Generalized Ricci flow I: Higher-derivative estimates for compact manifolds

A ^NALYSIS & PDE

msp

Volume 5 No. 4 2012

Y

L

GENERALIZED RICCI FLOW

I: HIGHER-DERIVATIVE ESTIMATES FOR COMPACT

MANIFOLDS

Vol. 5, No. 4, 2012

dx.doi.org/10.2140/apde.2012.5.747

msp

GENERALIZED RICCI FLOW

I: HIGHER-DERIVATIVE ESTIMATES FOR COMPACT MANIFOLDS

YILI

We consider a generalized Ricci flow with a given (not necessarily closed) three-form and establish higher-derivative estimates for compact manifolds. As an application, we prove the compactness theorem for this generalized Ricci flow. Similar results still hold for a more generalized Ricci flow.

1. Introduction 747

2. Short-time existence of GRF 752

3. Evolution of curvatures 754

4. Derivative estimates 756

5. Compactness theorem 764

6. Generalization 771

Acknowledgment 774

References 774

1. Introduction

Throughout this paper manifolds always mean smooth and closed (compact and without boundary) manifolds. LetMet.M/denote the space of smooth metrics on a manifoldM, andC¹.M/the set of all smooth functions onM. We denote byC the universal constants depending only on the dimension of M, which may take different values at different places.

An important and natural problem in differential geometry is to find a canonical metric on a given manifold. A classical example is the uniformization theorem (e.g.,[Chow and Knopf 2004]), which says that every smooth surface admits a unique conformal metric of constant curvature. To generalize to higher dimensional manifolds, Hamilton[1982]introduced a system of equations

@gij

@t D 2Rij; (1-1)

now called the Ricci flow, an analogue of the heat equation for metrics.

There are two ways to understand the Ricci flow: one way comes from the two-dimensional sigma model (see[Bakas 2007]), while another comes from Perelman’s energy functional[Perelman 2002]

defined by

F.g; f /D Z

M

RC jrfj²

e ^fdVg; .g; f /2Met.M/C¹.M/; (1-2)

MSC2010: 53C44, 35K55.

Keywords: Ricci flow, Generalized Ricci flow, BBS derivative estimates, compactness theorems, energy functionals.

747

whereR,r, anddVg, is the scalar curvature, Levi-Civita connection, and volume form ofg, respectively.

He showed that the Ricci flow is the gradient flow of(1-2)and the functional_Fis monotonic along this gradient flow. Precisely, under the following system

@gij

@t D 2Rij; @f

@t D R f C jrfj²; (1-3) we have

d

dt^F.g; f /D2 Z

M

ˇˇRijC rⁱr^jfˇ ˇ

2e ^fdVg0: (1-4)

Perelman’s energy functional plays an essential role in determining the structures of singularities of the Ricci flow and then the proof of Poincaré conjecture and Thurston’s generalization conjecture; for more details we refer readers to [Cao and Zhu 2006;Chow et al. 2006;2007;2008;2010;Kleiner and Lott 2008;Morgan and Tian 2007;Perelman 2002].

Ricci flow coupled with a one-form or a two-form. If we consider the two-dimensional nonlinear sigma model[Bakas 2007;Oliynyk et al. 2006], then we obtain a generalized Ricci flow that is the Ricci flow coupled with the evolution equation for a two-form. This flow can be also obtained from the point of view of Perelman-type energy functional.

Denoting by_A^p.M/the space ofp-forms onM, we consider the energy functional F^.1/WMet.M/A².M/C¹.M/!R

defined by

F^.1/.g;B; f /D Z

M

RC jrfj² ₁₂¹ jHj²

e ^fdVg; (1-5)

whereHDdB. As showed in[Oliynyk et al. 2006], the gradient flow ofF^.1/satisfies

@gij

@t D 2Rij 2rirjf C¹₂Hik`H<sub>j k`</sub>; (1-6)

@Bij

@t D3rkH^kij 3H^kijrkf; (1-7)

@f

@t D R f C¹₄jHj²; (1-8)

and under a family of diffeomorphisms the system(1-6)–(1-8)is equivalent to

@gij

@t D 2RijC¹₂Hik`Hj k`; (1-9)

@Bij

@t D3rkH^kij; (1-10)

@f

@t D R f C jrfj²C¹₄jHj²: (1-11) Using the adjoint operatord,Equation (1-10)can be written as

@Bij

@t D .dH/^ij; (1-12)

and therefore (because ofHDdB)

@H

@t D d dH DHLH; (1-13)

whereHLD .d dCdd/denotes the Hodge–Laplace operator.

The flow(1-9)–(1-10)can be interpreted as the connection Ricci flow[Streets 2008]. If we replace H DdBbyFDdA, i.e., replace a two-form by a one-form, then the flow(1-6)–(1-7)or(1-9)–(1-10)is exactly the Ricci Yang–Mills flow studied by Streets[2007]and Young[2008].

Ricci flow coupled with a one-form and a two-form. There is another generalized Ricci flow which connects to Thurston’s conjecture — roughly stating that a three-dimensional manifold with a given topology has a canonical decomposition into simple three-dimensional manifolds, each of which admits one, and only one, of eight homogeneous geometries: S³, the round three-sphere;R³, the Euclidean space;_H³, the standard hyperbolic space;_S²R;_H²R; Nil, the three-dimensional nilpotent Heisenberg group;SL.f 2;R/; Sol, the three -dimensional solvable Lie group. The proof of Thurston’s conjecture can be found in[Cao and Zhu 2006;Kleiner and Lott 2008;Morgan and Tian 2007;Perelman 2002].

To better understanding Thurston’s conjecture, Gegenberg and Kunstatter[2004]proposed a generalized flow by considering the modified3Dstringy theory. This flow is the Ricci flow coupled with evolution equations for a one-form and a two-form. As in(1-5), we define an energy functional

F^.2/WMet.M/A¹.M/A².M/C¹.M/!R by

F^.2/.g;A;B; f /D Z

M

RC jrfj² ₁₂¹ jHj^{2 1}₂jFj²

e ^fdVg; (1-14) whereH DdB, andF DdA. In[He et al. 2008], the authors showed that the gradient flow ofF^.2/ satisfies

@gij

@t D 2Rij 2rirjf C¹₂Hik`H<sub>j k`</sub>C2FikF_{j k}; (1-15)

@Ai

@t D2rjF^ji 2F^jirjf; (1-16)

@Bij

@t D3rkH^kij 3H^kijrkf; (1-17)

@f

@t D R f C¹₄jHj²C jFj²; (1-18)

and under a family of diffeomorphisms the system(1-15)–(1-18)is equivalent to

@gij

@t D 2RijC¹₂Hik`H<sub>j k`</sub>C2FikF_{j k}; (1-19)

@Ai

@t D2rjF^ji; (1-20)

@Bij

@t D3rkH^kij; (1-21)

@f

@t D R f C jrfj²C₄¹jHj²C jFj²: (1-22) Using again the adjoint operatord, we have

@F

@t DHLF; @H

@t DHLH: (1-23)

The flow(1-19)–(1-21)clearly contains the Ricci flow, the flow(1-9)–(1-10)or the connection Ricci flow, and the Ricci Yang–Mills flow; we expect this flow can give another proof of the Poincaré conjecture and Thurston’s generalization conjecture, with less analysis on singularities.

Main results. For convenience, we refer to GRF the generalized Ricci flow and RF.A;B/the Ricci flow coupled with a one-formAand a two-formB.

Let.M;g/denote ann-dimensional closed Riemannian manifold with a three-formH D fH_{ij k}g. In the first part of this paper we consider the following GRF onM:

@

@tgij.x;t/D 2Rij.x;t/C₂¹H<sub>ik`</sub>.x;t/Hjk`.x;t/; (1-24)

@

@tH.x;t/DHL;g.x;t/H.x;t/; H.x;0/DH.x/; g.x;0/Dg.x/: (1-25) It is clearly from(1-9)and(1-13)that the gradient flow of the energy functional_F^.1/is a special case of(1-24)–(1-25). The corresponding case thatH is closed is called the refined generalized Ricci flow (RGRF):

@

@tgij.x;t/D 2Rij.x;t/C1

2Hik`.x;t/Hjk`.x;t/; (1-26)

@

@tH.x;t/D d d_g.x;t/H.x;t/; H.x;0/DH.x/; g.x;0/Dg.x/: (1-27) Hered_g.x;t/is the dual operator ofd with respect to the metricg.x;t/.

Lemma 1.1. UnderRGRF,H.x;t/is closed if the initial valueH.x/is closed.

Proof. Since the exterior derivatived is independent of the metric, we have

@

@tdH.x;t/Dd @

@tH.x;t/Dd d d_g.x;t/H.x;t/ D0:

sodH.x;t/DdH.x/D0.

The closedness ofH is very important and has physical interpretation[Bakas 2007;Oliynyk et al.

2006]. Streets [2008] considered the connection Ricci flow in which H is the geometric torsion of connection.

Proposition 1.2. If.g.x;t/;H.x;t//is a solution ofRGRFand the initial valueH.x/is closed,then it is also a solution ofGRF.

Proof. FromLemma 1.1and the assumption we know thatH.x;t/are all closed. Hence

HL;g.x;t/H.x;t/D d d_g.x;t/H.x;t/:

For GRF, a basic and natural question is the existence. The short-time existence for RGRF has been established in [He et al. 2008], where the authors have already showed the short-time existence for RF.A;B/obviously including RGRF. In this paper, we prove the short- time existence for RGF.

Theorem 1.3. There is a unique solution toGRFfor a short time. More precisely,let.M;gij.x//be an n-dimensional closed Riemannian manifold with a three-formHD fHij kg,then there exists a constant T DT.n/ >0depending only onnsuch that the evolution system

@

@tgij.x;t/D 2Rij.x;t/C¹₂g^kp.x;t/g<sup>`q</sup>.x;t/H<sub>ik`</sub>.x;t/Hjpq.x;t/;

@

@tH.x;t/DHL;g.x;t/H.x;t/; H.x;0/DH.x/; g.x;0/Dg.x/;

has a unique solution.gij.x;t/;H_{ij k}.x;t//for a short time0tT.

After establishing the local existence, we are able to prove the higher derivatives estimates for GRF.

Precisely, we have the following

Theorem 1.4. Suppose that.g.x;t/;H.x;t//is a solution toGRFon a closed manifoldMⁿandKis an arbitrary given positive constant. Then for each˛ >0and each integerm1there exists a constant Cmdepending onm;n;maxf˛;1g,andKsuch that if

jRm.x;t/jg.x;t/K; jH.x/jg.x/K for allx2M andt 2Œ0; ˛=K,then

jr^{m 1}Rm.x;t/jg.x;t/C jr^mH.x;t/jg.x;t/ Cm

t^m=2 (1-28)

for allx2M andt 2.0; ˛=K.

As an application, we can prove the compactness theorem for GRF.

Theorem 1.5 (compactness for GRF). Let f.M_k;g_k.t/;H_k.t/;O_k/gk2N be a sequence of complete pointed solutions toGRFfort2Œ˛; !/30such that:

(i) There is a constantC0<1independent ofksuch that sup

.x;t/2Mk.˛;!/

ˇ

ˇRmg_k.x;t/ˇ ˇ_g

k.x;t/C0; sup

x2M_kjHk.x; ˛/jg_k.x;˛/C0: (ii) There exists a constant0>0satisfies

inj_gk.0/.O_k/0: Then there exists a subsequencefj_kgk2Nsuch that

.Mj_k;gj_k.t/;Hj_k.t/;Oj_k/!.M₁;g₁.t/;H₁.t/;O₁/;

converges to a complete pointed solution.M₁;g₁.t/;H₁.t/;O₁/;t 2Œ˛; !/toGRFask! 1.

In the second part of this paper, we consider the Ricci flow coupled with a one-form and a two-form.

This flow is the gradient flow of_F^.2/and takes the form

@

@tgij.x;t/D 2RijC¹₂Hik`.x;t/H<sub>j k`</sub>.x;t/C2Fik.x;t/F_{j k}.x;t/; (1-29)

@

@tAi.x;t/D2r^jF^ji.x;t/; Ai.x;0/DAi.x/; gij.x;0/Dgij.x/; (1-30)

@

@tBij.x;t/D3rkH^kij.x;t/; Bij.x;0/DBij.x/: (1-31) HereAD fAigandBD fBijgis a one-form and a two-form onM, respectively, andFDdA;HDdB.

For this flow, we can also prove the short-time existence, higher derivative estimates, and the compactness theorem.

The rest of this paper is organized as follows. In Section 2, we prove the short-time existence and uniqueness of the GRF for any given three-formH. InSection 3, we compute the evolution equations for the Levi-Civita connections, Riemann, Ricci, and scalar curvatures of a solution to the GRF. InSection 4, we establish higher derivative estimates for GRF, calledBernstein–Bando–Shi (BBS) derivative estimates (e.g., [Cao and Zhu 2006; Chow and Knopf 2004; Chow et al. 2007; 2008;2010;Morgan and Tian 2007;Shi 1989]). InSection 5, we prove the compactness theorem for GRF by using BBS estimates. In Section 6, based on the work of[He et al. 2008], the similar results are established for RF.A;B/.

2. Short-time existence of GRF

In this section we establish the short-time existence for GRF. Our method is standard: we use the DeTurck trick in Ricci flow to prove its short-time existence. We assume that M is an n-dimensional closed Riemannian manifold with metric

dzs²D zgij.x/dxⁱdx^j (2-1)

and with Riemannian curvature tensorf zR_{ij k`}g. We also assume thatHzD f zH_{ij k}gis a fixed three-form onM. In the following we put

hij WDH<sub>ik`</sub>Hjk`: (2-2)

Suppose the metrics

dys_t²D¹₂gyij.x;t/dxⁱdx^j (2-3) are the solutions of¹

@

@tgy_ij.x;t/D 2Ry_ij.x;t/C yh_ij.x;t/; gy_ij.x;0/D zg_ij.x/ (2-4) for a short time0t T. Consider a family of smooth diffeomorphisms't WM !M.0t T/of M. Let

ds²_t WD'tdsy²_t; 0tT (2-5)

1In the following computations we don’t need to use the evolution equation forH.x;t/, hence we only consider the evolution equation for metrics.

be the pull-back metrics ofdys²_t. For coordinates systemxD fx¹; : : : ;xⁿgonM, let

ds²_t Dgij.x;t/dxⁱdx^j (2-6)

and

y.x;t/D't.x/D fy¹.x;t/; : : : ;yⁿ.x;t/g: (2-7) Then we have

gij.x;t/D@y^˛

@xⁱ

@y^ˇ

@x^jgy_˛ˇ.y;t/: (2-8)

By the assumptiongy_˛ˇ.x;t/are the solutions of

@

@tgy_˛ˇ.x;t/D 2Ry_˛ˇ.x;t/C yh_˛ˇ.x;t/; gy_˛ˇ.x;0/D zg_˛ˇ.x/: (2-9) We useRij;Ryij;Rzij; €_ij^k;€y_ij^k;€z_ij^k; r;r;y rz; hij;hyij;hzij to denote the Ricci curvatures, Christoffel symbols, covariant derivatives, and products of the three-formH with respect togzij;gyij;gij respectively.

Then

@

@tgij.x;t/D@y^˛

@xⁱ

@y^ˇ

@x^j @

@tgy_˛ˇ.y;t/

C @

@xⁱ @y^˛

@t @y^ˇ

@x^jgy_˛ˇ.y;t/C @y^˛

@xⁱ

@

@x^j @y^ˇ

@t

y

g_˛ˇ.y;t/:

From(2-9)we have

@

@tgy_˛ˇ.y;t/D 2Ry_˛ˇ.y;t/C yh_˛ˇ.y;t/C@gy_˛ˇ

@y

@y

@t ; and

@

@tgij.x;t/D 2@y^˛

@xⁱ

@y^ˇ

@x^jRy_˛ˇ.y;t/C@y^˛

@xⁱ

@y^ˇ

@x^jhy_˛ˇ.y;t/ C@y^˛

@xⁱ

@y^ˇ

@x^j

@yg_˛ˇ

@y

@y

@t C @

@xⁱ @y^˛

@t @y^ˇ

@x^jgy_˛ˇ.y;t/C@y^˛

@xⁱ

@

@x^j @y^ˇ

@t

y

g_˛ˇ.y;t/:

Since

Rij.x;t/D@y^˛

@xⁱ

@y^ˇ

@x^jRy_˛ˇ.y;t/; hij.x;t/D@y^˛

@xⁱ

@y^ˇ

@x^jhy_˛ˇ.y;t/;

using[Shi 1989, §2, (29)], we obtain

@

@tgij.x;t/D 2Rij.x;t/Chij.x;t/C rⁱ @y^˛

@t

@x^k

@y^˛g_{j k}

C r^j @y^˛

@t

@x^k

@y^˛g_ik

: (2-10)

According to DeTurck trick, we definey.x;t/D't.x/by the equation

@y^˛

@t D @y^˛

@x^kg^ˇ.€_ˇ^k €z_ˇ^k /; y^˛.x;0/Dx^˛; (2-11) then(2-10)becomes

@

@tgij.x;t/D 2Rij.x;t/Chij.x;t/C riVjC rjVi; gij.x;0/D zgij.x/; (2-12)

where

ViDg_ikg^ˇ.€_ˇ^k €z_ˇ^k /: (2-13)

Lemma 2.1. The evolution equation(2-12)is a strictly parabolic system. Moreover,

@

@tgijDg^˛ˇrz_˛rz_ˇgij g^˛ˇgipgz^pqRz_j˛qˇ g^˛ˇgjpgz^pqRz_i˛qˇ

C¹₂g^˛ˇg^pq rzig_p˛ zrjg_qˇC2rzg_jp zrqg_iˇ 2rz˛g_jp zr_ˇg_iq 2rzjg_p˛ zr_ˇg_iq 2rzig_p˛ zr_ˇg_{j q} C¹₂g^˛ˇg^pqHi˛pH_jˇq:

Proof. It is an immediate consequence of Lemma 2.1 of[Shi 1989].

Now we can prove the short-time existence of GRF.

Theorem 2.2. There is a unique solution toGRFfor a short time. More precisely,let.M;gij.x//be an n-dimensional closed Riemannian manifold with a three-formHD fH_{ij k}g,then there exists a constant T DT.n/ >0depending only onnsuch that the evolution system

@

@tgij.x;t/D 2Rij.x;t/C¹₂g^kp.x;t/g<sup>`q</sup>.x;t/H<sub>ik`</sub>.x;t/Hjpq.x;t/;

@

@tH.x;t/DHL;g.x;t/H.x;t/; H.x;0/DH.x/; g.x;0/Dg.x/;

has a unique solution.gij.x;t/;H_{ij k}.x;t//for a short time0tT.

Proof. We proved that the first evolution equation is strictly parabolic byLemma 2.1. Form the Ricci identity, we haveHL;g.x;t/H DLB;g.x;t/HCRmH which is also strictly parabolic. Hence from the standard theory of parabolic systems, the evolution system has a unique solution.

3. Evolution of curvatures

The evolution equation for the Riemann curvature tensors to the usual Ricci flow (e.g., [Cao and Zhu 2006;Chow and Knopf 2004;Chow et al. 2007,2008;2010;Hamilton 1982;Morgan and Tian 2007;Shi 1989]) is given by

@

@tRij k`DRij k`C ij k`; (3-1) where

ij k`D2.Bij k` Bij`k Bi`j kCBikj`/ g<sup>pq</sup>.Rpj k`Rq`CRipk`RqjCRijp`RqkCRij kpRq`/;

andB_{ij k`</sub>Dg<sup>pr</sup>g<sup>qs</sup>RpiqjR<sub>r ks`}. From this we can easily deduce the evolution equation for the Riemann curvature tensors to GRF.

Letvij.x;t/be any symmetric2-tensor, we consider the flow

@

@tgij.x;t/Dv^ij.x;t/: (3-2)

Applying a formula in[Chow and Knopf 2004]to our casevijWD 2RijC¹₂hij withhijDH<sub>ik`</sub>Hjk`, we obtain

@

@tR_{ij k`</sub>D <sup>1</sup><sub>2</sub> 2rirkR<sub>j`}C¹₂rirkh_{j`</sub>C2rir<sub>`}R_{j k} ¹₂rir_`h_{j k}

C2rjrkR_{i`</sub> <sup>1</sup><sub>2</sub>rjrkh<sub>i`} 2rjr_{`</sub>R<sub>ik</sub>C<sup>1</sup><sub>2</sub>rjr<sub>`}h_ik C ¹₂g^pq

Rij kp. 2Rq`C<sup>1</sup><sub>2</sub>hq`/CRijp`. 2RqkC¹₂hqk/

D rirkRj` rir<sub>`</sub>Rj k rjrkRi`C rjr<sub>`</sub>Rik g^pq.Rij kpRq`CRijp`Rqk/ C ¹₄ rirkh_{j`</sub>C rir<sub>`}h_{j k}C rjrkh_{i`</sub> rjr<sub>`}h_ik

C ¹₄g^pq R_{ij kp}h_{q`</sub>CR<sub>ijp`}h_qk DR_{ij k`</sub>C2 B<sub>ij k`} B_{ij`k</sub> B<sub>ilj k</sub>CB<sub>ikj`}

g^pq.R_{pj k`</sub>R<sub>q`}CR_{ipk`</sub>RqjCR<sub>ijp`}R_qkCR_{ij kp}R_{q`</sub>/ C <sup>1</sup><sub>4</sub> rirkh<sub>j`}C rir_{`</sub>h<sub>j k</sub>C rjrkh<sub>i`} rjr_`h_ik

C ¹₄g^pq R_{ij kp}h_{q`</sub>CR<sub>ijp`}h_qk : Proposition 3.1. ForGRFwe have

@

@tR_{ij k`</sub>DR<sub>ij k`}C2.B_{ij k`</sub> B<sub>ij`k} B_{i`j k</sub>CB<sub>ikj`}/

g^pq.R_{pj k`</sub>R<sub>q`}CR_{ipk`</sub>RqjCR<sub>ijp`}R_qkCR_{ij kp}R_q`/

C ¹₄. rirkh_{j`</sub>C rir<sub>`}h_{j k}C rjrkh_{i`</sub> rjr<sub>`}h_ik/C¹₄g^pq R_{ij kp}h_{q`</sub>CR<sub>ijp`}h_qk : In particular:

Corollary 3.2. ForGRFwe have

@

@t RmDRmCRmRmCHHRmC

2

X

iD0

rⁱH r^{2 i}H: (3-3) Proof. FromProposition 3.1, we obtain

@

@t RmDRmCRmRmCr²hChRm: On the other hand,hDHH and

r²hD r.r.HH//D r.rHH/D r²HHC rH rH:

Combining these terms, we obtain the result.

Proposition 3.3. ForGRFwe have

@

@tRikDRikC2hRpiqk;Rpqi 2hRpi;Rpki C¹₄

hh<sub>`</sub>q;Ri`kqi C hRip;hkpi C¹₄

rirkjHj²Cg<sup>j`</sup>rir<sub>`</sub>h_{j k}Cg<sup>j`</sup>rjrkh<sub>i`</sub> h_ik : Proof. Since

@

@tR_ikDg^j` @

@tR<sub>ij k`</sub>C2g<sup>jp</sup>g<sup>`q</sup>R_{ij k`}Rpq

and

g^ijhij Dg^ijHipqHjpq

Dg^ijg^prg^qsHipqHj r sD jHj²; it follows that

g<sup>j`</sup>Œ rirkhj`C rir<sub>`</sub>hj kC rjrkhi` rjr<sub>`</sub>hikCg<sup>pq</sup>hq`Rij kpCg^pqhqkRijp`

D rirkjHj²Cg<sup>j`</sup>rir<sub>`</sub>h_{j k}Cg<sup>j`</sup>rjrkh<sub>i`</sub> h_ikCg<sup>j`</sup>g<sup>pq</sup>h<sub>q`</sub>R_{ij kp}Cg^pqh_qkRip:

From these identities, we get the result.

As a consequence, we obtain the evolution equation for scalar curvature.

Proposition 3.4. ForGRFwe have

@

@tRDRC2jRicj^{2 1}₂jHj²C¹₂hh_ij;R_iji C¹₂g^ikg<sup>j`</sup>rirjh<sub>k`</sub>: Proof. From the usual evolution equation for scalar curvature under the Ricci flow, we have

@

@tRDRC2jRicj²C¹₄g^ikŒhh<sub>`</sub>q;Ri`kqi C hRip;hkpi

C ¹₄g^ik rirkjHj²Cg<sup>j`</sup>rir<sub>`</sub>hj kCg<sup>j`</sup>rjrkhi` hik

DRC2jRicj²C¹₄hhij;Riji C¹₄hRip;hipi

1

4jHj²C¹₄g^ikg<sup>j`</sup>rir<sub>`</sub>h_{j k}C¹₄g^ikg<sup>j`</sup>rjrkh<sub>i`</sub> ¹₄jHj²:

Simplifying the terms, we obtain the required result.

4. Derivative estimates

In this section we are going to prove BBS estimates. At first we review several basic identities of commutators Œ;r and Œ@=@t;r. If A DA.t/ is a t-dependency tensor, and @gij=@t Dv^ij, then applying the well-known formulas stated in[Chow and Knopf 2004]on GRF we have

@

@trRmD r @

@t RmCRmr.RmCHH/

D r.RmCRmRmCHHRmCr²HHCrHrH/CRmrRmCHrHRm D.rRm/C X

iCjD0

rⁱRmr^jRmC X

iCjCkD0

rⁱHr^jHr^kRmC X

iCjD0C2

rⁱHr^jH: (4-1) More generally:

Proposition 4.1. ForGRFand any nonnegative integer`we have

@

@tr^{`</sup>RmD.r<sup>`}Rm/C X

iCjD`

rⁱRmr^jRmC X

iCjCkD`

rⁱH r^jH r^kRmC X

iCjD`C2

rⁱH r^jH: (4-2)

Proof. For`D1, this is(4-1). Suppose(4-2)holds for1; : : : ; `. By induction on`, for`C1we have

@

@tr^`C1Rm D @

@tr.r^`Rm/

D r @

@t.r^{`</sup>Rm/C r<sup>`}Rmr.RmCHH/ D r

.r^`Rm/C X

iCjD`

rⁱRmr^jRmC X

iCjCkD`

rⁱH r^jH r^kRmC X

iCjD`C2

rⁱH r^jH

C r^{`</sup>RmrRmCH rH r<sup>`}Rm D.r^{`C1</sup>Rm/C rRmr<sup>`}RmCRmr^`C1Rm

C X

iCjD`

r^iC1Rmr^jRmCrⁱRmr^jC1Rm

C X

iCjCkD`

r^iC1H r^jH r^kRmCrⁱH r^jC1H r^kRmCrⁱH r^jH r^kC1Rm

C X

iCjD`C2

.r^iC1H r^jHC rⁱH r^jC1H/CH rH r^lRm:

Simplifying these terms, we obtain the required result.

As an immediate consequence, we have an evolution inequality forjr^lRmj². Corollary 4.2. ForGRFand any nonnegative integer`we have

@

@tjr^{`</sup>Rmj<sup>2</sup>jr<sup>l</sup>Rmj<sup>2</sup> 2jr<sup>`C1}Rmj²CC X

iCjD`

jrⁱRmj jr^jRmj jr^`Rmj CC X

iCjCkD`

jrⁱHj jr^jHj jr^kRmj jr^`Rmj CC X

iCjD`C2

jrⁱHj jr^jHj jr^`Rmj; (4-3)

whereC represents universal constants depending only on the dimension ofM. Next we derive the evolution equations for the covariant derivatives ofH. Proposition 4.3. ForGRFand any positive integer`we have

@

@tr^{`</sup>H D.r<sup>`}H/C X

iCjD`

rⁱH r^jRmC X

iCjCkD`

rⁱH r^jH r^kH: (4-4) Proof. From the Bochner formula, the evolution equation forH can be rewritten as

@

@tH DHCRmH: (4-5)

For`D1, we have

@

@trHD r @

@tHCH r.RmCHH/

D r.HCRmH/CH rRmCHH rH D r.H/CH rRmCrHRmCHH rH D.rH/C rRmHC rHRmCHH rH:

Using(4-2)and the same argument, we can prove the evolution equation for higher covariant derivatives.

Similarly, we have an evolution inequality forjr^`Hj².

Corollary 4.4. ForGRFand for any positive integerl we have

@

@tjr^lHj²jr^{`</sup>Hj<sup>2</sup> 2jr<sup>`C1}Hj² CC X

iCjD`

jrⁱHj jr^jRmj jr^`Hj CC X

iCjCkD`

jrⁱHj jr^jHj jr^kHj jr^lHj; (4-6) while

@

@tjHj²jHj² 2jrHj²CC jRmj jHj²: (4-7) Theorem 4.5. Suppose that.g.x;t/;H.x;t//is a solution toGRFon a closed manifoldMⁿfor a short time0t T andK1;K2are arbitrary given nonnegative constants. Then there exists a constantCn

depending only onnsuch that if

jRm.x;t/jg.x;t/K1; jH.x/jg.x/K2

for allx2M andt 2Œ0;T,then

jH.x;t/jg.x;t/K2e^CnK1t (4-8) for allx2M andt 2Œ0;T.

Proof. Since

@

@tjHj²jHj²CCnjRmj jHj²jHj²CCnK1jHj²;

using the maximum principle, we obtainu.t/u.0/e^CnK1t, whereu.t/D jHj². The main result in this section is the following estimates for higher derivatives of Riemann curvature tensors and three-forms. Some special cases were proved in [Streets 2007;2008;Young 2008].

Theorem 4.6. Suppose that.g.x;t/;H.x;t//is a solution toGRFon a compact manifoldMⁿandKis an arbitrary given positive constant. Then for each˛ >0and each integerm1there exists a constant Cmdepending onm;n;maxf˛;1g,andKsuch that if

jRm.x;t/jg.x;t/K; jH.x/jg.x/K

for allx2M andt 2Œ0; ˛=K,then

jr^{m 1}Rm.x;t/jg.x;t/C jr^mH.x;t/jg.x;t/ Cm

t^m=2 (4-9)

for allx2M andt 2.0; ˛=K.

Proof.In the following computations we always letC be any constants depending onn;m;maxf˛;1g,and K, which may take different values at different places. From the evolution equations andTheorem 4.5, we have

@

@tjRmj²jRmj² 2jrRmj²CCCCjr²Hj CCjrHj²;

@

@tjHj²jHj² 2jrHj²CC;

@

@tjrHj²jrHj² 2jr²Hj²CCjrRmj jrHj CCjrHj²: Consider the functionuDtjrHj²CjHj²CtjRmj². Directly computing, we obtain

@

@tuu 2tjr²Hj²CC tjr²Hj C.C 2 /jrHj²CCCC 2tjrRmj²CC t jrRmj jrHj uC2.C / jrHj²CC.1C /:

If we chooseDC, then_@^@_tuuCC which implies thatuC e^{C t} sinceu.0/C. With this estimate we are able to bound the first covariant derivative of Rm and the second covariant derivative ofH. In order to control the termjrRmj², we should use the evolution equations ofjHj²,jrHj²andjr²Hj²to cancel with the bad terms, i.e.,jr²Rmj²,jr²Hj², andjr³Hj², in the evolution equation ofjrRmj²:

@

@tjrRmj²

jrRmj² 2jr²Rmj²CCjrRmj²C C

t¹⁼²jrRmj CC jrRmj jr³Hj C C

t¹⁼²jr²Hj jrRmj;

@

@tjr²Hj²jr²Hj² 2jr³Hj²CCjr²Rmjjr²HjC C

t¹⁼²jrRmjjr²HjCCjr²Hj²CC

t jr²Hj:

As above, we define

uWDt².jr²Hj²C jrRmj²/Ctˇ.jrHj²C jRmj²/CjHj²; and therefore, @u

@t uCC. Motivated by cases formD1andmD2, for generalm, we can define a function

uWDt^m.jr^mHj²C jr^{m 1}Rmj²/C

m 1

X

iD1

ˇitⁱ.jrⁱHj²C jr^{i 1}Rmj²/CjHj²;

whereˇi and are positive constants determined later. In the following, we always assumem3.

Suppose thatjr^{i 1}Rmj C jrⁱHj Ci

tⁱ⁼², fori D1;2; : : : ;m 1. For suchi, from Corollary 4.4, we have

@

@tjrⁱHj²jrⁱHj² 2jr^iC1Hj²CC

i

X

jD0

jr^jHj jr^{i j}Rmj jrⁱHj CC

i

X

jD0 i j

X

`D0

jr^jHj jr^{i j `</sup>Hj jr<sup>`}Hj jrⁱHj

jrⁱHj² 2jr^iC1Hj²CCjrⁱHj

i

X

jD0

Cj

t^j2 C_{i jC1}

t^{i jC2 1} CCjrⁱHj

i

X

jD0 i j

X

`D0

Cj

t^j2 C_{i j `</sub> t<sup>i j2 1</sup> C<sub>`}

t^l2

jrⁱHj² 2jr^iC1Hj²C Ci

t^iC21jrⁱHj CCi

tⁱ²jrⁱHj:

Similarly, fromCorollary 4.2we also have

@

@tjr^{i 1}Rmj²jr^{i 1}Rmj² 2jrⁱRmj²CC

i 1

X

jD0

jr^jRmjjr^{i 1 j}Rmjjr^{i 1}Rmj

CC

i 1

X

jD0 i 1 j

X

`D0

jr^jHj jr^{i 1 j `</sup>Hj jr<sup>`}Rmj jr^{i 1}Rmj

CC

iC1

X

jD0

jr^jHj jr^{iC1 j}Hj jr^{i 1}Rmj

jr^{i 1}Rmj² 2jrⁱRmj²CC jr^{i 1}Rmj

i 1

X

jD0

CjC1

t^jC21 Ci j

t^i2j

CC jr^{i 1}Rmj

i 1

X

jD0 i 1 j

X

`D0

C_j

t^j2 C<sub>i 1 j `</sub> t<sup>i 12j `</sup> C_`C1

t^`C21

CC jr^{i 1}Rmj

i

X

jD1

Cj

t^j2 CiC1 j

t^{iC12 j} CC jr^iC1Hj Ci

t²ⁱ

jr^{i 1}Rmj² 2jrⁱRmj²C Ci

t^iC21 jr^{i 1}Rmj CCi

t²ⁱjr^iC1Hj CCi

t²ⁱ jr^{i 1}Rmj:

The evolution inequality foruis now given by

@u

@t mt^{m 1}.jr^mHj²C jr^{m 1}Rmj²/C

m 1

X

iD1

iˇit^{i 1}.jrⁱHj²C jr^{i 1}Rmj²/

Ct^m @

@tjr^mHj²C @

@tjr^{m 1}Rmj²

C

m 1

X

iD1

ˇitⁱ @

@tjrⁱHj²C @

@tjr^{i 1}Rmj²

C @

@tjHj²:

It’s easy to see that the second term is bounded by

m 1

X

iD1

iˇit^{i 1}Ci

tⁱ D

m 1

X

iD1

iˇiCit ¹;

but this bound depends on t and approaches to infinity when t goes to zero. Hence we use the last second term to control this bad term. The evolution inequality for the third term is the combination of the inequalities

@

@tjr^mHj²

jr^mHj² 2jr^mC1Hj²CC

m

X

iD0

jrⁱHj jr^{m i}Rmj jr^mHj CC

m

X

iD0 m i

X

jD0

jr^jHj jr^{m i j}Hj jrⁱHj jr^mHj jr^mHj² 2jr^mC1Hj²CCjr^mHj²CC jr^mRmj jr^mHj C Cm

t^mC21jr^mHj CCm

t^m2 jr^mHj and

@

@tjr^{m 1}Rmj²jr^{m 1}Rmj² 2jr^mRmj²CC

m 1

X

iD0

jrⁱRmj jr^{m 1 i}Rmj jr^{m 1}Rmj CC

m 1

X

iD0 m 1 i

X

jD0

jr^jHj jr^{m 1 i j}Hj jrⁱRmj jr^{m 1}Rmj

CC

mC1

X

iD0

jrⁱHj jr^{mC1 i}Hj jr^{m 1}Rmj

jr^{m 1}Rmj² 2jr^mRmj²CCjr^{m 1}Rmj²C C

t¹² jr^mHj jr^{m 1}Rmj CCjr^mC1Hjjr^{m 1}Rmj C Cm

t^mC21jr^{m 1}Rmj CCm

t^m2 jr^{m 1}Rmj:

Therefore we have

@u

@t mt^{m 1}.jr^mHj²C jr^{m 1}Rmj²/C

m 1

X

iD1

iˇit^{i 1}.jrⁱHj²C jr^{i 1}Rmj²/ Ct^m

jr^mHj² 2jr^mC1Hj²C C

t^mC21jr^mHj CCjr^mHj² C Cjr^mRmj jr^mHj Cjr^{m 1}Rmj²

2jr^mRmj²C C

t^mC21jr^{m 1}Rmj CCjr^{m 1}Rmj² C C

t¹⁼²jr^mHj jr^{m 1}Rmj CCjr^mC1Hj jr^{m 1}Rmj

C

m 1

X

iD1

ˇitⁱ Ci

t^iC21jr^{i 1}Rmj CjrⁱHj² 2jr^iC1Hj² Cjr^{i 1}Rmj²C Ci

t^iC21jrⁱHj CCi

tⁱ²jr^iC1Hj 2jrⁱRmj²

C .jHj² 2jrHj²CC/

u 2t^mjr^mC1Hj²CC t^mjr^mC1Hj jr^{m 1}Rmj 2t^mjr^mRmj²CC t^mjr^mRmj jr^mHj C

m 2

X

iD0

.iC1/ˇiC1tⁱ.jr^iC1Hj²C jrⁱRmj²/ 2

m 1

X

iD1

ˇitⁱ.jr^iC1Hj²C jrⁱRmj²/ 2jrHj²CC

CC t^{m 1}jr^mHj²CC t^{m 1}jr^{m 1}Rmj²Cj t^m21jr^mHj CC t^m21jr^{m 1}Rmj CC t^{m 12}jr^mHj jr^{m 1}Rmj CC t^mjr^mC1Hj jr^{m 1}Rmj

C

m 1

X

iD1

ˇiCit²ⁱjr^iC1Hj C

m 1

X

iD1

ˇiCitⁱ²¹.jrⁱHj²C jr^{i 1}Rmj/:

Choosing

.iC1/ˇiC1Dˇi; ˇiD A

i !; i 0; whereAis constant which is determined later, and noting that

m 1

X

iD1

ˇiCitⁱ⁼²jr^iC1Hj 1 2

m 1

X

iD1

ˇitⁱjr^iC1Hj²C1 2

m 1

X

iD1

ˇiC_i²

and

m 1

X

iD1

ˇiCitⁱ²¹.jrⁱHj C jr^{i 1}Rmj/

ˇ1C₁.jrHj C jRmj/C

m 2

X

iD1

ˇiC1C_iC1t²ⁱ.jr^iC1Hj C jrⁱRmj/

ˇ1C₁.jrHj C jRmj/C

m 2

X

iD1

ˇiC1C_iC1

tⁱjr^iC1Hj²CtⁱjrⁱRmj²

2ˇiC1CiC1=ˇi CˇiC1CiC1

ˇi

ˇ1C1.jrHj C jRmj/C1 2

m 2

X

iD1

ˇitⁱ.jr^iC1Hj²C jrⁱRmj²/C

m 2

X

iD1

ˇ²_iC1C_i²_C1 ˇi ;