Long time existence and bounded scalar curvature in the Ricci-harmonic flow

LONG TIME EXISTENCE AND BOUNDED SCALAR CURVATURE IN THE RICCI-HARMONIC FLOW

YI LI

ABSTRACT. In this paper we study the long time existence of the Ricci-harmonic flow in terms of scalar curvature and Weyl tensor which extends Cao’s result [6]

in the Ricci flow. In dimension four, we also study the integral bound of the “Rie- mann curvature” for the Ricci-harmonic flow generalizing a recently result of Si- mon [38].

1. INTRODUCTION

In this paper we study the long time existence of the Ricci-harmonic flow (RHF) (1.1) ∂_tg

(

t

) = −

2Ric_g(t)

+

2α

(

t

)∇

_g(t)φ

(

t

) ⊗ ∇

_g(t)φ

(

t

)

, ∂_tφ

(

t

) =

∆_g(t)φ

(

t

)

. introduced in [25, 26, 30, 32], whereg

(

t

)

is a family of Riemannian metric,φ

(

t

)

is a family of functions, andt

∈ [

0,T

)

(withT

≤ +

_{∞). Here}α

(

t

)

is a time-dependent positive constant. In particular, we may choose α

(

t

) ≡

αa positive constant. If all functionsφ

(

t

) ≡

0, we obtain the Ricci flow (RF) introduced by Hamilton in his famous paper [23] and definitely used by Perelman [34, 35, 36] on his work about the Poincar´e conjecture. The flow equations (1.1) come from static Einstein vacuum equations arising in the general relativity, and also arise as dimensional reductions of RF in higher dimensions [28].

For the RF, Hamilton [23] showed that the short-time existence and (1.2) T

<

_∞

= ⇒

lim sup

t→T

maxM

|

Rm_g(t)

|

²_g(t)

=

_∞.

Later, Sesum [37] improved Hamilton’s result as (1.3) T

<

∞

= ⇒

lim sup

t→T

maxM

|

Ric_g(t)

|

²_g(t)

=

∞

by blow-up argument. For the integral bounds, Ye [47] and Wang [42] indepen- dently proved that

(1.4) T

<

_∞

= ⇒

_{Z T}

0 Z

M

|

Rm_g(t)

|

_g(t)^m+22dV_g(t)dt _m²₊₂

=

_∞.

2010Mathematics Subject Classification. Primary 53C44.

Key words and phrases. Ricci-harmonic flow, curvature pinching estimates, bounded scalar curvature.

Yi Li is partially sponsored by Shanghai Sailing Program (grant) No. 14YF1401400 and NSFC (grant) No. 11401374. Yi Li is also supported in part by the Fonds National de la Recherche Luxembourg (FNR) under the OPEN scheme (project GEOMREV O14/7628746).

1

Moreover, Wang [42] proved another version for RH that (1.5) Ric_g(t)

≥ −

C, T

<

_∞

= ⇒

_{Z T}

0 Z

M

|

R_g(t)

|

_g(t)^m+22dV_g(t)dt _m²₊₂

=

_∞.

HereCis a uniform constant. For other works on integral bounds, see for example [8, 29, 43, 46].

A well-known conjecture (see [6]) about the extension of RF states that (1.6) Is it true for lim sup

t→T

maxM R_g(t)

=

_∞?Here T

<

_∞.

This conjecture was settled for K¨ahler-Ricci flow by Zhang [48] and for type-I maximal solution of RF by Enders-M ¨uller-Topping [11]. Cao [6] proved the fol- lowing

(1.7)

T

<

∞

= ⇒

either lim sup_t→T

maxMR_g(t)

=

∞, or lim sup_t→T

maxMR_g(t)

<

∞but lim sup_t→T^|Wg_R^(t)|g(t)

g(t)

=

∞, whereW_g(t)denotes the Weyl tensor ofg

(

t

)

.

For 4DRF, Simon [38] and Bamler-Zhang [5] independently proved (1.8) T

<

∞,

|

R_g(t)

| ≤

C

= ⇒

Z

M

|

Ric_g(t)

|

²_g(t)dV_g(t), Z

M

|

Rm_g(t)

|

²_g(t)dV_g(t)

≤

C⁰ by different methods (for earlier work see [41]).

On the other hand, for the Ricci-harmonic flow (RHF) we have the following results. Whenm

=

dimM

≥

3 andT

< +

∞. M ¨uller [30, 32] showed that (1.2) is also true for RHF. Recently, Cheng and Zhu [9] extended Sesum’s result [37] to Ricci-harmonic flow, that is, (1.3) is true for RHF. For more results about the RHF, see [1, 2, 3, 4, 7, 9, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 30, 31, 32, 33, 39, 40, 44, 45, 49].

The first main result is an extension of Cao’s result [6] to RHF.

Theorem 1.1. Let

(

M,g

(

t

)

,φ

(

t

))

_t∈[0,T) be a solution to RHF (1.1) withα

(

t

) ≡

αa positive constant on a closed manifold M with m

=

_dimM

≥

3and T

< +

_{∞. Either} one has

lim sup

t→T

maxM R_g(t)

=

_∞ or

lim sup

t→T

maxM R_g(t)

<

∞but lim sup

t→T

maxM

|

W_g(t)

|

_g(t)

+ |∇

²_g(t)φ

(

t

)|

²_g(t) R_g(t)

!

=

∞.

Here W_g(t)is the Weyl part ofRm_g(t).

The second main result focuses on the 4DRHF. Write

Sic_g(t):

=

Ric_g(t)

−

α

∇

_g(t)φ

(

t

) ⊗ ∇

_g(t)φ

(

t

)

, S_g(t):

=

R_g(t)

−

α

|∇

_g(t)φ

(

t

)|

²_g(t). According to Theorem 2.2 below we can find a uniform constantCsuch thatS_g(t)

+

C

>

0 for allt

∈ [

0,T

)

.

Theorem 1.2. Let

(

M,g

(

t

)

,φ

(

t

))

_t∈[0,T)be a solution to RHF (1.1) on a closed manifold M with m

=

dimM

=

4, T

≤ +

_∞,α

(

t

) ≡

αa positive constant. Choose a uniform constant C in Theorem 2.2 such that S_g(t)

+

C

>

0. Then

Z

M

|

Sic_g(s)

|

_g(s)dV_g(s)

≤

2c0

(

M,g

(

0

)

,φ

(

0

)

,s

) +

^C

2Vol

(

M,g

(

s

))

(1.9)

+

1148e^36Cs Z _s

0 Z

MS²_g(t)dV_g(t)dt, Z _s

0 Z

M

|

Sic_g(t)

|

²_g(t)dV_g(t)dt

≤

8c0

(

M,g

(

0

)

,φ

(

0

)

,s

) +

^C

2

4 Z _s

0 Vol

(

M,g

(

t

))

dt

+

4592e^36Cs

Z _s 0

Z

MS²_g(t)dV_g(t)dt, (1.10)

for all s

∈ [

0,T

)

. Here, A1

=

maxM

|∇

_g(0)φ

(

0

)|

²_g(0)and c0

(

M,g

(

0

)

,φ

(

0

)

,s

)

=

^256π

2χ

(

M

)

36C

e^36Cs

−

1

+

^104α

2A²₁Vol

(

M,g

(

0

))

35C

e^35Cs

−

e^Cs (1.11)

+

e^37CsαA₁Vol

(

M,g

(

0

)) +

e^36Cs Z

M

|

Sic_g(0)

|

²_g(0) S_g(0)

+

C dV_g(0).

According to Theorem 2.3 below and following [38], we consider the basic as- sumption (BA) for a solution

(

M,g

(

t

)

,φ

(

t

))

_t∈[0,T)to RHF:

(a) Mis a connected and closed 4-dimensional smooth manifold,

(b)

(

M,g

(

t

)

,φ

(

t

))

_t∈[0,T)is a solution to RHF withα

(

t

) ≡

αa positive constant, (c) T

< +

∞,

(d) maxM×[0,T)

|

S_g(t)

| ≤

1.

The upper bound 1 in condition (d) is not essential, since we can rescale the pair

(

g

(

t

)

,φ

(

t

))

so that the condition (d) is always satisfied. Furthermore, since

|∇

_g(t)φ

(

t

)|

²_g(t)

≤

A₁

(by (3.6)) it follows that condition (d) is equivalent to the uniform bound forR_g(t). Theorem 1.3. If

(

M,g

(

t

)

,φ

(

t

))

_t∈[0,T)satisfiesBA, then

Z

M

|

Sic_g(s)

|

²_g(s)dV_g(s)

≤

b

(

M,g

(

0

)

,φ

(

0

)

,s

)

, (1.12)

Z _s 0

Z

M

|

Sic_g(t)

|

⁴_g(t)dV_g(t)dt

≤

b

(

M,g

(

₀

)

_,φ

(

₀

)

_,s

)

_, (1.13)

for any s

∈ [

0,T

)

. Here

b

(

M,g

(

0

)

,φ

(

0

)

,s

)

:

=

9e^88s Z

M

|

Sic_g(0)

|

²_g(0)dV_g(0)

+

¹¹⁵²

88 π²χ

(

M

)

e^88s

−

1

+

⁴⁶⁸

86

(

αA₁

)

²_Vol

(

M,g

(

₀

))

e^88s

−

e^2s (1.14)

+

9

(

αA1

)

Vol

(

M,g

(

0

))

e^90s.

Define

c

(

M,g

(

0

)

,φ

(

0

)

,T

)

:

=

9e^90T _Z

M

|

Sic_g(0)

|

²_g(0)dV_g(0)

+

π²

|

χ

(

M

)|

+ [(

αA1

)

²

+ (

αA1

)]

Vol

(

M,g

(

0

))

. (1.15)

Then

|

b

(

M,g

(

0

)

,φ

(

0

)

,T

)| ≤

c

(

M,g

(

0

)

,φ

(

0

)

,T

)

. Theorem 1.3 now yields Theorem 1.4. If

(

M,g

(

t

)

,φ

(

t

))

_t∈[0,T)satisfiesBA, then

sup

t∈[0,T) Z

M

|

Sic_g(t)

|

²_g(t)dV_g(t)

≤

c

(

M,g

(

0

)

,φ

(

0

)

,T

) < +

∞, (1.16)

sup

t∈[0,T) Z

M

|

Sm_g(t)

|

²_g(t)dV_g(t)

≤

32π²χ

(

M

) +

8c

(

M,g

(

₀

)

_,φ

(

₀

)

_,T

)

(1.17)

+

₁₃

(

αA₁

)

²_Vol

(

M,g

(

₀

))

e^2T

< +

_∞.

Remark1.5. For a time-dependent tensor field, we always omit time variable in its components. Although in Theorem 1.2–Theorem 1.4, we assume thatα

(

t

) ≡

αis a positive constant, the same results also hold forα

(

t

) >

0 and ˙α

(

t

) <

0 (see Section 4).

The proof of Theorem 1.1 is based on a “curvature pinching estimate” for RHF (see Theorem 2.2). The new ingredient in the proof of Theorem 2.2 is an intro- duction of “Riemann curvature tensor” Sm_g(t)for RHF, so that we can express the Weyl tensorW_g(t)in terms of Sm_g(t).

The proofs of Theorem 1.2–Theorem 1.4 follow from the method of Simon [38].

As in [38] we define Z_ijk:

=

∇

_iS_jk

(

S_g(t)

+

C

) −

S_jk

∇

_iS_g(t)

, Z_g(t):

= (

Z_ijk

)

, f :

=

|

Sic_g(t)

|

²_g(t) S_g(t)

+

C . Analogous to [38], we can show that

d dt

Z

Mf dV_g(t)

=

Z

M

−

2

|

Z

|

²

(

S

+

C

)

³

−

2f²

+

4Sm

(

Sic, Sic

)

S

+

C

−

f S

−

2α ∆φ Sic

S

+

C

− ∇

²φ

2

+

2α

|∇

²φ

|

²

dV_g(t).

The main difference is the last term on the right-hand side of the above equation.

To control the integral of

|∇

²φ

|

² we make use the evolution equation for

|∇

φ

|

² (see (3.6)) so that

2 Z _t

0 Z

M

|∇

²φ

|

²dV_g(s)ds

+

Z

M

|∇

φ

|

²dV_g(t)

≤

e^CtA1Vol

(

M,g

(

0

))

.

The above estimate not only controls the space-time integral of

|∇

²φ

|

², but also an uniform bound for the integral of

|∇

φ

|

². These two estimates play essential role in the following proof. In dimension four, the famous Gauss-Bonnet-Chern formula (3.10) should transform to (3.13), where the terms involving

|∇

φ

|

²can be bounded by the above discussion. A modification of [38] is now applied to the RHF.

2. CURVATURE PINCHING ESTIMATE FORRHF

Consider a solution

(

M,g

(

t

)

,φ

(

t

))

_t∈[0,T)to RHF with coupling time-dependent constantα

(

t

)

(2.1) ∂tg

(

t

) = −

2Ric_g(t)

+

_2α

(

t

)∇

_g(t)φ

(

t

) ⊗ ∇

_g(t)φ

(

t

)

_, ∂tφ

(

t

) =

_∆g(t)φ

(

t

)

_. Let

g(t):

=

∂_t

−

∆_g(t). As in [25, 26, 30, 32] we define the following notions

Sic_g(t) :

=

Ric_g(t)

−

α

(

t

)∇

_g(t)φ

(

t

) ⊗ ∇

_g(t)φ

(

t

)

, (2.2)

S_g(t) :

=

tr_g(t)Sic_g(t)

=

R_g(t)

−

α

(

t

)

∇

_g(t)φ

(

t

)

²

g(t). (2.3)

Hereα

(

t

)

is a family of time-dependent functions. Motivated by RF, we introduce a “Riemann curvature” type for RHF

(2.4) Sijk`:

=

Rijk`

−

^α 2

gj`

∇

_iφ

∇

_kφ

+

gk`

∇

_iφ

∇

_jφ

. Our notation forSijk`implies that

S_ij:

=

g<sup>k`</sup>S<sub>ik`j</sub>

=

R_ij

−

α

∇

_iφ

∇

_jφ

=

g<sup>k`</sup>S<sub>kij`</sub>

which coincides with the components of Sic_g(t). The corresponding tensor field for S_ijk`is denoted by Sm_g(t).

Lemma 2.1. For a solution

(

_M,g

(

_t

)

_,φ

(

_t

))

_t∈[0,T)with a time-dependent coupling func- tionα

(

t

)

, we have

g(t)S_g(t)

=

2

|

Sic_g(t)

|

²_g(t)

+

2α

(

t

)

∆_g(t)φ

(

t

)

²

g(t)

−

α˙

(

t

)

∇

_g(t)φ

(

t

)

²

g(t), (2.5)

g(t)Sic_g(t)

=

2Sm_g(t)

(

Sic_g(t),

·) −

2Sic²_g(t)

+

2α

(

t

)

_∆g(t)φ

(

t

)∇

²_g(t)φ

(

t

) −

α˙

(

t

)∇

_g(t)φ

(

t

) ⊗ ∇

_g(t)φ

(

t

)

(2.6)

whereSic²_g(t)

= (

SikSj`g<sup>k`</sup>

)

_ijandSm_g(t)

(

Sic_g(t),

·) = (

Skij`S<sup>k`</sup>

)

_ij.

Proof. The first equation can be found in [32], Corollary 4.5. In the same corollary we also have

∂tS_ij

=

_∆g(t),LS_ij

+

_2α∆g(t)φ

(

t

)∇

_i

∇

_jφ

−

α˙

∇

_iφ

∇

_jφ.

Here∆_g(t),Ldenotes the Lichnerowicz Laplacian with respect tog

(

t

)

defined by

∆g(t),LS_ij

=

_∆g(t)S_ij

+

2R<sub>kij`</sub>S<sup>k`</sup>

−

R_ikS_j^k

−

R_jkS_i^k. Then

g(t)S_ij

=

2R<sub>kij`</sub>S<sup>k`</sup>

−

R_ikS_j^k

−

R_jkS_i^k

+

_2α∆g(t)φ

(

t

)∇

_i

∇

_jφ

−

α˙

∇

_iφ

∇

_jφ.

PluggingSij

=

Rij

−

α

∇

_iφ

∇

_jφinto the above equation yields the second desired

equation.

As a corollary of Lemma 2.1 we have (see [32], Corollary 5.2)

(2.7) min

M S_g(t)

≥

min

M S_g(0), providedα

(

t

) ≥

0 and ˙α

(

t

) ≤

0.

Theorem 2.2. (Curvature pinching estimate)Let

(

M,g

(

t

)

,φ

(

t

))

_t∈[0,T)be a solution to RHF on a closed manifold M with m

=

dimM

≥

3, T

≤ +

_∞,α

(

t

) ≥

0andα˙

(

t

) ≤

0.

There exist uniform constants C₁,C2,C, depending only on m,g

(

0

)

,φ

(

0

)

,α

(

0

)

such that (2.8) S_g(t)

+

C

>

0,

|

Sin_g(t)

|

_g(t)

S_g(t)

+

C

≤

C1

+

C2 max

M×[0,t]

v u u t

|

W_g(s)

|

_g(s)

+ |∇

²_g(s)φ

(

s

)|

²_g(s) S_g(s)

+

C

whereSin_g(t)

=

Sic_g(t)

−

^S_m^g(t)g

(

t

)

is the trace-free part ofSic_g(t)and W_g(t)is the Weyl tensor field of g

(

t

)

.

Proof. The first inequality follows from (2.7). Let f :

= |

Sin_g(t)

|

²_g(t)

(

S_g(t)

+

C

)

^{γ, γ}

>

_0.

Since Sin_g(t)

= (

Sic_g(t)

+

_m^Cg

(

t

)) − (

^Sg(t_m^)+C

)

g

(

t

)

, it follows that f

=

|

Sic_g(t)

+

^C_mg

(

t

)|

²_g(t)

(

S_g(t)

+

C

)

^γ

−

¹

m

(

S_g(t)

+

C

)

^2−γ. In the following we always omit the subscriptstandg

(

t

)

and set

Sic⁰_g(t):

=

Sic_g(t)

+

^C

mg

(

t

)

, S⁰_g(t):

=

S_g(t)

+

C

=

tr_g(t)Sic⁰_g(t).

In the following we always omit the subscripts g

(

t

)

and t. Using the equation (3.21) in [10] we have

|

Sic⁰

|

²

(

S⁰

)

^γ

=

¹

(

S⁰

)

^γ

|

Sic⁰

|

²

−

γ

|

Sic⁰

|

²

(

S⁰

)

^γ+1

S

−

γ

(

γ

+

1

) |

Sic⁰

|

²

(

S⁰

)

^γ+2

|∇

S⁰

|

²

+

^2γ

(

S⁰

)

^γ+1

h∇|

Sic⁰

|

²,

∇

S⁰

i

. It is clear from (2.6) that

|

Sic

|

²

= −

2

|∇

Sic

|

²

+

4Sm

(

Sic, Sic

) +

2

Sic, 2α∆φ

∇

²φ

−

α˙

∇

φ

⊗ ∇

φ

(

t

)

. Therefore

|

Sic⁰

|

²

=

|

Sic

|

²

+

^C

2

m

+

^2C mS

= |

Sic

|

²

+

^2C m

h2

|

Sic

|

²

+

2α

|

_∆φ

|

²

−

α˙

|∇

φ

|

²ⁱ

= −

2

|∇

Sic⁰

|

²

+

4Sm

(

Sic, Sic

) +

^4C m

|

Sic

|

²

+

2D

Sic⁰, 2α∆φ

∇

²_φ

−

α˙

∇

φ

⊗ ∇

φ E

= −

2

|∇

Sic⁰

|

²

+

4Sm

(

Sic⁰, Sic⁰

) −

^4C

m

|

Sic⁰

|

²

+

^4C

2

m²S⁰

+

2

h

Sic⁰, 2α∆φ

∇

²φ

−

α˙

∇

φ

⊗ ∇

φ

i

and

|

Sic⁰

|

²

(

S⁰

)

^γ

= −

²

(

S⁰

)

^γ

|∇

Sic⁰

|

²

−

^2γ

|

Sic⁰

|

⁴

(

S⁰

)

^γ+1

+

⁴

(

S⁰

)

^γSm

(

Sic⁰, Sic⁰

)

−

γ

(

γ

+

1

) |

Sic⁰

|

²

|∇

S

|

²

(

S⁰

)

^γ+2

+

^2γ

(

S⁰

)

^γ+1

h∇|

Sic⁰

|

²,

∇

S⁰

i +

^4C

2

m² S⁰

(

S⁰

)

^γ

−

^4C m

(

1

−

γ

)

S⁰

+

¹₂γC

(

S⁰

)

^γ+1

|

Sic⁰

|

²

+

²

(

S⁰

)

^γ

h

Sic⁰, 2α∆φ

∇

²φ

−

α˙

∇

φ

⊗ ∇

φ

i

−

^γ

|

Sic⁰

|

²

(

S⁰

)

^γ+1

h2α

|

_∆φ

|

²

−

α˙

|∇

φ

|

²ⁱ, according to

S⁰

=

2

|

Sic⁰

|

²

−

^4C

mS⁰

+

^2C

2

m

+

2α

|

_∆φ

|

²

−

α˙

|∇

φ

|

². Using the identity

∇ |

Sic⁰

|

²

(

S⁰

)

^γ,

∇

S⁰

=

¹

(

S⁰

)

^γ

h∇|

Sic⁰

|

²,

∇

S⁰

i −

^γ

(

S⁰

)

^γ+1

|∇

S

|

²

|

Sic⁰

|

², we arrive at

|

Sic⁰

|

²

(

S⁰

)

^γ

=

^2γ S⁰

∇ |

Sic⁰

|

²

(

S⁰

)

^γ,

∇

S⁰

−

²

(

S⁰

)

^γ+2

|

S⁰

∇

Sic⁰

|

²

+

^γ

(

γ

−

1

)

(

S⁰

)

^γ+2

|

Sic⁰

|

²

|∇

S⁰

|

²

+

⁴

(

S⁰

)

^γSm

(

Sic⁰, Sic⁰

) −

^2γ

(

_S⁰

)

^γ+1

|

Sic⁰

|

⁴

+

^4C

2

m² S⁰

(

_S⁰

)

^γ

−

^4C m

(

1

−

γ

)

S⁰

+

¹₂γC

(

s⁰

)

^γ+1

|

Sic⁰

|

²

+

²

(

S⁰

)

^γ

h

Sic⁰, 2α∆φ

∇

²φ

−

α˙

∇

φ

⊗ ∇

φ

i

−

^γ

|

Sic⁰

|

²

(

S⁰

)

^γ+1 h

2α

|

∆φ

|

²

−

α˙

|∇

φ

|

²ⁱ. On the other hand, the following two identities

|

S⁰

∇

Sic⁰

|

²

= |

S⁰

∇

_iS⁰_jk

−

S⁰_jk

∇

_iS⁰

+

S⁰_jk

∇

_iS⁰

|

²

= |

S⁰

∇

_iS⁰_jk

−

S⁰_jk

∇

_iS⁰

|

²

− |

Sic⁰

|

²

|∇

S⁰

|

²

+

S⁰

h∇|

Sic⁰

|

²,

∇

S⁰

i

,

h∇|

Sic⁰

|

²,

∇

S⁰

i =

∇

(

S⁰

)

^γ

|

Sic⁰

|

²

(

S⁰

)

^γ

,

∇

S⁰

=

^γ

S⁰

|

Sic⁰

|

²

|∇

S⁰

|

²

+ (

S⁰

)

^γ

∇ |

Sic⁰

|

²

(

S⁰

)

^γ,

∇

S⁰

implies

|

Sic⁰

|

²

(

S⁰

)

^γ

=

²

(

γ

−

1

)

S⁰

∇ |

Sic⁰

|

²

(

S⁰

)

^γ,

∇

S⁰

−

²

(

S⁰

)

^γ+2

S⁰

∇

_iS⁰_jk

−

S⁰_jk

∇

_iS⁰

2

− (

2

−

γ

)(

γ

−

1

)

(

S⁰

)

^γ+2

|

Sic⁰

|

²

|∇

S⁰

|

²

+

⁴

(

S⁰

)

^γSm

(

Sic⁰, Sic⁰

) −

^2γ

|

Sic⁰

|

⁴

(

S⁰

)

^γ+1

+

^4C

2

m² S⁰

(

S⁰

)

^γ

−

^4C m

(

1

−

γ

)

S⁰

+

¹₂γC

(

S⁰

)

^γ+1

|

Sic⁰

|

²

+

²

(

S⁰

)

^γ

h

Sic⁰, 2α∆φ

∇

²φ

−

α˙

∇

φ

⊗ ∇

φ

i −

^γ

|

Sic⁰

|

²

(

S⁰

)

^γ+1

h2α

|

_∆φ

|

²

−

α˙

|∇

φ

|

²ⁱ. It is clear that

(

S⁰

)

^2−γ

= (

2

−

γ

)(

S⁰

)

^1−γ

S⁰

− (

2

−

γ

)(

1

−

γ

)(

S⁰

)

^−γ

|∇

S⁰

|

² and

|

Sic

|

²

= |

Sic⁰

|

²

+

^C_m²

−

^2CS_m⁰. These yield

(

S⁰

)

^2−γ

=

²

(

γ

−

1

)

S⁰

D

∇(

S⁰

)

^2−γ_,

∇

S⁰E

+

₂

(

₂

−

γ

)(

S⁰

)

^1−γ

|

Sic⁰

|

²

− (

2

−

γ

)(

γ

−

1

)

(

S⁰

)

²

(

S⁰

)

^2−γ

|∇

S⁰

|

²

+ (

2

−

γ

)(

S⁰

)

^1−γ 2C²

m

−

^4C mS⁰

+ (

2

−

γ

)(

S⁰

)

^1−γ2α

|

∆φ

|

²

−

α˙

|∇

φ

|

². Then

f

=

²

(

γ

−

1

)

S⁰

h∇

f,

∇

S⁰

i −

²

(

S⁰

)

^γ+2

|

S⁰

∇

_iS⁰_jk

−

S⁰_jk

∇

_iS⁰

|

²

− (

2

−

γ

)(

γ

−

1

)

(

S⁰

)

²

|∇

S⁰

|

²f

+

_Q1

+

_Q2

+

_Q3 (2.9)

where

Q1 :

= −

²

(

2

−

γ

)

m

(

S⁰

)

^1−γ

|

Sic⁰

|

²

+

⁴

(

S⁰

)

^γSm

(

Sic⁰, Sic⁰

) −

^2γ

|

Sic⁰

|

⁴

(

S⁰

)

^{1+γ ,} (2.10)

Q2 :

=

^4C m

"

CS⁰

m

(

S⁰

)

²

− (

1

−

γ

)

S⁰

+

¹₂γC

(

S⁰

)

^γ+1

|

Sic⁰

|

²

−

²

−

γ 2m

C

−

2S⁰

(

S⁰

)

^γ−1

# (2.11)

Q3 :

=

²

(

S⁰

)

^γ

h

Sic⁰, 2α∆φ

∇

²φ

−

α˙

∇

φ

⊗ ∇

φ

i

−

^2α

|

_∆φ

|

²

−

α˙

|∇

φ

|

²

(

S⁰

)

^γ+1

γ

|

Sic⁰

|

²

+

²

−

γ m

(

S⁰

)

²

. (2.12)

Observe that

(2.13) Q2

=

^4C m²

C

S⁰

+

^C

(

S⁰

)

^γ−1

+

¹

(

S⁰

)

^γ−2

+

m f

γ

−

1

−

^γC sS⁰

. TheQ1terms can be written as

Q1

=

²

(

S⁰

)

^γ+1

γ

−

2

m

|

Sic⁰

|

²

(

s⁰

)

²

+

2S⁰Sm

(

Sic⁰, Sic⁰

) −

γ

|

Sic⁰

|

⁴

=

²

(

S⁰

)

^γ+1

(

2

−

γ

)|

Sic⁰

|

²

|

Sin

|

²

−

2

|

Sic⁰

|

⁴

−

S⁰Sm

(

Sic⁰, Sic⁰

)

, (2.14)

where Sin

=

_Sic

−

_m^Sg

=

_Sic⁰

−

_m^Cg

−

_m^Sg

=

_Sic⁰

−

^S_m⁰g

=

_{: Sin}⁰. Recall the decom- position of Rm that

R_ijk`

=

¹

m

−

2

(

R_i`g_jk

+

R_jkg_i`

−

R_ikg_j`

−

R_j`g_ik

) −

^R

(

g_i`g_jk

−

g_ikg_j`

)

(

m

−

1

)(

m

−

2

) +

W_ijk`

whereW_{ijk`</sub>stands for the Weyl tensor field. Then S<sub>ijk`}

=

W_ijk`

+

¹

m

−

2

(

S⁰_i`g_jk

+

S⁰_jkg_i`

−

S⁰_ikg_j`

−

S⁰_j`g_ik

) −

^S

0

(

gi`gjk

−

gikgj`

) (

m

−

2

)(

m

−

1

)

+

^C

m

(

m

−

1

)(

m

−

2

) (

gi`gjk

−

gikgj`

) −

^C

m

(

m

−

2

) (

gjkgi`

−

gj`gik

)

+

^α

m

−

2

(

g_jk

∇

_iφ

∇

_`φ

+

g_i`

∇

_jφ

∇

_kφ

−

g_j`

∇

_iφ

∇

_kφ

−

g_ik

∇

_jφ

∇

_`φ

)

(2.15)

−

^α

|∇

φ

|

²

(

m

−

2

)(

m

−

1

) (

g_i`g_jk

−

g_ikg_j`

) −

^α

2

(

g_j`

∇

_iφ

∇

_kφ

+

g_k`

∇

_iφ

∇

_jφ

)

. Together with (2.15), we get

Sm

(

Sic⁰, Sic⁰

) =

¹ m

−

2

2m

−

1

m

−

1 S⁰

|

Sic⁰

|

²

−

2Sic⁰³

−

^S

03

m

−

1

+

W

(

Sic⁰, Sic⁰

)

−

¹ m

−

1

C m

+

^α

m

−

2

|∇

φ

|

²

|

S⁰

|

²

− |

Sic⁰

|

² (2.16)

+

^2α m

−

2

DS⁰Sic⁰

−

^m

2Sic⁰²,

∇

φ

⊗ ∇

φ E

where Sic⁰³

=

S⁰_ijS^0j_kS^0ki. Substituting (2.16) into (2.14) we arrive at Q1

=

²

(

S⁰

)

^γ+1

(

₂

−

γ

)|

Sic⁰

|

²

|

Sin

|

²

−

2Q4

+

2S⁰W

(

_Sic⁰_{, Sic}⁰

)

−

² m

−

1

C m

+

^α

m

−

2

|∇

φ

|

²

S⁰³

−

S⁰

|

Sic⁰

|

² (2.17)

+

^2α m

−

2

D

S⁰²Sic⁰

−

^m

2S⁰Sic⁰²,

∇

φ

⊗ ∇

φ E

where

(2.18) Q4

= |

Sic⁰

|

⁴

−

^S

0

m

−

2

2m

−

1

m

−

1 S⁰

|

Sic⁰

|

²

−

2Sic⁰³

−

^S

03

m

−

1

. The first term

(

2

−

γ

)|

Sic⁰

|

²

|

Sin

|

²

−

_2Q4

+

2S⁰W

(

Sic⁰, Sic⁰

)

on the right-hand side of (2.17) can be written as

−

γ

|

Sin

|

⁴

+

2

(

m²

+

2m

−

2

)

m

(

m

−

1

)(

m

−

2

) −

^γ

m

S⁰²

|

Sin

|

²

+

⁴

m²

(

m

−

2

)

^S

04

−

⁴

m

−

2S⁰Sic⁰³

+

2S⁰W

(

Sic⁰, Sic⁰

)

by using the fact that

|

Sic⁰

|

²

= |

Sin

|

²

+

^S_m⁰², where Sic⁰³is equals to

Sic⁰³

=

_Sin³

+

^3S

0

m

|

Sin

|

²

+

^S

03

m²,

and therefore

−

γ

|

Sin

|

⁴

+

2

(

m

−

2

)

m

(

m

−

1

) −

^γ

m

S⁰²

|

Sin

|

²

−

⁴

m

−

2S⁰Sin³

+

2S⁰W

(

Sin, Sin

)

, because ofW

(

Sic⁰, Sic⁰

) =

W

(

Sin, Sin

)

. Finally, we obtain

f

=

2

(

γ

−

1

)h∇

f,

∇

lnS⁰

i −

²

(

S⁰

)

^γ

|∇

_iS⁰_jk

−

S⁰_jk

∇

_ilnS⁰

|

²

− (

2

−

γ

)(

γ

−

1

)|∇

lnS⁰

|

²f

+

_Q1

+

_Q2

+

_Q3 (2.19)

where

Q1

=

²

(

S⁰

)

^γ+1

−

γ

(

S⁰

)

^2γf²

+

2m

−

4 n

(

m

−

1

) −

^γ

m

(

S⁰

)

^γ+2f

−

^4S

04

m

−

2 sin³

S⁰³

+

₂

(

S⁰

)

³W

Sin S⁰ ,Sin

S⁰

−

² m

−

1

C

m

+

^α

|∇

φ

|

² m

−

2

m

−

1

m S⁰³

− (

S⁰

)

^γ+1f

+

^2α m

−

2

DS⁰²Sic⁰

−

^m

2S⁰Sic⁰²,

∇

φ

⊗ ∇

φ E

, Q2

=

^4C

m² C

S⁰

+

^C

(

S⁰

)

^γ+1

+

¹

(

S⁰

)

^γ−2

+

m f

γ

−

1

−

^γC 2S⁰ , Q3

=

²

(

S⁰

)

^γ

h

Sic⁰, 2α∆φ

∇

²φ

−

α˙

∇

φ

⊗ ∇

φ

i

−

^2α

|

_∆φ

|

²

−

α˙

|∇

φ

|

²

(

S⁰

)

^γ+1

γ

(

S⁰

)

^γf

+

^2S

02

m

. from (2.9)–(2.13) and (2.17)–(2.18).

In particular, whenγ

=

2, we have f

= |

Sin

|

²/S⁰²

= |

Sic⁰

|

²/S⁰²and (2.20)

f

=

2

h∇

f,

∇

lnS⁰

i −

2

∇

Sin

S⁰

2

+

Q1

+

Q2

+

Q3

where

Q1

=

4S⁰

−

f²

−

^f

m

(

m

−

1

) −

² m

−

2

Sin³ S⁰³

+

¹

S⁰W Sin

S⁰ ,Sin S⁰

−

¹ S⁰

C

m

+

^α

|∇

φ

|

² m

−

2

1 m

−

^f

m

−

1

+

¹ S⁰

α m

−

2

*Sin²

S⁰² ,

∇

φ

⊗ ∇

φ +

+

¹ S⁰

α

|∇

φ

|

² 2m

(

m

−

2

)

, Q2

=

^4C

m² C

S⁰

+

^C S⁰³

+

m f

1

−

^C

S⁰

, Q3

=

²

S⁰

h

_{Sin, 2α∆φ}

∇

²φ

−

α˙

∇

φ

⊗ ∇

φ

i −

2α

|

_∆φ

|

²

−

α˙

|∇

φ

|

²f

.

Sinceα

(

t

) ≥

0 and ˙α

(

t

) ≤

0, it follows from [32], Proposition 5.5, that

|∇

φ

|

²is bounded from above by a uniform constant. Consequently

Q1

≤

4S⁰

−

f²

−

^f

m

(

m

−

1

) +

²

m

−

2f^3/2

+

Cm

|

W

|

S⁰ f

+

C0f

, Q2

≤

4S⁰

(

C₀

+

C₀f

)

_,

Q3

≤

4S⁰

C0

+

C0

|∇

²φ

|

² S⁰ f^1/2

where Cm

=

C

(

m

) >

0 and C0

=

C

(

m,g

(

0

)

,φ

(

0

)

,α

(

0

)) >

0. Without loss of generality, we may assume thatf

≥

1. In this case, we have

f

≤

2

h∇

f,

∇

lnS⁰

i +

4S⁰f

−

f

−

¹

m

(

m

−

1

) +

² m

−

2f^1/2

+

C₀

+

C₀

|

W

| + |∇

²φ

|

²

S⁰ (2.21)

Applying the maximum principle to (2.21) yields f

−

²

m

−

2f^1/2

+

¹

m

(

m

−

1

) −

C₀

−

C₀

|

W

| + |∇

²φ

|

² S⁰

≤

0

at the point where f achieves its maximum. Thus we obtain (2.8).

As an immediate consequently of Theorem 2.2 we obtain the following theorem that is an extension of Cao’s result ([6], Corollary 3.1).

Theorem 2.3. Let

(

M,g

(

t

)

,φ

(

t

))

_t∈[0,T)be a solution to RHF with nonincreasingα

(

t

) ∈ [

α,α

]

,0

<

α

≤

α

< +

∞, on a closed manifold M with m

=

dimM

≥

3and T

< +

∞.

Either one has

lim sup

t→T

maxM R_g(t)

=

_∞ or

lim sup

t→T

maxM R_g(t)

<

∞but lim sup

t→T

maxM

|

W_g(t)

|

_g(t)

+ |∇

²_g(t)φ

(

t

)|

²_g(t) R_g(t)

!

=

∞.

Proof. Suppose now that lim sup

t→T

maxM R_g(t)

<

_∞and lim sup

t→T

maxM

|

W_g(t)

|

_g(t)

+ |∇

²_g(t)φ

(

t

)|

²_g(t) R_g(t)

!

<

_∞.

In this case bothR_g(t)and

|

W_g(t)

|

_g(t)

+ |∇

²_g(t)φ

(

t

)|

²_g(t)are uniformly bounded. The- orem 2.2 then implies that

|

Sin_g(t)

|

_g(t) is uniformly bounded. Since Sin_g(t)

=

Sic_g(t)

−

^Sg_m^(t)g

(

t

)

, it follows that Sic_g(t)is uniformly bounded. However, the as- sumption onα

(

t

)

tells us that

|∇

_g(t)φ

(

t

)|

²_g(t)is uniformly bounded (e.g., [32], Propo- sition 5.5). Thus Ric_g(t) is uniformly bounded, contradicting with the fact (1.2).

Therefore we prove the theorem.

3. 4D RICCI-HARMONIC FLOW WITH BOUNDEDS_g(t): I

Let the constantCbe given in Theorem 2.2 and we assume thatαis a positive constant so that ˙α

(

t

) ≡

0, andm

=

dimM

=

4. As in [38] we define

(3.1) Z_ijk:

=

∇

_iS_jk

(

S_g(t)

+

C

) −

S_jk

∇

_iS_g(t)

, Z_g(t):

= (

Z_ijk

)

. In the proof of Theorem 2.2, we actually have proved

|

Sic

|

²

S

+

C

= −

2

|

Z

|

²

(

S

+

C

)

³

−

2

|

Sic

|

⁴

(

S

+

C

)

²

+

4Sm

(

Sic, Sic

)

S

+

C

−

¹

(

S

+

C

)

²

2α

|

∆φ

|

²

−

α˙

|∇

φ

|

²

|

Sic

|

² (3.2)

−

2

(

S

+

C

)

^D_{Sic, 2α∆φ}

∇

²φ

−

α˙

∇

φ

⊗ ∇

φ E

. The bracket in the right-hand side of (3.2) can be expressed as

2α

Sic∆φ

− (

S

+

C

)∇

²φ

2

− (

S

+

C

)

²

|∇

²φ

|

²

because of ˙α

≡

0. Therefore the identity (3.2) is equal to

f

= −

2

|

Z

|

²

(

S

+

C

)

³

−

2f²

+

₄^Sm

(

Sic, Sic

)

S

+

C

−

2α ∆φ Sic

S

+

C

− ∇

²φ

2

+

2α

|∇

²φ

|

², (3.3)

where

(3.4) f :

= |

Sic

|

²

S

+

C

which differs from the previous one in the proof of Theorem 2.2. Integrating (3.3) overMyields

d dt

Z

Mf dV_g(t)

=

Z

M

−

2

|

Z

|

²

(

S

+

C

)

³

−

2f²

+

4Sm

(

Sic, Sic

)

S

+

C

−

f S

−

2α ∆φ Sic

S

+

C

− ∇

²φ

2

+

2α

|∇

²φ

|

²

dV_g(t). (3.5)

To control the integral of

|∇

²φ

|

²we recall the evolution equation for

|∇

φ

|

²(see [32], Proposition 4.3):

(3.6)

|∇

φ

|

²

= −

2α

|∇

φ

⊗ ∇

φ

|

²

−

2

|∇

²φ

|

². In particular, we see that

(3.7)

|∇

φ

|

²

≤

max

M

|∇

φ

|

² t=0

=

:A1. Moreover we have

d dt

Z

M

|∇

φ

|

²dV_g(t)

≤

Z

M

h

−(

S

+

C

)|∇

φ

|

²

−

2

|∇

²φ

|

²

+

C

|∇

φ

|

²ⁱdV_g(t)