g∞)|2∞ for Kähler metrics (cf.
[Le3]). Thus
j 6=j1,··· ,jm0
Volg∞(Uj,r) ≥ R
j 6=j1,··· ,jm0
Volg∞(Uj,r)− CR
Volg∞(Nj\Uj, r2 )
js=j1,··· ,jm0
+ 2|W+(g∞)|2∞)dv∞
≥ R2∞
j 6=j1,··· ,jm0
Volg∞(Uj,r)− CR
Volg∞(Nj\Uj, r2 )
js=j1,··· ,jm0
∞Volg∞(Ujs,r).
MAXIMUM SOLUTIONS OF NORMALIZED RICCI FLOWS ON 4-MANIFOLDS 21
Note that, for r ≫ 1,
1 ≫ 3CR2∞
Volg∞(Nj\Uj, r2 ) ≥
js=j1,··· ,jm0
∞Volg∞(Ujs,r).
A contradiction. Thus, for all j,
|Ωj |2∞dv∞ 6= 0, and∇∞Ωj ≡ 0, |Ωj |∞ ≡ cont. 6= 0.
Thus we obtain the conclusion.
Proof of Theorem 5.1. First, assume that diamg(tk)(M) −→ ∞, when k −→ ∞. By
Proposition 3.2 and Proposition 4.1, there exists a m ∈ N, and a sequence of points
{xj,k ∈ M}, k ∈ N, j = 1, · · · , m, satisfying that, by passing to a subsequence,
(M, g(tk+t), x1,k, · · · , xm,k), t ∈ [0,∞), converges to {(N1, g∞, x1,∞), · · · , (Nm, g∞, xm,∞)}
in them-pointed Gromov-Hausdorff sense, when k −→ ∞, where (Nj , g∞) j = 1, · · · , m
are complete Einstein 4-orbifolds with finite isolated orbifold points {qi}. The scalar
curvature of g∞ is
R∞ = lim
λM(g(t)), and V = Volg0(M) =
Volg∞(Nj).
By Lemma 5.2, g∞ is a Kähler-Einstein metric in the non-singular part of
Then by the same arguments as in Section 4 of [Ti], g∞ is actually a Kähler-Einstein
orbifold metric. Furthermore, in the non-singular part of
Nj , {g(tk+ t)}, t ∈ [0,∞),
C∞-converges to g∞ by Proposition 4.1.
If diamgk(M) < C for a constant C in-dependent of k, we can also obtain the
conclusion by the similar, but much easier, arguments as above.
Theorem 5.4. Let (M, c) be a smooth compact closed oriented 4-manifold with a Spinc-
structure c. Assume that the first Chern class c1(c) of c is a monopole class of M
satisfying c21(c)[M ] = 2χ(M) + 3τ(M) > 0, and χ(M) = 3τ(M). If M admits a
solution g(t), t ∈ [0,∞) to (1.3) with |R(g(t))| ≤ 12, then
λM(g(t)) = −
32π2c21(c)[M ].
Furthermore, if |Ric(g(t))| ≤ 3, the Kähler-Einstein metric g∞ in Theorem 5.1 is a
complex hyperbolic metric.
Proof. Let V = V olg(t)(M). By the Chern-Gauss-Bonnet formula and the Hirzebruch
signature theorem,
(5.11) 2χ(M)− 3τ(M) ≥ 1
R(g(t))2 + 2|W−(g(t))|2 − 1
|Rico(g(t))|2dvg(t),
where W− is the anti-self-dual Weyl tensor. Note that
(5.12)
R(g(t))2dvg(t) ≥ R(g(t))2V −→ R
∞V = lim
λM(g(t))
22 F. FANG, Y. ZHANG, AND Z. ZHANG
when t −→ ∞, by Schwarz inequality and Lemma 3.1. By (5.11), (5.12), Lemma 3.1
and Theorem 1.1 in [FZ],
2χ(M)− 3τ(M) ≥ lim inf
|W−(g(t))|2dvg(t) +
λM(g(t))
≥ lim inf
|W−(g(t))|2dvg(t) +
c21(c)[M ]
= lim inf
|W−(g(t))|2dvg(t) +
(2χ(M) + 3τ(M)).
Since χ(M) = 3τ(M), we obtain
λM(g(t)) = −
32π2c21(c)[M ],
lim inf
|W−(g(t))|2dvg(t) = 0.
Now, assume that |Ric(g(t))| ≤ 3. Let tk, Nj , gk, and g∞ be the same as above.
For any j and compact subset U of the regular part of Nj,
|W−(g∞)|2∞dv∞ ≤ lim inf
|W−(g(tk))|2kdvk = 0,
since g(tk)
L2,p−→ g∞ on U . Hence g∞ is a Kähler-Einstein metric with W−(g∞) ≡ 0.
This implies that g∞ is a complex hyperbolic metric (cf. [Le1]). The desired result
follows.
Proofs of Theorem 1.1 and Theorem 1.2. By the work of Taubes [Ta], if (M,ω) is a
compact symplectic manifold with b+2 (M) > 1, the spin
c-structure induced by ω is a
monopole class. Moreover, since in this situation c21(c)[M ] = 2χ(M)+3τ(M), Theorem
1.1 (resp. Theorem 1.2) is an obvious consequence of Theorem 5.1 (resp. Theorem
5.4). �
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[Pe1] G.Perelman, The entropy formula for the Ricci flow and its geometric applications,
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[STW] N. Sesum, G. Tian and X.D. Wang, Notes on Perelman’s paper on the entropy formula for
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871-896.
Department of Mathematics, Capital Normal University, Beijing, P.R.China
E-mail address : ffang@nankai.edu.cn
Department of Mathematics, Capital Normal University, Beijing, P.R.China
Nankai Institute of Mathematics, Weijin Road 94, Tianjin 300071, P.R.China
http://arxiv.org/abs/math/0608439
http://arxiv.org/abs/math/0609254
http://arxiv.org/abs/math/0608504
http://arxiv.org/abs/math/0211159
http://arxiv.org/abs/math/0402238
	1. Introduction
	2. Preliminaries 
	2.1. Monopole class
	2.3. Kato's inequality
	2.5. Chern-Gauss-Bonnet formula and Hirzebruch signature formula
	2.6. Curvature estimates for 4-manifolds
	3.  The limiting behavior of Ricci flow 
	4. Smooth convergence on the regular part
	5. Proofs of Theorems 1.1 and 1.2
	References

  We consider maximum solution $g(t)$, $t\in [0, +\infty)$, to the normalized
Ricci flow. Among other things, we prove that, if $(M, \omega) $ is a smooth
compact symplectic 4-manifold such that $b_2^+(M)>1$ and let
$g(t),t\in[0,\infty)$, be a solution to (1.3) on $M$ whose Ricci curvature
satisfies that $|\text{Ric}(g(t))|\leq 3$ and additionally $\chi(M)=3 \tau
(M)>0$, then there exists an $m\in \mathbb{N}$, and a sequence of points
$\{x_{j,k}\in M\}$, $j=1, ..., m$, satisfying that, by passing to a
subsequence, $$(M, g(t_{k}+t), x_{1,k},..., x_{m,k})
\stackrel{d_{GH}}\longrightarrow (\coprod_{j=1}^m N_j, g_{\infty},
x_{1,\infty}, ...,, x_{m,\infty}),$$ $t\in [0, \infty)$, in the $m$-pointed
Gromov-Hausdorff sense for any sequence $t_{k}\longrightarrow \infty$, where
$(N_{j}, g_{\infty})$, $j=1,..., m$, are complete complex hyperbolic orbifolds
of complex dimension 2 with at most finitely many isolated orbifold points.
Moreover, the convergence is $C^{\infty}$ in the non-singular part of
$\coprod_1^m N_{j}$ and
$\text{Vol}_{g_{0}}(M)=\sum_{j=1}^{m}\text{Vol}_{g_{\infty}}(N_{j})$, where
$\chi(M)$ (resp. $\tau(M)$) is the Euler characteristic (resp. signature) of
$M$.

Introduction
Let (M, g) be a compact Riemannian manifold. The Perelman λ-functional
(1.1) λM(g) = inf
f∈C∞(M)
{F(g, f) :
e−fdvolg = 1}
where F(g, f) =
(Rg + |∇f |2)e−fdvolg and Rg is the scalar curvature of g. Note
that λM(g) is the lowest eigenvalue of the operator −4△ + Rg. By [Pe1] the gradient
flow of the Perelman λ-functional is the Hamilton’s the Ricci-flow evolution equation
(1.2)
g(t) = −2Ric(g(t))
The normalized Ricci flow equation on an n-manifold M reads
(1.3)
g(t) = −2Ric(g(t)) + 2R
where Ric (resp. R) denotes the Ricci tensor (resp. the average scalar curvature
). Note that (1.2) and (1.3) differ only by a change of scale in space and time,
and the volume Vol(g(t)) is constant in t. If dimM = n, λM(g) = λM(g)Volg(M)
is invariant up to rescaling the metric. Perelman [Pe1] has proved that λM(g(t)) is
non-decreasing along the Ricci flow g(t) whenever λM(g(t)) ≤ 0. This leads to the
The first author was supported by NSF Grant 19925104 of China, 973 project of Foundation Science
of China, and the Capital Normal University.
http://arxiv.org/abs/0704.0714v1
2 F. FANG, Y. ZHANG, AND Z. ZHANG
Perelman invariant λM by taking supremum of λM(g) in the set of all Riemannian
metrics on M .
By [AIL] the Perelman invariant λM is equal to the Yamabe invariant whenever
λM ≤ 0, after the earlier estimations (cf. [An5] [Pe2] [Le4] [FZ] and [Kot]). In
particular, if (M, g) is a smooth compact oriented 4-manifold with a Spinc-structure
c which is a monopole class (i.e., the associated Seiberg-Witten equation possesses
an irreducible solution) so that that c21(c)[M ] > 0, by [FZ] λM ≤ −
32π2c21(c)[M ].
Moreover, g is a Kähler-Einstein metric of negative scalar curvature if and only if
λM(g) = −
32π2c21(c)[M ]. However, there are plenty of 4-manifolds where the Perel-
man invariant λM = −
32π2c21(c)[M ] but do not admit any Kähler Einstein metric.
It is natural to study 4-manifolds with these extremal property. For such a 4-manifold
M , to seek for an ”optimal” Riemannian metric on M with respect to the Perelman
functional λM : M → R, we want to consider a maximal solution g(t) which is a solu-
tion of the Ricci flow (1.3). We call a longtime solution g(t), t ∈ [0,+∞), to the Ricci
flow (1.3) a maximum solution if lim
λM(g(t)) = λM . For a compact 3-manifold, by
Perelman [Pe2] all solutions of the Ricci flow (1.2) with surgery exist for longtime and
are maximum solutions, provided λM ≤ 0. In the paper [FZZ] obstructions are found
for the longtime solutions with bounded curvature to (1.3).
In this paper we are going to study the maximum solutions of (1.3) with bounded
Ricci curvatures instead. To avoid technique terminology we only state our results
for symplectic 4-manifolds by using the celebrated work of Taubes [Ta]: if (M,ω) is
a compact symplectic manifold with b+2 (M) > 1 (the dimension of self-dual harmonic
2-forms of M), the spinc-structure induced by ω is a monopole class. Moreover, in
this situation c21(c)[M ] = 2χ(M) + 3τ(M), where χ(M) (resp. τ(M)) is the Euler
characteristic (resp. signature) of M .
Theorem 1.1. Let (M,ω) be a smooth compact symplectic 4-manifold satisfying that
b+2 (M) > 1 and 2χ(M) + 3τ(M) > 0. If g(t), t ∈ [0,∞), is a solution to (1.3) such
that |Ric(g(t))| ≤ 3, and
λM(g(t)) = −
32π2(2χ(M) + 3τ(M)),
then there exists an m ∈ N, and sequences of points {xj,k ∈ M}, j = 1, · · · , m,
satisfying that, by passing to a subsequence,
(M, g(tk + t), x1,k, · · · , xm,k)
dGH−→ (
Nj, g∞, x1,∞, · · · , , xm,∞),
t ∈ [0,∞), in the m-pointed Gromov-Hausdorff sense for any sequence tk −→ ∞, where
(Nj , g∞), j = 1, · · · , m, are complete Kähler-Einstein orbifolds of complex dimension 2
with at most finitely many isolated orbifold points. The scalar curvature (resp. volume)
of g∞ is
−Volg0(M)−
32π2(2χ(M) + 3τ(M)) (resp. Volg0(M) =
Volg∞(Nj))
Moreover, the convergence is C∞ in the non-singular part of
1 Nj.
MAXIMUM SOLUTIONS OF NORMALIZED RICCI FLOWS ON 4-MANIFOLDS 3
We first remark that, if the diameters diamg(tk)(M) possess a uniform upper bound,
then m = 1, and N1 is a compact Kähler-Einstein orbifold. Secondly, if the Ricci curva-
ture bound in the above theorem is replaced by a uniform bound of sectional curvature,
then every (Nj, g∞), j = 1, · · · , m are complete Kähler-Einstein manifolds. By the
same arguments as in [An5][An6],
j=1Nj can weakly embed in M ,
j=1Nj ⊂⊂ M ,
i.e. for any compact subset K ⊂
j=1Nj , there is a smooth embedding FK : K −→ M .
Furthermore, there exists a sufficiently large compact subset K ⊂
j=1Nj such that
M\K admits an F-structure of positive rank. This type geometric decomposition seems
very useful to understand the diffeomorphism type of 4-manifolds.
Theorem 1.2. Let (M,ω) be a smooth compact symplectic 4-manifold such that b+2 (M) >
1 and let g(t), t ∈ [0,∞), be a solution to (1.3) such that |R(g(t))| ≤ 12. If in addition
χ(M) = 3τ(M) > 0, then
λM(g(t)) = −
32π2(2χ(M) + 3τ(M))
Moreover, if |Ric(g(t))| ≤ 3, the Kähler-Einstein metric g∞ in Theorem 1.1 is complex
hyperbolic.
To conclude the section we point out that the main result in Theorem 1.1 (resp.
Corollary 1.2) holds if the manifold is not symplectic but a compact oriented 4-manifold
with a monopole class c1 (i.e. with a spin
c-structure with non-vanishing Seiberg-Witten
invariant) so that c21 = 2χ(M) + 3τ(M) > 0.
2. Preliminaries
2.1. Monopole class. Let (M, g) be a compact oriented Riemannian 4-manifold with
a Spinc structure c. Let b+2 (M) denote the dimension of the space of self-dual harmonic
2-forms in M . Let S±
denote the Spinc-bundles associated to c, and let L be the
determinant line bundle of c. There is a well-defined Dirac operator
DA : Γ(S+c ) −→ Γ(S−c )
Let c : ∧∗T ∗M −→ End(S+
) denote the Clifford multiplication on the Spinc-
bundles, and, for any φ ∈ Γ(S±
), let
q(φ) = φ⊗ φ− 1
|φ|2id.
The Seiberg-Witten equations read
(2.1)
DAφ = 0
c(F+A ) = q(φ)
where A is an Hermitian connection on L, and F+A is the self-dual part of the curvature
of A.
A solution of (2.1) is called reducible if φ ≡ 0; otherwise, it is called irreducible. If
(φ,A) is a resolution of (2.1), one calculates easily that
(2.2) |F+A | =
|φ|2,
4 F. FANG, Y. ZHANG, AND Z. ZHANG
The Bochner formula reads
(2.3) 0 = −2△|φ|2 + 4|∇Aφ|2 +Rg|φ|2 + |φ|4,
where Rg is the scalar curvature of g.
The Seiberg-Witten invariant can be defined by counting the irreducible solutions
of the Seiberg-Witten equations (cf. [Le2]).
Definition 2.2. ([K1]) Let M be a smooth compact oriented 4-manifold. An element
α ∈ H2(M,Z)/torsion is called a monopole class ofM if and only if there exists a Spinc-
structure c on M with first Chern class c1 ≡ α(mod torsion), so that the Seiberg-Witten
equations have a solution for every Riemannian metric g on M .
By the celebrated work of Taubes [Ta], if (M,ω) is a compact symplectic 4-manifold
with b+2 (M) > 1, the canonical class of (M,ω) is a monopole class.
2.3. Kato’s inequality. Let (M, g) be a Riemannian Spinc-manifold of dimension n,
the following Kato inequality is useful.
Proposition 2.4. (Proposition 2.2 in [BD]) Let φ be a harmonic Spinc-spinor on
(M, g), i.e. DAφ = 0, where DA is the Dirac operator and A is an Hermitian connection
on the determinant line bundle. Then
(2.4) |∇|φ||2 ≤ n− 1
|∇Aφ|2 ≤ |∇Aφ|2
at all points where φ is non-zero. Moreover, |∇|φ||2 = |∇Aφ|2 occurs only if ∇Aφ ≡ 0.
Note that the arguments in the proof of Proposition 2.2 in [BD] can be used to
prove this proposition without any change, where the same conclusion was derived for
Spin-spinor φ. For any ǫ > 0, let |φ|2ǫ = |φ|2+ǫ2. If φ is harmonic, by above proposition,
(2.5) |∇|φ|ǫ|2 ≤
|∇|φ||2 ≤ n− 1
|∇Aφ|2 ≤ |∇Aφ|2
at points where φ(p) 6= 0. Since {p ∈ M : φ(p) 6= 0} is dense in M for harmonic φ, we
conclude that (2.5) holds everywhere in M .
2.5. Chern-Gauss-Bonnet formula and Hirzebruch signature formula. Let
(M, g) be a compact closed oriented Riemannian 4-manifold, χ(M) and τ(M) are the
Euler number and the signature of M respectively. The Chern-Gauss-Bonnet formula
and the Hirzebruch signature theorem say that
(2.6) χ(M) =
+ |Wg|2 −
|Rico|2)dvg, and
(2.7) τ(M) =
(|W+g |2 − |W−g |2)dvg,
where Rico = Ric(g)− Rg
g is the Einstein tensor, W+g and W
g are the self-dual and
anti-self-dual Weyl tensors respectively (cf. [B]). If g is a Kähler-Einstein metric, then
(2.8) R2g = 24|W+g |2,
MAXIMUM SOLUTIONS OF NORMALIZED RICCI FLOWS ON 4-MANIFOLDS 5
(cf. [Le3]) which will be used in the proof of Theorem 1.1.
By Chern-Gauss-Bonnet formula, one has an L2-bound of the curvature operator
Rm(g) by the bounds of Ricci curvature, i.e. if |Ric(g)| < C, then
(2.9)
|Rm(g)|2dvg ≤ 8π2χ(M) + C1V olg(M),
where C and C1 are constants independent of (M, g).
Let (N, g) be a complete Ricci-flat Einstein 4-manifold. Assume that
(2.10)
|Rm(g)|2dvg < ∞, and Volg(Bg(x, r)) ≥ Cr4,
for all r > 0, a point x ∈ N , and a positive constant C. By Theorem 2.11 of [N], (N, g)
is ALE. (i.e, Asymptotically Locally Euclidean space) of order 4. It is well-known that
N is asymptotic to the cone on the spherical space form S3/Γ, where Γ ⊂ SO(4) is a
finite group. The Chern-Gauss-Bonnet formula implies that
(2.11) χ(N) =
|Rm(g)|2dvg +
(cf. [N] and [An1]).
2.6. Curvature estimates for 4-manifolds. Now let’s recall a result of [CT], which
is important to the proof of Theorem 1.1. Let (M, g) be a complete Riemannian 4-
manifold. A subset U ⊂ M such that for all p ∈ U , sup
Bg(p,1)
Ric(g) ≥ −3, is called
̺-collapsed if for all p ∈ U ,
V olg(Bg(p, 1)) ≤ ̺.
By Theorem 0.1 in [CG], there is a constant ε4 such that if U is ̺-collapsed with
sectional curvature |Kg| ≤ 1 and ̺ ≤ ε4, then U carries an F-structure of positive
rank.
Theorem 2.7. (Remark 5.11 and Theorem 1.26 in [CT]) There exist constants δ > 0,
c > 0 such that: if (M, g) is a complete Riemannian 4-manifold with |Ric(g)| ≤ 3 and
|Rm(g)|2dvg ≤ C,
and if E ⊂ M is a bounded open subset such that T1(E) = {x ∈ M : dist(x, E) ≤ 1} is
ε4-collapsed with
Bg(x,1)
|Rm(g)|2dvg ≤ δ (for all T1(E)),
then ∫
|Rm(g)|2dvg ≤ cV olg(A0,1(E)),
where A0,1(E) = T1(E)\E.
6 F. FANG, Y. ZHANG, AND Z. ZHANG
3. The limiting behavior of Ricci flow
In this section we study the limiting behavior of Ricci-flow with bounded Ricci
curvatures on 4-manifolds. We will assume in this section that M is a smooth closed
oriented 4-manifold with λM < 0, and g(t), t ∈ [0,+∞), is a longtime solution of the
normalized Ricci flow (1.3) with bounded Ricci-curvature. By normalization we may
assume that |Ric(g(t))| ≤ 3. By (2.9) there is a constant C independent of t such that
|Rm(g(t))|2dvg(t) ≤ C.
Let us denoted by V the volume Volg(0)(M) = Volg(t)(M), and R̆(g(t)) = min
R(g(t))(x)
the minimum of the scalar curvature of g(t). It is easy to see that R̆(g(t)) ≤ λMV −
Lemma 3.1. (3.1.1) lim
λM(g(t)) = lim
R(g(t)) = lim
R̆(g(t)) = R∞
(3.1.2) lim
|R(g(t))−R(g(t))|dvg(t) = 0,
(3.1.3) lim
|Rico(g(t))|2dvg(t) = 0.
Proof. By Perelman [Pe1] λM(g(t)) is a non-decreasing function on t, therefore the limit
λM(g(t)) exists since λM < 0. Now let us denote by R∞ the limit lim
λM(g(t)).
Note that R∞ ≤ λMV −
2 < 0. To prove (3.1.1), we first prove that both lim
R(g(t))
and lim
R̆(g(t)) exist and take values R∞. By the same arguments as in the proof of
Proposition 2.6 and Lemma 2.7 of [FZZ] we get that
R(g(t))− R̆(g(t)) = 0.
Observe that R(g(t)) ≥ λM(g(t)) ≥ R̆(g(t)) (cf. [KL] (92.3)). Therefore lim
R(g(t)) =
R∞ = lim
R̆(g(t)). This proves (3.1.1).
Note that∫
|R(g(t))− R(g(t))|dvg(t) ≤
(R(g(t))− R̆(g(t)))dvg(t) +
(R(g(t))− R̆(g(t)))dvg(t)
= 2(R(g(t))− R̆(g(t)))V
(3.1.2) follows from (3.1.1).
By Lemma 3.1 in [FZZ],
|Rico(g(t))|2dvg(t)dt < ∞,
and, by Lemma 1 in [Ye], we have
|Rico(g(t))|2dvg(t) ≤ −2
|∇Rico(g(t))|2dvg(t)+4
|Rm||Rico(g(t))|2dvg(t) < D,
where D is a constant independent of t. By the same argument as in the proof of
Proposition 2.6 in [FZZ] (3.1.3) follows. �
MAXIMUM SOLUTIONS OF NORMALIZED RICCI FLOWS ON 4-MANIFOLDS 7
The following is the main result of this section, which is an analogy of Theorem 10.5
in [CT], where the same conclusion was derived for closed oriented Einstein 4-manifolds
with the same negative Einstein constant. The key point in our case is to use Lemma
3.1 to get non-collapsing balls and to prove the limiting metric is an Einstein metric
(cf. Lemma 3.3 and Lemma 3.4 below).
Proposition 3.2. Let M be a smooth closed oriented 4-manifold with λM < 0. If
g(t), t ∈ [0,∞) is a solution to (1.3) such that |Ric(g(t))| ≤ 3, and {tk} is a sequence
of times tends to infinity such that
diamgk(M) −→ ∞,
when k −→ ∞, where gk = g(tk), then there exists an m ∈ N, and sequences of points
{xj,k ∈ M}, j = 1, · · · , m, satisfying that, by passing to a subsequence,
(M, gk, x1,k, · · · , xm,k)
dGH−→ (
Nj , g∞, x1,∞, · · · , , xm,∞)
in the m-pointed Gromov-Hausdorff sense for k → ∞, where (Nj , g∞) j = 1, · · · , m are
complete Einstein 4-orbifolds with at most finitely many isolated orbifold points {qi}.
The scalar curvature (resp. volume) of g∞ is
R∞ = lim
λM(g(t)), (resp. V = Volg0(M) =
Volg∞(Nj)).
Furthermore, in the regular part of Nj, {gk} converges to g∞ in both L2,p (resp. C1,α)
sense for all p < ∞ (resp. α < 1).
We divide the proof of Proposition 3.2 into several useful lemmas.
A key result in the paper [CT] shows that, for any compact oriented Einstein 4-
manifold (X, g) with Einstein constant −3, there exists a constant C depending only
on the Euler number of X , and a point x ∈ X such that Volg(Bg(x, 1)) ≥ CVolg(X) (cf.
Theorem 0.14 [CT]). Cheeger-Tian remarked that the same result continues to hold
for 4-manifolds which are sufficiently negatively Ricci pinched. The following lemmas
is an analogy of the result for the metric gk in Proposition 3.2.
Lemma 3.3. There exists a constant v > 0, and a sequence {xk} ⊂ M such that
Volgk(Bgk(xk, 1)) ≥ v.
Proof. Let ε4 > 0 be the critical constant of Cheeger-Tian (cf. §1 [CT]), i.e., if X is a
Riemannian 4-manifold which is ε4-collapsed with locally bounded curvature, then X
carries an F-structure of positive rank. We may assume that, for all x ∈ M and gk,
Volgk(Bgk(x, 1)) < ε4. By a standard covering argument, for any k, there exist finitely
many points q1, · · · , ql such that E = M\
i=1Bgk(qi, 1) satisfies the hypothesis of
Theorem 2.7. Moreover, l ≤ Cδ−1 where C and δ are the constants in Theorem 2.7.
Therefore, by Theorem 2.7 we conclude that, there is a constant C1 independent of k
such that
(3.1)
|R(gk)|2dvk ≤ 6
|Rm(gk)|2dvk ≤ C1
Volgk(Bgk(qi, 1)).
8 F. FANG, Y. ZHANG, AND Z. ZHANG
On the other hand, by Lemma (3.1.2)
(R(gk)
2 − R(gk)2)dvk| ≤ 24
|R(gk)− R(gk)|dvk
k→∞−→ 0.
Therefore
(3.2)
∞Volgk(E)−
R(gk)
2dvk ≤ R(gk)2Volgk(E)−
R(gk)
(R(gk)
2 − R(gk)2)dvk
for sufficiently large k since R∞ ≤ λMV −
2 < 0. By inserting (3.1) we get that
∞(V −
V olgk(Bgk(qi, 1))−
∞Volgk(E)−
Volgk(Bgk(qi, 1)),
V ≤ C2
Volgk(Bgk(qi, 1)),
where C2 is a constant independent of k. Therefore, there is at least a ball among the
l balls whose volume is at least V
. The desired result follows. �
Assuming that diamgk(M) → ∞ for k → ∞, by using the technique developed in
[An3], the analogue of Theorem 3.3 in [An2] holds (cf. Theorem 2.3 in [An4]), i.e.
there exist a sequence of points {xk} ⊂ M such that, by passing to a subsequence,
{(M, gk, xk)}
dGH−→ (N∞, g∞, x∞)
where N∞ is a 4-orbifold with only isolated orbifold points {qi}, g∞ is a complete C0
orbifold metric, and g∞ is a C
1,α ∩L2,p Riemannian metric on the regular part of N∞,
for all p < ∞ and α < 1. Furthermore, {gk} converges to g∞ in the L2,p (resp. C1,α)
sense on the regular part of N∞, i.e. for any r ≫ 1 and k, there is a smooth embedding
Fk,r : Bg∞(x∞, r)\
i Bg∞(qi, r
−1) ⊂ N∞ → M such that, by passing to a subsequence,
F ∗k,rgk converge to g∞ in both L
2,p and C1,α senses.
Lemma 3.4. g∞ is an Einstein orbifold metric with scalar curvature R∞.
Proof. We first prove that g∞ is an Einstein metric with scalar curvature R∞ on the
regular part of N∞. Since F
k,rgk converge to g∞ in the L
2,p(resp. C1,α) sense on
Bg∞(p∞, r)\
i Bg∞(qi, r
−1), for any r, by Lemma 3.1, we obtain that
Bg∞ (p∞,r)\
i Bg∞(qi,r
|Rico(g∞)|2dv∞ ≤ lim
|Rico(gk)|2dvk = 0,
MAXIMUM SOLUTIONS OF NORMALIZED RICCI FLOWS ON 4-MANIFOLDS 9
Bg∞ (p∞,r)\
i Bg∞ (qi,r
|R(g∞)− R∞|dv∞ ≤ lim
|R(gk)− R(gk)|dvk = 0.
Therefore g∞ is a C
1,α Riemannian metric on Bg∞(p∞, r)\
i Bg∞(qi, r
−1) which sat-
isfies the Einstein equation in the weak sense. By elliptic regularity theory, g∞ is a
smooth Einstein metric with scalar curvature R∞.
Since g∞ is a C
0-orbifold metric, i.e. for any orbifold point qi ∈ N∞, there is
a neighborhood Ui ∼= B(0, r)/Γ of qi such that g̃∞ is a C0-Riemannian metric on
B(0, r) ⊂ R4 where Γ ⊂ SO(4) is a finite subgroup acting freely on S3, and g̃∞|B(0,r)\{0}
is the pull-back metric of g∞. Note that g̃∞ is a smooth Einstein metric on B(0, r)\{0}
satisfying that
B(0,r)
|Rm(g̃∞)|2dveg∞ < C < ∞. By the arguments as in [An1] and
[Ti], g̃∞ is a C
∞ Einstein metric on B(0, r) (cf. the proof of Theorem C in [An1], and
Section 4 in [Ti]). Hence g∞ is an Einstein orbifold metric. �
By the discussion before Lemma 3.4 we may choose ℓ sequences of points {xj,k} ⊂
M , j = 1, · · · , ℓ, such that distgk(xi,k, xj,k)
k→∞−→ ∞ for any i 6= j, and
(3.3) {(M, gk, x1,k, · · · , xℓ,k)}
dGH−→ (
Nj , g∞, x1,∞, · · · , xℓ,∞)
where (Nj , g∞, xj,∞), j = 1, · · · , ℓ are complete Einstein 4-orbifolds with only isolated
singular points and scalar curvatures R∞. Furthermore, {gk} converges to g∞ in both
L2,p (resp. C1,α) sense on the regular parts of Nj , j = 1, · · · , ℓ. Note that
(3.4) V ≥
Volg∞(Nj).
Lemma 3.5. The number of orbifold points of
Nj is less than a constant depending
only on the Euler characteristic χ(M).
Proof. For each orbifold point q ∈ Nj , there exist a sequence {qk} ⊂ M , and two
constants r ≫ r1 > 0 such that:
(3.5.1) q ∈ Bg∞(xj,∞, r);
(3.5.2) Bg∞(q, r1)\Bg∞(q, σ) lies in the regular part of Bg∞(xj,∞, r) for any σ < r1;
(3.5.3) (Bgk(qk, r1)\Bgk(qk, σ), gk)
C1,α−→ (Bg∞(q, r1)\Bg∞(q, σ), g∞).
By the definition of harmonic radius (cf. [An3]), the harmonic radii of all points in
Bgk(qk, r1)\Bgk(qk, σ) have a uniform lower bound, saying µ > 0, a constant depending
on σ but independent of k.
Clearly, there is a positive constant v0 (e.g.,
Volg∞(Bg∞(xj,∞, r))) such that
Volgk(Bgk(xj,k, r)) ≥ v0. Note that the Sobolev constants CS,k of Bgk(xj,k, r) are
bounded from below by a constant depending only on v0, r (cf. [An2] and [Cr]). There-
fore, by [An2] again we get that Volgk(Bgk(qk, s)) ≥ Cs4 for any s ≪ 1, where C is
independent of k.
10 F. FANG, Y. ZHANG, AND Z. ZHANG
Let us denote by rh,k the infimum of the harmonic radii of gk in the ball Bgk(qk, r1).
Note that rh,k
k→∞−→ 0 since q is a orbifold point (cf. [An3]). Therefore, there is a point
q̄k ∈ Bgk(qk, σ) so that rh(q̄k) = rh,k for sufficiently large k.
Consider the normalized balls (Bgk(qk, r1), r
h,kgk), which have harmonic radii at
least 1. By passing to a subsequence if necessary,
(Bgk(qk, r1), r
h,kgk, q̄k)
C1,α−→ (W, ḡ∞, q̄)
where (W, ḡ∞) is a complete Ricci-flat 4-manifold satisfying that
(3.5) Volḡ∞(Bḡ∞(q̄, r)) ≥ Cr4
for any r > 0. It is obvious that
|Rm(ḡ∞)|2dvḡ∞ ≤ lim inf
|Rm(gk)|2dvk ≤ C.
Therefore (W, ḡ∞) is an Asymptotically Locally Euclidean space (cf. Theorem 2.11 in
[N] or [An1]), which is asymptotic to a cone of S3/Γ where Γ ⊂ SO(4) is a finite group
acting freely on S3. By the Chern-Gauss-Bonnet formula
(3.6) χ(W ) =
|Rm(ḡ∞)|2dvḡ∞ +
|Γ| .
By [An1] W is isometric to R4, provided |Γ| = 1. Since the harmonic radius of ḡ∞ at
q̄ is 1, hence ḡ∞ can not be the Euclidean metric. Hence |Γ| ≥ 2. It is easy to verify
that χ(W ) ≥ 1. By (3.6) we get that
|Rm(ḡ∞)|2dvḡ∞ ≥ 4π2.
This proves that every orbifold point contributes to lim inf
|Rm(gk)|2dvk at least
4π2. By the rescaling invariance of the integral we conclude that the number of orbifold
points β ≤ C
The following lemma is an analogue of a result in Cheeger-Tian [CT].
Lemma 3.6. ℓ < χ(M)+β+1, where β := #{number of orbifold points in Lemma 3.5}.
Proof. Suppose not, i.e, ℓ ≥ χ(M) + β + 1, by definition there are at least χ(M) + 1
components of
1Nj which are smooth complete non-compact Einstein 4-manifolds of
finite volume, for simplicity saying N1, · · · , Ns, where s ≥ χ(M) + 1. By Theorem 4.5
in [CT], for each 1 ≤ j ≤ s,
|Rm(g∞)|2dvg∞ ≥ 8π2.
Since (M, gk, xk,j)
L2,p−→ (Nj , g∞, x∞,j), by Chern-Gauss-Bonnet formula and (3.1.3) in
Lemma 3.1 we get that
8π2χ(M) = lim
|Rm(gk)|2dvgk ≥
|Rm(g∞)|2dvg∞ ≥ 8π2(χ(M) + 1).
A contradiction. �
MAXIMUM SOLUTIONS OF NORMALIZED RICCI FLOWS ON 4-MANIFOLDS 11
Let m denote the maximal value of all possible choice of the base point sequences
in (3.3), which has a upper bound by Lemma 3.6.
Lemma 3.7. Let Mk,r = M\
Bgk(xj,k, r). For sufficiently large r, there is a
constant C independent of r such that
(3.7) lim
Volgk(Mk,r) ≤ C
Volg∞(Nj\Bg∞(xj,∞,
(3.8)
Volg∞(Nj) = V.
Proof. We may choose r ≫ 1 such that, for any y ∈
j=1(Nj\Bg∞(xj,∞, r − 1)),
Volg∞(Bg∞(y, 1)) ≤ 12ε4, where ε4 > 0 is the critical constant of Cheeger-Tian (cf.
proof of Lemma 3.3 or §1 [CT] ).
Now we claim that there is a constant k0 ≫ 1 such that, for any k > k0 and any
x ∈ Mk,r, Volgk(Bgk(x, 1)) ≤ ε4.
If it is false, without loss of generality we may assume a sequence of points {yk} ⊂
Mk,r such that
(3.9) Volgk(Bgk(yk, 1)) > ε4
Observe that the distance distgk(yk, xj,k) → ∞ as k → ∞ for all 1 ≤ j ≤ m. Otherwise,
assuming distgk(yk, xj,k) < ρ for some j and ρ > 0, we get that F
j,k,ρ(yk) → y∞ ∈
Bg∞(xj,∞, ρ)\Bg∞(xj,∞, r − 1), and so
(3.10) Volgk(Bgk(yk, 1)) → Volg∞(Bg∞(y∞, 1)) ≤
when k → ∞, since F ∗j,k,ρgk C1,α-converges to gj,∞, where
(3.11) Fj,k,ρ : Bg∞(xj,∞, ρ)\
Bg∞(qi, ρ
−1) ⊂ N∞ → M
is a smooth embedding so that F ∗j,k,ρgk converges to g∞ in the C
1,α-sense (cf. the
discussion before Lemma 3.4). A contradiction to (3.9).
Note that (M, gk, yk)
dGH−→ (N∞, g∞, y∞) where N∞ is a complete 4-orbifold different
from each of Nj, 1 ≤ j ≤ m. This violates the choice of maximality of m. Hence we
have proved the claim.
By a standard covering argument, for any k, there exist finitely many points
z1,k, · · · , zI,k such that Ek,r = Mk,r\
i=1Bzi,k(1) satisfies the hypothesis of Theorem
2.7, where I is independent of k. By Theorem 2.7, there is a constant C independent
of k such that
|R(gk)|2dvk ≤ 6
|Rm(gk)|2dvk ≤ C(
Volgk(Bgk(zi,k, 1))+Volgk(A0,1(Mk,r))).
By Lemma 3.1, for k ≫ 1, we have
(3.12)
|R(gk)− R(gk)|dvk <
|R(gk)− R(gk)|dvk −→ 0.
12 F. FANG, Y. ZHANG, AND Z. ZHANG
By (3.2) we get
∞Volgk(Ek,r)−
R(gk)
2dvk ≤
(R(gk)
2 − R(gk)2)dvk
|R(gk)− R(gk)|dvk,
Since Volgk(Ek,r) ≥ Volgk(Mk,r) −
Volgk(Bgk(zi,k, 1)), by the above together we get
immediately that
(3.13)
Volgk(Mk,r) ≤ C(
Volgk(Bgk(zi,k, 1))+Volgk(A0,1(Mk,r)))+24
|R(gk)−R(gk)|dvk.
If distgk(zi,k, xj,k) → ∞ for all 1 ≤ j ≤ m, by the same argument as above we get that
Volgk(Bgk(zi,k, 1)) → 0
when k → ∞. Otherwise, there exists a subsequence ks → ∞ and an index j such that
distgks (zi,ks, xj,ks) < ρ
for some constant ρ. In both cases, we obtain
lim sup
Volgk(Bgk(zi,ks, 1)) ≤
Volg∞(Nj\Bg∞(xj,∞,
for r ≫ ρ. Therefore, by (3.12) and (3.13) we conclude immediately (3.7).
By (3.7) it follows that lim
k,r→∞
Volgk(Mk,r) → 0. Hence (3.8) follows. �
By now Proposition 3.2 follows by the above lemmas.
4. Smooth convergence on the regular part
The main result of this section is the following:
Proposition 4.1. Let M be a closed 4-manifold satisfying that λ̄M < 0 and let
g(t), t ∈ [0,∞), be a solution to the normalized Ricci flow equation (1.3) on M with
uniformly bounded Ricci curvature. If (M, g(tk), pk)
dGH−→ (N∞, g∞, p∞), where tk → ∞
and N∞ is a 4-dimensional orbifold, and g(tk)
C1,α−→ g∞ on the regular part R of N∞ (the
compliment of the orbifold points), then, by passing to a subsequence, for all t ∈ [0,∞),
(M, g(tk + t), pk)
dGH−→ (N∞, g∞(t), p∞), where g∞(t) is a family of smooth metrics on
R solving the normalized Ricci flow equation on R with g∞(0) = g∞. Moreover, the
convergence is smooth on R× [0,∞).
In [Se] the convergence of Kähler-Ricci flow on compact Kähler manifolds with
bounded Ricci curvature was studied. It seems that the arguments in [Se] could be
applied to prove Proposition 4.1, but the authors can not follow completely her line.
Therefore, we give a quite different approach, where we first give a curvature estimate
of the Ricci flow similar to Perelman’s pseudolocality theorem. Using this curvature
MAXIMUM SOLUTIONS OF NORMALIZED RICCI FLOWS ON 4-MANIFOLDS 13
estimation we prove the limit Ricci flow exists on R × [0,∞). Finally, we prove that
R is exactly the regular part of every subsequence limit of (M, g(tk + t), pk), for all
t ∈ [0,∞). It deserves to point out that our approach works only in dimension 4.
We now give a curvature estimate for the Ricci flow which is an analogy of Perel-
man’s pseudolocality theorem (cf. [Pe1] Thm. 10.1). The difference is that here we
use the hypothesis of local almost Euclidean volume growth, instead of the almost
Euclidean isoperimetric estimate. The proof is much easier than that of Perelman’s
pseudolocality theorem.
Theorem 4.2. There exist universal constants δ0, ǫ0 > 0 with the following property.
Let g(t), t ∈ [0, (ǫP r0)2], be a solution to the Ricci flow equation (1.2) on a closed
n-manifold M and x0 ∈ M be a point. If the scalar curvature
R(x, t) ≥ −r−20 whenever distg(t)(x0, x) ≤ r0,
and the volume
Volg(t)(Bg(t)(x, r)) ≥ (1− δ0)Vol(B(r)) for all Bg(t)(x, r) ⊂ Bg(t)(x0, r0),
where B(r) denotes a ball of radius r in the n-Euclidean space and Vol(B(r)) denotes
its Euclidean volume, then the Riemannian curvature tensor satisfies
|Rm|g(t)(x, t) ≤ t−1, whenever distg(t)(x0, x) < ǫ0r0, and 0 < t ≤ (ǫ0r0)2.
In particular, |Rm|g(t)(x0, t) ≤ t−1 for all time t ∈ (0, (ǫ0r0)2].
Proof. We use Claim 1 and Claim 2 of Theorem 10.1 in [Pe1] and adopt a contradiction
argument. For any given small constants ǫ, δ > 0, set ǫ0 = ǫ, δ0 = δ, then there is a
solution to the Ricci flow equation (1.2), say (M, g(t)), not satisfying the conclusion
of the theorem. After a rescaling, we may assume that r0 = 1. Denote by M̄ the
non-empty set of pairs (x, t) such that |Rm|g(t)(x, t) > t−1, then as in Claim 1 and
Claim 2 of Theorem 10.1 in [Pe1], we can choose another space time point (x̄, t̄) ∈ M̄
with 0 < t̄ ≤ ǫ2, distg(t̄)(x0, x̄) < 110 , such that |Rm|g(t)(x, t) ≤ 4Q whenever
t̄− 1
Q−1 ≤ t ≤ t̄, distg(t̄)(x̄, x) ≤
(100nǫ)−1Q−1/2,
where Q = |Rm|g(t̄)(x̄, t̄). It is remarkable that from the proof of Claim 2 of Theorem
10.1 in [Pe1], each such a space time point (x, t) satisfies
distg(t)(x, x0) < distg(t̄)(x0, x̄) + (100nǫ)
−1Q−1/2 <
+ (100n)−1 <
Now choosing sequences of positive numbers ǫk → 0 and δk → 0, we obtain a
sequence of solutions (Mk, gk(t)), t ∈ [0, ǫ2k] and a sequence of points x0,k, x̄k ∈ Mk
and times t̄k, with each satisfying the assumptions of the theorem and the properties
described above. In particular, we have that Qk = |Rmk|gk(t̄k)(x̄k, t̄k) → ∞. Consider
the sequence of pointed Ricci flow solutions
(Bgk(t̄k)(x̄k,
(100nǫk)
k ), Qkgk(Q
k t+ t̄k), x̄k), t ∈ [−
, 0].
Using Hamilton’s compactness theorem for solutions to the Ricci flow, we can extract
a subsequence which converge to a complete Ricci flow solution (M∞, g∞(t), x̄∞), t ∈
, 0], with |Rm∞|g∞(0)(x̄∞, 0) = 1.
14 F. FANG, Y. ZHANG, AND Z. ZHANG
By assumption, the balls
Bgk(t̄k)(x̄k,
(100nǫk)
k ) ⊂ Bgk(t)(x0,k,
for any t ∈ [t̄− 1
Q−1, t̄], so the scalar curvature Rk(x, t) ≥ −1 for t ∈ [t̄− 12nQ
−1, t̄] and
x ∈ Bgk(t̄k)(x̄k, 110(100nǫk)
k ) and Volgk(t)(Bgk(t)(x, r)) ≥ (1−δk)Vol(B(r)) for any
metric ball Bgk(t)(x, r) ⊂ Bgk(t̄k)(x̄k, 110(100nǫk)
k ), t ∈ [t̄− 12nQ
−1, t̄]. Passing to
the limit, we see that g∞(t) has scalar curvature R∞ ≥ 0 everywhere and local volume
Volg∞(t)(Bg∞(t)(z, r)) ≥ Vol(B(r)) for any balls Bg∞(t)(z, r) at time t ∈ (− 12n , 0]. Then
the local variation formula of volume implies that R∞ ≡ 0 on M∞×(− 12n , 0], see [STW]
for details. By the evolution of the scalar curvature ∂
R∞ = △R∞ + 2|Ric∞|2, we get
that Ric∞ ≡ 0 over M∞ × (− 12n , 0]. Then the Bishop-Gromov volume comparison
theorem implies that g∞(t) are flat solutions to the Ricci flow, which contradicts the
fact that |Rm∞|(x̄∞, 0) = 1. This ends the proof of the theorem. �
The next lemma provides a comparison of the curvature of the normalized and un-
normalized Ricci flow. By assumption, there is C̄ < ∞ such that |Ric| ≤ C̄ everywhere
along the flow (M, g(t)). Note that by Lemma 3.1, there is some time T < ∞ such that
2R∞ ≤ R(g(t)) ≤ 12R∞ < 0 whenever t > T . Fix any such a time t̄ > T and let h(t)
and h̃(t̃) be the solutions to the normalized and unnormalized Ricci flow with initial
metric h(0) = h̃(0) = g(t̄) respectively, where t̃ = t̃(t) is the corresponding rescaled
time for t. Denote by Rmt̄, Rict̄, Rt̄ and R̃mt̄, R̃ict̄, R̃t̄ the corresponding Riemann-
ian curvature, Ricci curvature and scalar curvature of them, where |Rict̄| ≤ C̄ since
h(t) = g(t̄+ t). Then we have
Lemma 4.3. The solution h̃(t̃) exists for all time t̃ ∈ [0,∞). Furthermore, there exist
constants C and τ depending on λ̄M and C̄, such that
t ≤ t̃ ≤ Ct, |R̃mt̄|(x, t̃) ≤ |Rmt̄|(x, t) ≤ C|R̃mt̄|(x, t̃), whenever t ≤ τ.
Proof. The solution h(t) has average scalar curvature R(t̄ + t) ≤ 1
R∞ < 0, so h̃(t̃)
also has average scalar curvature R̃ < 0. From the evolution d
lnVol(h̃(t̃)) = −R̃, the
volume Vol(h̃(t̃)) increases strictly in t̃, so to normalize it, we need to compress the
space and time. Thus t̃ ≥ t and |R̃mt̄|(x, t̃) ≤ |Rmt̄|(x, t) for all (x, t). So h̃(t̃) exists
for all time.
The last assertion means that the scaling factor from normalized Ricci flow to the
unnormalized one is less than C on the time interval [0, τ ]. Consider the evolution of
average scalar curvature R̃(t̃):
(2|R̃ict̄|2 − R̃2t̄ )dvk
V olh̃( t̃)(M)
for some constant Λ = Λ(C̄), since |R̃ict̄| ≤ |Rict̄| ≤ C̄, |R̃t̄| ≤ |Rt̄| ≤ C̄, |R̃| ≤ |R| ≤ C̄.
Note that the initial value R̃(0) = R(g(t̄)) ≤ 1
R∞, so there is some constant τ̃ = τ̃(Λ)
such that R̃(t̃) ≤ 1
R∞ for t̃ ∈ [0, τ̃ ]. Thus the scaling factor from normalized Ricci
MAXIMUM SOLUTIONS OF NORMALIZED RICCI FLOWS ON 4-MANIFOLDS 15
flow to the unnormalized one, which equals
R(h(t))
R̃(t̃)
, is less than 8 on the time interval
t̃ ∈ [0, τ̃ ]. Now the result follows, by setting τ = τ̃
and C = 8. �
The following lemma gives the estimation of the local volume along the Ricci flow.
As in [Se], the proof uses Theorem A 1.5 of [CC]. By assumption, we have a solution
(M, g(t)) to the normalized Ricci flow (1.3) and a sequence of times tk → ∞ and points
pk such that (M, g(tk), pk)
dGH−→ (N∞, g∞, p∞) with g(tk)
C1,α−→ g∞ on the regular part R
of the orbifold N∞. For the space M or N∞, let Rǫ,ρ be the set of points x such that
dGH(B(x, r), B(r)) < ǫr for any r ≤ u, where u ≥ ρ is some constant depending on x.
Here and after, B(r) denotes a ball of radius r in 4-Euclidean space and B(x, r) the
metric ball of radius r with center x in a metric space. A weak version is WRǫ,ρ, the
set of points x such that there is u ≥ ρ with dGH(B(x, u), B(u)) < ǫu.
Lemma 4.4. For each q ∈ R, choose a sequence qk ∈ M that converge to q. Then for
any ǫ > 0, there exist k0, η, ρ > 0 such that
Vol(Bg(tk+t)(q
k, r)) ≥ (1− ǫ)Vol(B(r)), ∀r < ρ, k0 < k,
whenever Bg(tk+t)(q
k, r) ⊂ Bg(tk)(qk, ρ) and t ∈ [−η, η].
Proof. By the boundedness of Ricci tensor, there is a universal constant Λ = Λ(C̄) > 1
such that Bg(t)(p,Λ
−1r) ⊂ Bg(s)(p, r) ⊂ Bg(t)(p,Λr) for all t, s ∈ [tk − 1, tk + 1], p ∈ M
and r > 0. By Theorem A.1.5 of [CC], for fixed ǫ > 0, there are δ = δ(ǫ, n), ρ =
ρ(ǫ, n) > 0 such that x ∈ WRδ,r implies Vol(Bg(t)(x, r)) ≥ (1 − ǫ)Vol(B(r)) for each
r ≤ ρ and x ∈ M . So by definition, it suffice to show q′k ∈ Rδ,ρ with respect to each
metric g(t), t ∈ [tk − η, tk + η], whenever q
k ∈ Bg(tk)(qk,Λρ), for some constant η > 0.
The constant ρ may be modified by a smaller one if necessary.
Using Theorem A.1.5 of [CC] again, for fixed δ as above, there is δ1 = δ1(δ, n) > 0
such that qk ∈ WR
(Λ2+1)ρ
implies q
k ∈ Rδ,ρ for any q
k ∈ Bg(t)(qk,Λ2ρ). So it re-
duces to show qk ∈ WR
(Λ2+1)ρ
with respect to each time t ∈ [tk − η, tk + η] for some
η > 0 small enough. In fact, as showed in [Se], dGH(Bg(tk)(qk, ρ1), B(ρ1)) <
for some small number ρ1 and all k large enough. By the boundedness of Ricci ten-
sor again, there is a constant η ≤ 1 such that for each time t ∈ [−η, η], we have
dGH(Bg(tk+t)(qk, ρ1), Bg(tk)(qk, ρ1)) <
δ1ρ1 for all k. Thus dGH(Bg(tk+t)(qk, ρ1), B(ρ1)) <
δ1ρ1 for each t ∈ [−η, η]. Now the result follows by setting ρ = (1−δ)ρ1Λ2+1 . �
Note that in the proof, the constant δ1 = δ1(ǫ, n), so the constant η depends only on
ǫ, n and C̄. By assumption, there is a compact exhaustion {Ki}∞i=1 of R and a sequence
of smooth embeddings Fi : Ki → M such that Fi(p∞) = pi and F ∗i g(ti) converges to
g∞ in the local C
1,α sense. Following the lines described in [Se], we can prove
Lemma 4.5. Denote by Ki,k = Fk(Ki), then for any ǫ > 0 and i, there are k0, η, ρ > 0
such that
Vol(Bg(tk+t)(q
k, r)) ≥ (1− ǫ)Vol(B(r)), ∀q
k ∈ Ki,k, k0 < k, t ∈ [−η, η] and r < ρ.
Now we are ready to prove the Proposition 4.1.
16 F. FANG, Y. ZHANG, AND Z. ZHANG
Proof of Proposition 4.1. Assume that p∞ ∈ Ki for each i. Set ǫ = δ0 in the the
previous lemma, where δ0 is just the constant in Theorem 4.2, then for one fixed Ki,
there exist k0, η, ρ > 0 such that Vol(Bg(tk+t)(q, r)) ≥ (1 − δ0)Vol(B(r)) whenever
q ∈ Ki,k, k0 < k, t ∈ [−η, η] and r < ρ. Modifying ρ and η by smaller constants, we
assume (ǫ0ρ)
2 ≤ 2η < τ , where τ and ǫ0 are constants in Lemma 4.3 and Theorem 4.2
respectively.
Let hk(t̃) be the corresponding solutions to the unnormalized Ricci flow equation
with initial value hk(0) = g(tk−η), then Vol(Bhk(t̃)(q, r)) ≥ (1−δ0)Vol(B(r)) whenever
q ∈ Ki,k, r < ρ, k0 < k and t̃ satisfying t(t̃) ∈ [0, 2η], since the inequality Vol(B(q, r)) ≥
(1− δ0)Vol(B(r)) is scale invariant and Bhk(t̃) ⊂ Bg(tk+t(t̃))(q, r) for k large enough such
that tk ≥ T + η for T chosen as above. Denote by R̃mk the Riemannian curvature
tensor of hk, then by Theorem 4.2 and Lemma 4.3, we have
|Rm|(q, tk + t) ≤ C|R̃mk|(q, t̃) ≤ C(t̃)−1 ≤ C(t− tk + η)−1,
for all q ∈ Ki,k. Hence |Rm|(q, t) is uniformly bounded on Ki,k × [tk − η2 , tk +
By Hamilton’s compactness theorem of Ricci flow solution, {(Ki,k, g(tk+ t), pk)}∞k=1
converge along a subsequence to a solution to the normalized Ricci flow (Ki,∞, gi,∞(t), pi,∞), t ∈
), in the local C∞ sense. When we consider the time t = 0, then using a diag-
onalization argument, a subsequence of {(Ki,k, g(tk), pk)}i,k will converge in the local
C∞ sense to a smooth Riemannian manifold (K∞, g∞, p∞), which is just (R, g∞), by
the uniqueness of the limit space.
For fixed i, there is a family of metrics gi,∞(t), t ∈ (−η2 ,
), on Ki. As showed
in [Se], we translate the time by η
, say considering the sequence {(Ki,k, g(tk + η4 +
t), pk)}k, and repeat the above argument, then obtain that {(Ki,k, g(tk + t), pk)}k
loc−→
(Ki,∞, gi,∞(t), pi,∞) along another subsequence, on the time interval t ∈ (−η2 ,
The essential point is that the estimate dGH(Bg(tk)(qk, ρ1), B(ρ1)) <
δ1ρ1 in the proof
of Lemma 4.4 holds for some constant ρ1, simultaneously the time tk is replaced by
, but the constant η in Lemma 4.5 is fixed in this procedure. Iterating this process
infinite times we obtain the convergence on Ki for all t ∈ [0,∞). Then do the same
thing for each Ki, i = 1, 2, · · · , and after a diagonalization argument, we get that a
subsequence of {(Ki,k, g(tk + t), pk)}k, say (Ki,ki, g(tki + t), pki)
loc−→ (R, g∞(t), p∞) for
all t ∈ [0,∞), with g∞(0) = g∞.
We finally show that the completion of R with respect to the metric g∞(t), say R̄t,
is just N∞, for each time t ∈ [0,∞). Denote by S = N∞\R the set of singular points
of (N∞, g∞(0)), then it suffice to show that R̄t = R ∪ S for fixed time t. Assume
S = {ql}Ql=1, where Q ≤ β for β = β(M) by Lemma 3.5, and let ε > 0 be any
small constant such that Bg∞(0)(qi, ε) ∩ Bg∞(0)(qj , ε) = ∅ whenever i 6= j. Denote by
Kε = R\
Bg∞(0)(pl, ε), then using |Ric∞| ≤ C̄ on R× [0,∞) and by the evolution
of the distance function, we obtain dGH((R\Kε, g∞(t)),S) ≤ e2C̄tε and consequently
R̄t = R ∪ S, by letting ε → 0. �
5. Proofs of Theorems 1.1 and 1.2
The main result of this section is the following
MAXIMUM SOLUTIONS OF NORMALIZED RICCI FLOWS ON 4-MANIFOLDS 17
Theorem 5.1. Let (M, c) be a smooth oriented closed 4-manifold with a Spinc-structure
c. Assume that the first Chern class c1(c) of c is a monopole class of M satisfying that
(5.1) c21(c)[M ] ≥ 2χ(M) + 3τ(M) > 0.
Let g(t), t ∈ [0,∞), be a solution to (1.3) so that |Ric(g(t))| ≤ 3, and
(5.2) lim
λM(g(t)) = −
32π2c21(c)[M ].
Then there exists an m ∈ N, and sequences of points {xj,k ∈ M}, j = 1, · · · , m,
satisfying that, by passing to a subsequence,
(M, g(tk + t), x1,k, · · · , xm,k)
dGH−→ (
Nj, g∞, x1,∞, · · · , , xm,∞),
t ∈ [0,∞), in the m-pointed Gromov-Hausdorff sense for any tk → ∞, where (Nj , g∞)
j = 1, · · · , m are complete Kähler-Einstein orbifolds of complex dimension 2 with at
most finitely many isolated orbifold points {qi}. The scalar curvature (resp. volume)
of g∞ is
−Volg0(M)−
32π2c21(c)[M ] (resp. V = Volg0(M) =
Volg∞(Nj)).
Furthermore, in the regular part of Nj, {g(tk + t)} converges to g∞ in C∞-sense.
Comparing with Proposition 3.2, Theorem 5.1 shows that the Einstein orbifolds are
actually Kähler Einstein orbifolds under the additional assumptions. The key point in
the proof is that the sequence of the self-dual parts of the curvatures of the connections
on the determinant line bundles given by the irreducible solutions in the Seiberg-Witten
equations converges to a non-trivial parallel self-dual 2-form on every component Nj ,
which is a candidate of the Kähler form.
Let (M, c) and g(t) be the same as in Thoerem 5.1, and let V , m, tk, xj,k, R̆(g(t)),
gk, g∞, Nj and Fj,k,r be the same as in Section 3. Assume that, for each k, (φk, Ak) is
an irreducible solution to the Seiberg-Witten equations (2.1). Let | · |k denote the norm
with respect to the metric gk = g(tk). The following lemma shows that the L
2-norms
of the self-dual parts F+Ak tends to zero.
Lemma 5.2.
|∇kF+Ak |
kdvk = 0,
where ∇k is the connection on Λ2T ∗(M) induced by Levi-civita connection.
Proof. The Bochner formula implies that
0 = −1
∆k|φk|2k + |∇Akφk|2k +
R(gk)
|φk|2k +
|φk|4k,
By taking integration we get that
(5.3)
(|∇Akφk|2k +
R(gk)
|φk|2k)dvk = −
|φk|4kdvk.
18 F. FANG, Y. ZHANG, AND Z. ZHANG
Since λM(gk) is the lowest eigenvalue of the operator −4△k+R(gk), for any 1 ≫ ǫ > 0,
by definition
(5.4) λM(gk)
|φk|2k,ǫdvk ≤
(4|∇|φk|k,ǫ|2 +R(gk)|φk|2k,ǫ)dvk,
where | · |2k,ǫ = | · |2k + ǫ2. By Kato’s inequality (cf. (2.5)) and letting ǫ → 0,
λM(gk)
|φk|2kdvk ≤
(4|∇Akφk|2k +R(gk)|φk|2k)dvk = −
|φk|4kdvk ≤ 0.
As λM(gk) ≤ 0, by Schwarz inequality,
λM(gk)(
|φk|4k,ǫdvk)
2 = λM(gk)Volgk(M)
|φk|4k,ǫdvk)
2 ≤ λM(gk)
|φk|2k,ǫdvk.
Therefore
λM(gk)(
|φk|4k,ǫdvk)
(4|∇|φk|k,ǫ|2 +R(gk)|φ|2k,ǫ)dvk.
(5.5) 4
(|∇Akφk|2k − |∇|φk|k,ǫ|2)dvk ≤ −
|φk|4kdvk − λM(gk)(
|φk|4k,ǫdvk)
From (2.5), |∇|φk|k,ǫ|2 ≤ 34 |∇
Akφk|2k. Hence, by letting ǫ −→ 0, we have
(5.6)
|∇Akφk|2kdvk ≤ −((
|φk|4kdvk)
2 + λM(gk))(
|φk|4kdvk)
If c+1,k denotes the self-dual part of the harmonic form representing the first Chern class
c1(c) of c, by the Seiberg-Witten equation we get that
(5.7)
|φk|4kdvk = 8
|F+Ak |
2dvk ≥ 32π2[c+1,k]2[M ] ≥ 32π2c21(c)[M ].
Note that, by the standard estimates for Seiberg-Witten equations,
−R̆(gk) ≥ |φk|2k
and, by Theorem 1.1 in [FZ],
32π2c21(c)[M ] + λM(gk) is non-positive. Hence
(5.8)
|∇Akφk|2kdvk ≤ −(
32π2c21(c)[M ] + λM(gk))(
|φk|4kdvk)
≤ R̆(gk)V
32π2c21(c)[M ] + λM(gk)) −→ 0,
when k −→ ∞, by (5.2) and Lemma 3.1.
By the second one of the Seiberg-Witten equations again (cf. [Le2]),
(5.9) |∇kF+Ak |
|φk|2k|∇Akφk|2k,
where ∇Ak is the connection on Γ(S
) induced by the Levi-civita connection. Hence
|∇kF+Ak |
kdvk ≤
|R̆(g(tk))|
|∇Akφk|2kdvk −→ 0,
when k −→ ∞. �
MAXIMUM SOLUTIONS OF NORMALIZED RICCI FLOWS ON 4-MANIFOLDS 19
Regard F+Ak as self-dual 2-forms of g
k on Uj,r = Bg∞(xj,∞, r)\
i Bg∞(qi,j, r
where g′k = F
j,k,r+1gk, and qi,j are the orbifold points of Nj. Since
(5.10) |F+Ak |
|φk|4k ≤
R̆(gk)
2 ≤ C,
where C is a constant independent of k, F+Ak ∈ L
1,2(g′k), and
‖F+Ak‖L1,2(g′k) ≤ C
where C ′ is a constant independent of k. Note that ‖ · ‖L1,2(g∞) ≤ 2‖ · ‖L1,2(g′k) for k ≫ 1
since g′k
C1,α−→ g∞ on Uj,r. Thus, by passing to a subsequence, F+Ak
L1,2−→ Ωj ∈ L1,2(g∞), a
self-dual 2-form with respect to g∞.
Lemma 5.3. For any j, Ωj is a smooth self-dual 2-form on Uj,r\∂Uj,r such that
∇∞Ωj ≡ 0, and |Ωj |∞ ≡ cont. 6= 0, where ∇∞ is the connection induced by the
Levi-civita connection of g∞. Hence, g∞ is a Kähler metric with Kähler form
|Ωj |
on Uj,r.
Proof. By Lemma 5.2
|∇∞Ωj |2∞dv∞ = lim
|∇∞F+Ak |
∞dv∞ ≤ 2 lim
|∇kF+Ak |
kdvk = 0.
It is easy to see that Ωj is a weak solution of the elliptic equation ∇∞Ωj = 0 on Uj,r.
By elliptic equation theory, Ωj is a smooth self-dual 2-form on Uj,r\∂Uj,r, ∇∞Ωj ≡ 0,
and |Ωj|∞ ≡ cont..
Now we claim that, for any j and r ≫ 1,
|Ωj |2∞dv∞ 6= 0. If not, there exist
js, s = 1, · · · , m0, m0 ≤ m, such that
Ujs,r
|Ωjs|2∞dv∞ ≡ 0. By Lemma 3.1, R∞ =
R(gk) = lim
R̆(gk) = λMV
2 , which is the scalar curvature of g∞, i.e. R∞ =
R(g∞). Note that, by (5.10) and Lemma 3.7,
|Ωj |2∞dv∞ = lim
|F+Ak |
R̆(gk)
2Volg′
(Uj,r)
∞Volg∞(Uj,r),
|F+Ak |
kdvk −
|F+Ak |
kdvk| ≤
R̆(gk)
2Volgk(M\
Fk,j,r(Uj,r))
Volg∞(Nj\Uj, r2 ),
and, by Lemma 3.1,
(R(gk)
2 − R2∞)dvk| ≤ 24 lim
(|R(gk)−R(gk)|+ |R∞ − R(gk)|)dvk = 0,
20 F. FANG, Y. ZHANG, AND Z. ZHANG
where C is a constant in-dependent of k. Hence, we obtain
j 6=j1,··· ,jm0
Volg∞(Uj,r) ≥
8|Ωj|2∞dv∞ = lim
8|F+Ak|
≥ lim
8|F+Ak |
kdvk − CR
Volg∞(Nj\Uj, r2 )
≥ 32π2c21(c)[M ]− CR
Volg∞(Nj\Uj, r2 ).
The last inequality is obtained by (5.7). Thus, by (5.1),
j 6=j1,··· ,jm0
Volg∞(Uj,r) ≥ 32π2(2χ(M) + 3τ(M))− CR
Volg∞(Nj\Uj, r2 ).
By the Chern-Gauss-Bonnet formula and the Hirzebruch signature theorem,
2χ(M) + 3τ(M) ≥ 1
R(gk)
2 + 2|W+(gk)|2k)dvk −
|Rico(gk)|2dvk.
By Lemma 3.1, and the fact that g′k
L2,p−→ g∞ on Uj,r, we obtain that
j 6=j1,··· ,jm0
Volg∞(Uj,r) ≥
+ 2|W+(g∞)|2∞)dv∞
−CR2∞
Volg∞(Nj\Uj, r2 ).
Note that, on any Uj,r, j 6= j1, · · · , jm0 , ∇∞Ωj ≡ 0, |Ωj|∞ ≡ cont. 6= 0, and Ωj
is a self-dual 2-form. Thus g∞ is a Kähler metric with Kähler form
|Ωj |
on Uj,r,
j 6= j1, · · · , jm0 . It is well known that R
∞ = 24|W+(g∞)|2∞ for Kähler metrics (cf.
[Le3]). Thus
j 6=j1,··· ,jm0
Volg∞(Uj,r) ≥ R
j 6=j1,··· ,jm0
Volg∞(Uj,r)− CR
Volg∞(Nj\Uj, r2 )
js=j1,··· ,jm0
+ 2|W+(g∞)|2∞)dv∞
≥ R2∞
j 6=j1,··· ,jm0
Volg∞(Uj,r)− CR
Volg∞(Nj\Uj, r2 )
js=j1,··· ,jm0
∞Volg∞(Ujs,r).
MAXIMUM SOLUTIONS OF NORMALIZED RICCI FLOWS ON 4-MANIFOLDS 21
Note that, for r ≫ 1,
1 ≫ 3CR2∞
Volg∞(Nj\Uj, r2 ) ≥
js=j1,··· ,jm0
∞Volg∞(Ujs,r).
A contradiction. Thus, for all j,
|Ωj |2∞dv∞ 6= 0, and∇∞Ωj ≡ 0, |Ωj |∞ ≡ cont. 6= 0.
Thus we obtain the conclusion.
Proof of Theorem 5.1. First, assume that diamg(tk)(M) −→ ∞, when k −→ ∞. By
Proposition 3.2 and Proposition 4.1, there exists a m ∈ N, and a sequence of points
{xj,k ∈ M}, k ∈ N, j = 1, · · · , m, satisfying that, by passing to a subsequence,
(M, g(tk+t), x1,k, · · · , xm,k), t ∈ [0,∞), converges to {(N1, g∞, x1,∞), · · · , (Nm, g∞, xm,∞)}
in them-pointed Gromov-Hausdorff sense, when k −→ ∞, where (Nj , g∞) j = 1, · · · , m
are complete Einstein 4-orbifolds with finite isolated orbifold points {qi}. The scalar
curvature of g∞ is
R∞ = lim
λM(g(t)), and V = Volg0(M) =
Volg∞(Nj).
By Lemma 5.2, g∞ is a Kähler-Einstein metric in the non-singular part of
Then by the same arguments as in Section 4 of [Ti], g∞ is actually a Kähler-Einstein
orbifold metric. Furthermore, in the non-singular part of
Nj , {g(tk+ t)}, t ∈ [0,∞),
C∞-converges to g∞ by Proposition 4.1.
If diamgk(M) < C for a constant C in-dependent of k, we can also obtain the
conclusion by the similar, but much easier, arguments as above.
Theorem 5.4. Let (M, c) be a smooth compact closed oriented 4-manifold with a Spinc-
structure c. Assume that the first Chern class c1(c) of c is a monopole class of M
satisfying c21(c)[M ] = 2χ(M) + 3τ(M) > 0, and χ(M) = 3τ(M). If M admits a
solution g(t), t ∈ [0,∞) to (1.3) with |R(g(t))| ≤ 12, then
λM(g(t)) = −
32π2c21(c)[M ].
Furthermore, if |Ric(g(t))| ≤ 3, the Kähler-Einstein metric g∞ in Theorem 5.1 is a
complex hyperbolic metric.
Proof. Let V = V olg(t)(M). By the Chern-Gauss-Bonnet formula and the Hirzebruch
signature theorem,
(5.11) 2χ(M)− 3τ(M) ≥ 1
R(g(t))2 + 2|W−(g(t))|2 − 1
|Rico(g(t))|2dvg(t),
where W− is the anti-self-dual Weyl tensor. Note that
(5.12)
R(g(t))2dvg(t) ≥ R(g(t))2V −→ R
∞V = lim
λM(g(t))
22 F. FANG, Y. ZHANG, AND Z. ZHANG
when t −→ ∞, by Schwarz inequality and Lemma 3.1. By (5.11), (5.12), Lemma 3.1
and Theorem 1.1 in [FZ],
2χ(M)− 3τ(M) ≥ lim inf
|W−(g(t))|2dvg(t) +
λM(g(t))
≥ lim inf
|W−(g(t))|2dvg(t) +
c21(c)[M ]
= lim inf
|W−(g(t))|2dvg(t) +
(2χ(M) + 3τ(M)).
Since χ(M) = 3τ(M), we obtain
λM(g(t)) = −
32π2c21(c)[M ],
lim inf
|W−(g(t))|2dvg(t) = 0.
Now, assume that |Ric(g(t))| ≤ 3. Let tk, Nj , gk, and g∞ be the same as above.
For any j and compact subset U of the regular part of Nj,
|W−(g∞)|2∞dv∞ ≤ lim inf
|W−(g(tk))|2kdvk = 0,
since g(tk)
L2,p−→ g∞ on U . Hence g∞ is a Kähler-Einstein metric with W−(g∞) ≡ 0.
This implies that g∞ is a complex hyperbolic metric (cf. [Le1]). The desired result
follows.
Proofs of Theorem 1.1 and Theorem 1.2. By the work of Taubes [Ta], if (M,ω) is a
compact symplectic manifold with b+2 (M) > 1, the spin
c-structure induced by ω is a
monopole class. Moreover, since in this situation c21(c)[M ] = 2χ(M)+3τ(M), Theorem
1.1 (resp. Theorem 1.2) is an obvious consequence of Theorem 5.1 (resp. Theorem
5.4). �
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Department of Mathematics, Capital Normal University, Beijing, P.R.China
E-mail address : ffang@nankai.edu.cn
Department of Mathematics, Capital Normal University, Beijing, P.R.China
Nankai Institute of Mathematics, Weijin Road 94, Tianjin 300071, P.R.China
http://arxiv.org/abs/math/0608439
http://arxiv.org/abs/math/0609254
http://arxiv.org/abs/math/0608504
http://arxiv.org/abs/math/0211159
http://arxiv.org/abs/math/0402238
	1. Introduction
	2. Preliminaries 
	2.1. Monopole class
	2.3. Kato's inequality
	2.5. Chern-Gauss-Bonnet formula and Hirzebruch signature formula
	2.6. Curvature estimates for 4-manifolds
	3.  The limiting behavior of Ricci flow 
	4. Smooth convergence on the regular part
	5. Proofs of Theorems 1.1 and 1.2
	References
