New York Journal of Mathematics New York J. Math.

New York Journal of Mathematics

New York J. Math.19(2013) 565–582.

Local Calabi and curvature estimates for the Chern–Ricci flow

Morgan Sherman and Ben Weinkove

Abstract. Assuming local uniform bounds on the metric for a solu- tion of the Chern–Ricci flow, we establish local Calabi and curvature estimates using the maximum principle.

Contents

1. Introduction 565

2. Preliminaries 567

3. Local Calabi estimate 569

4. Local curvature bound 574

5. Higher order estimates 578

References 580

1. Introduction

Let (M,ˆg) be a Hermitian manifold. TheChern–Ricci flow starting at ˆg is a smooth flow of Hermitian metrics g=g(t) given by

(1.1) ∂

∂tgi¯=−R^C_i¯, gi¯|_t=0 = ˆgi¯,

whereR^C_i¯ :=−∂_i∂_¯log detgis theChern–Riccicurvature ofg. If ˆgis K¨ahler, then the Chern–Ricci flow coincides with the K¨ahler–Ricci flow.

The Chern–Ricci flow was introduced by Gill [11] and further investi- gated by Tosatti and the second-named author [25, 26]. This flow has many of same properties as the K¨ahler–Ricci flow. For example: on manifolds with vanishing first Bott–Chern class the Chern–Ricci flow converges to a Chern–Ricci flat metric [11]; on manifolds with negative first Chern class, the Chern–Ricci flow takes any Hermitian metric to the K¨ahler–Einstein metric [25]; when M is a compact complex surface and ˆg is ∂∂-closed,

Received May 20, 2013.

2010Mathematics Subject Classification. 53C44, 53C55 (primary).

Key words and phrases. Chern–Ricci flow, K¨ahler–Ricci flow, Chern connection, Chern curvature, Calabi estimate.

Supported in part by NSF grant DMS-1105373. Part of this work was carried out while the second-named author was a member of the mathematics department of the University of California, San Diego.

ISSN 1076-9803/2013

565

M. SHERMAN AND B. WEINKOVE

the Chern–Ricci flow exists until either the volume of the manifold goes to zero or the volume of a curve of negative self-intersection goes to zero [25]; if in addition M is nonminimal with nonnegative Kodaira dimension, the Chern–Ricci flow shrinks exceptional curves in finite time [26] in the sense of Gromov–Hausdorff. These results are closely analogous to results for the K¨ahler–Ricci flow [3, 10, 23, 20, 21].

In this note, we establish local derivative estimates for solutions of the Chern–Ricci flow assuming local uniform bounds on the metric, generalizing our previous work [18] on the K¨ahler–Ricci flow. Our estimates are local, so we work in a small open subset ofCⁿ. WriteB_r for the ball of radius r centered at the origin in Cⁿ, and fix T <∞. We have the following result (see Section 2 for more details about the notation).

Theorem 1.1. Fix r with 0 < r < 1. Let g(t) solve the Chern–Ricci flow (1.1) in a neighborhood ofBr for t∈[0, T]. AssumeN >1 satisfies

(1.2) 1

Nˆg≤g(t)≤Nˆg on B_r×[0, T].

Then there exist positive constantsC, α, β depending only on ˆg such that:

(i) |∇g|ˆ ²_g≤ CN^α

r² onB_r/2×[0, T],where∇ˆ is the Chern connection of ˆ

g.

(ii) |Rm|²_g ≤ CN^β

r⁴ onB_r/4×[0, T], forRmthe Chern curvature tensor of g.

Note that the estimates are independent of the timeT and so the results hold also for time intervals [0, T) or [0,∞). The dependence of the constants on ˆg is as follows: up to three derivatives of torsion of ˆg and one derivative of the Chern curvature of ˆg (see Remarks 3.1 and 4.1). We call the bound (i) a local Calabi estimate [2] (see [29] for a similar estimate in the elliptic case).

As a consequence of Theorem 1.1, we have local derivative estimates for g to all orders:

Corollary 1.2. With the assumptions of Theorem 1.1, for any ε >0 with 0< ε < T, there exist constantsC_m, α_m andγ_m for m= 1,2,3, . . . depend- ing only on ˆg and εsuch that

|∇ˆ^m

Rg|²_ˆg≤ C_mN^αm

r^γm onB_r/8×[ε, T], where ∇ˆ_R is the Levi-Civita covariant derivative associated to g.ˆ

Note that our assumption (1.2) often holds for the Chern–Ricci flow on compact subsets away from a subvariety. For example, this always occurs for the Chern–Ricci flow on a nonminimal complex surface of nonnegative Kodaira dimension [25, 26]. It has already been shown by Gill [11] that local derivative estimates exist using the method of Evans–Krylov [9, 14] adapted

to this setting. The purpose of this note is to give a direct maximum princi- ple proof of Gill’s estimates, and in the process identify evolution equations for the Calabi quantity |∇g|ˆ ²_g and the Chern curvature tensor R_i¯kl, which were previously unknown for this flow. In addition, we more precisely de- termine the form of dependence on the constants N and r. We anticipate that this may be useful, for example in generalizations of arguments of [21].

In the case when ˆgis K¨ahler, so thatg(t) solves the K¨ahler–Ricci flow, the above result follows from results of the authors in [18]. The more general case we deal with here leads to many more difficulties, arising from the torsion tensors of g and ˆg. For these reasons, our conclusions here are slightly weaker: for example, we cannot obtain the small values (α= 3 and β = 8) in the estimates of (i) and (ii) that we achieved in [18].

Acknowledgements. The second-named author thanks Valentino Tosatti and Xiaokui Yang for some helpful discussions.

2. Preliminaries

In this section we introduce the basic notions that we will be using throughout the paper. We largely follow notation given in [25]. Given a Hermitian metric g we write ∇ for the Chern connection associated to g, which is characterized as follows. Define Christoffel symbols Γ^l_ik=g^¯sl∂_ig_k¯s. LetX =X^{l ∂}_∂zl be a vector field and leta=a_kdz^k be a (1,0) form. Then (2.1) ∇_iX^l=∂iX^l+ Γ^l_irX^r, ∇_iaj =∂iaj−Γ^r_ijar.

We can, in a natural way, extend∇to act on any tensor. Note that∇makes g parallel: i.e., ∇g = 0. Similarly we let ˆ∇ denote the Chern connection associated to ˆg.

Define the torsion tensor T of gby

(2.2) T_ij^k= Γ^k_ij−Γ^k_ji.

We note thatg is K¨ahler precisely when T = 0. We write T¯ı¯

k¯ := Γ_¯^¯k_ı¯−Γ^¯k_¯¯ı := Γ^k_ij−Γ^k_ji

for the components of the tensor T. We lower and raise indices using the metric g. For example, T^ij_k=g^aig^bjg_klT_ab^l.

We define theChern curvature tensorofgto be the tensor written locally as

(2.3) R_i¯k^l =−∂_¯Γ^l_ik. Then

(2.4) R_i¯k¯l=−∂_i∂_¯g_k¯l+g^sr¯ ∂_ig_k¯s∂_¯g_r¯l.

Again we have lowered an index using the metric g. Note thatR_i¯k¯l =R_j¯ılk¯

holds.

M. SHERMAN AND B. WEINKOVE

The commutation formulas for the Chern connection are given by [∇_i,∇_¯]X^l=Ri¯klX^k, [∇_i,∇_¯]X^k=−R_i¯^k¯¯lX^l (2.5)

[∇_i,∇_¯]a_k=−R_i¯k^la_l, [∇_i,∇_¯]a_l=R_i¯^¯k¯la_k.

Becausegis not assumed to be a K¨ahler metric theBianchi identities will not necessarily hold forR_i¯k¯l. However their failure to hold can be measured with the torsion tensorT defined above:

R_i¯k¯l−R_k¯i¯l =−∇_¯T_ik¯l

(2.6)

R_i¯k¯l−R_i¯lk¯=−∇_iT_¯¯lk

R_i¯k¯l−R_k¯li¯=−∇_¯T_ik¯l− ∇_kT_¯¯li=−∇_iT_¯¯lk− ∇¯lT_ik¯

∇_pR_i¯k¯l− ∇_iR_p¯k¯l =−T_pi^rR_r¯k¯l

∇_q¯R_i¯k¯l− ∇_¯R_i¯qk¯l =−T_q¯¯^¯sR_i¯sk¯l.

These identities are well-known (see [27] for example). Indeed, it is routine to verify the first line, and the second and third lines follow directly from it.

Furthermore the fifth line follows directly from the fourth. For the fourth line we calculate:

∇_pRi¯kl=−∇_p(∂¯Γ^l_ik) =−∂_p∂¯Γ^l_ik−Γ^l_pr∂¯Γ^r_ik+ Γ^r_pi∂¯Γ^l_rk+ Γ^r_pk∂¯Γ^l_ir. Swapping thep and iindices, subtracting, and combining terms, we find

∇_pR_i¯k^l− ∇_iR_p¯k^l =−T_pi^rR_r¯k^l+∂¯

∂iΓ^l_pk−∂pΓ^l_ik+ Γ^l_irΓ^r_pk−Γ^l_prΓ^r_ik

. Now one checks that the quantity in parentheses vanishes.

We define the Chern–Ricci curvature tensorR^C_i¯ by (2.7) R^C_i¯ =g^¯lkR_i¯k¯l=−∂_i∂_¯log detg.

Note that√

−1R^C_i¯dzⁱ∧dz^¯is a real closed (1,1) form. We will suppose that g=g(t) satisfies theChern–Ricci flow:

(2.8) ∂

∂tgi¯=−R^C_i¯, gi¯|_t=0= ˆgi¯,

fort∈[0, T] for some fixed positive time T. We will use ˆ∇, ˆΓ^l_ik, ˆT_ik^l, ˆR_i¯kl, etc., to denote the corresponding quantities with respect to the metric ˆg.

Define a real (1,1) form ω=ω(t) by ω=√

−1g_i¯dzⁱ∧dz^¯ and similarly for ˆ

ω. From (2.8) we have that

(2.9) ω= ˆω+η(t)

for a closed (1,1) formη. Hence

(2.10) T_ik¯l= ˆT_ik¯l.

Here we raise and lower indices of ˆT using the metric ˆg, in the same manner as forg above. Note that T_ik¯l=g_r¯lTikr =∂ig_kl−∂kg_il and ˆT_ik¯l= ˆg_r¯lTˆikr=

∂iˆg_kl−∂_kˆg_il.

It is convenient to introduce the tensor Ψikl = Γ^l_ik−Γˆ^l_ik. We raise and lower indices of Ψ using the metricg, and write Ψ_¯ık^l for the components of Ψ. We note here that Ψ can be used to switch between the connections ∇ and ˆ∇. For example given a tensor of the form Xij we have

(2.11) ∇_pX_i^j−∇ˆ_pX_i^j =−Ψ_pi^rX_r^j + Ψ_pr^jX_i^r. Observe that

(2.12) ∇_¯Ψ_ik^l=−R_i¯k^l+ ˆR_i¯k^l.

We write ∆ for the “rough Laplacian” ofg, ∆ =∇^q¯∇_q¯, where∇^q¯=g^qp¯ ∇_p. Finally note that we will write all norms| · | with respect to the metricg.

3. Local Calabi estimate

In this section we prove part (i) of Theorem 1.1. We consider the Calabi- type [2, 28] quantity

(3.1) S:=|Ψ|²=|∇g|ˆ ².

Our goal in this section is to uniformly bound S on the set B_r/2, which we will do using a maximum principle argument. First we compute its evolution. Calculate

∆S=g^qp¯ ∇_p∇_q¯

g^¯aig^¯bjg_k¯cΨ_ij^kΨ_ab^c

=g^qp¯ g^ai¯g^¯bjg_k¯c∇_p

∇_¯qΨijkΨ_ab^c+ Ψijk∇_qΨ_ab^c

=|∇Ψ|²+|∇Ψ|²+g^¯aig^¯bjgk¯c

∆ΨijkΨabc

+ Ψijk(∆Ψabc+g^qp¯ Rp¯qarΨrbc+g^qp¯ Rp¯qbrΨarc−g^qp¯ Rp¯qrcΨabr)

=|∇Ψ|²+|∇Ψ|²+ 2Re

(∆Ψ_ij^k)Ψ^ij_k

+ (R_p^p_rⁱΨ^rj_k+R_p^p_r^jΨ^ir_k−R_p^p_k^rΨ^ij_r)Ψ_ij^k. From (2.12) we have

(3.2) ∆Ψ_ij^k=−∇^¯qR_i¯qj^k+∇^q¯Rˆ_iqj¯^k.

For the time derivative ofS, first compute (cf. [17] in the K¨ahler case),

(3.3) ∂

∂tΨ_ij^k= ∂

∂tΓ^k_ij =−∇_i(R^C)_j^k. Then

∂

∂tS = ∂

∂t

g^¯aig^¯bjg_k¯cΨ_ij^kΨ_ab^c

= ∂

∂tg^¯ai

ΨijkΨ¯aj k+

∂

∂tg^¯bj

ΨijkΨⁱ¯bk

+ ∂

∂tg_k¯c

Ψ_ij^kΨ^ij¯c+ 2Re ∂

∂tΨ_ij^k

Ψ^ij_k

M. SHERMAN AND B. WEINKOVE

= (R^C)^¯aiΨijkΨ¯aj

k+ (R^C)^¯bjΨijkΨⁱ¯bk

−(R^C)k¯cΨijkΨ^ij¯c−2Re

(∇_i(R^C)jk)Ψ^ijk

.

Therefore ∂

∂t−∆

S = − |∇Ψ|²− |∇Ψ|²+ R^ri¯_p^p−R_p^p¯ri

Ψ_ij^kΨ_r¯^j_k + R^rj¯ _p^p−R_p^p¯rj

Ψ_ij^kΨⁱ_rk¯ −(R_{k¯r p}^p−R_p^p_k¯r) Ψ_ij^kΨ^ijr¯

−2Re h

∇_iRjk

pp+ ∆Ψijk Ψ^ij_k

i .

By (2.6) we can re-write the terms involving a difference in curvature using the torsion tensor T. For the term in square brackets we compute, using (3.2) and again (2.6) that

∇_iR_j^k_p^p+ ∆Ψ_ij^k =∇_i

R_p^p_j^k+∇_jT^pk_p+∇^pT_pj^k

− ∇^q¯R_iqj¯^k+∇^q¯Rˆ_i¯qj^k

=

∇_pR_i^p_j^k−T_ip^rR_r^p_j^k+∇_i∇_jT^pk_p+∇_i∇^pT_pj^k

− ∇^q¯Ri¯qjk+∇^q¯Rˆi¯pjk

=−T_ip^rR_r^p_j^k+∇_i∇_jT^pk_p+∇_i∇^pT_pj^k+∇^q¯Rˆ_iqj¯^k. Hence S satisfies the following evolution equation

∂

∂t−∆

S= (3.4)

− |∇Ψ|²− |∇Ψ|² + ∇_rT_q¯^i¯q+∇_q¯T^q¯_rⁱ

Ψ_ij^kΨ^rj_k+ ∇_rT_q¯^jq¯+∇_q¯T^q¯_r^j

Ψ_ij^kΨ^ir_k

− ∇_kT_q¯^r¯q+∇_q¯T^q¯_k^r

Ψ_ij^kΨ^ij_r

−2Re h

∇_i∇_jT^pkp+∇_i∇_¯qT^q¯jk−TiprRrp

jk+g^qp¯ ∇_pRˆi¯qjk Ψ^ij_k

i . There are similar calculations to (3.4) in the literature which generalize Calabi’s argument [2, 28]: in the elliptic Hermitian case [6, 29]; in the case of the K¨ahler–Ricci flow (see also [18]) in [3, 17]; and in other settings [27, 24, 22].

For the remainder of this section we will writeCfor a constant of the form CN^α forC and αdepending only on ˆg. Our goal is to show thatS ≤C/r². The constant C will be used repeatedly and may change from line to line, and we may at times useC⁰ orC₁, etc.

We would like to bound the right-hand side of (3.4). First, from (2.10) and (2.11) we have, for example,

(3.5) ∇_¯aT_ij^k=g^¯lk( ˆ∇_a¯Tˆ_ij¯l−Ψ_¯a¯lr¯Tˆ_ijr¯).

This and similar calculations show that the third and fourth lines of (3.4) can be bounded by C(S^3/2+ 1). Next we address the terms in the last line of the evolution equation for S.

• Building on (3.5) we find

∇_a∇_bT_¯ı¯^¯k=g^kl

∇_a( ˆ∇_bTˆ_¯ı¯l−Ψ_bl^rTˆ_¯ı¯r) (3.6)

=g^¯kl

∇ˆ_a∇ˆ_bTˆ_¯ı¯l−Ψ_ab^r∇ˆ_rTˆ_¯ı¯l−Ψ_al^r∇ˆ_bTˆ¯ı¯r

−(∇_aΨblr) ˆT¯ı¯r−Ψblr∇ˆ_aTˆ¯ı¯r+ ΨblrΨarsTˆ¯ı¯s

, and hence |∇_i∇_jT^pkp|can be bounded by C(S+|∇Ψ|+ 1).

• Similarly,

∇_a∇_bTijk

(3.7)

=g^lk∇_a( ˆ∇_bTˆ_ijk−Ψ_bk^qTˆijq)

=g^lk

∇ˆ_a∇ˆ_bTˆ_ijk−Ψ_ai^p∇ˆ_bTˆ_pjk−Ψ_aj^p∇ˆ_bTˆ_ipk−(∇_aΨ_bk^q) ˆT_ijq

−Ψ_bk^q( ˆ∇_aTˆ_ijq−Ψ_ai^pTˆ_pjq−Ψ_aj^pTˆ_ipq) ,

and so |∇_i∇_q¯T^q¯jk|can be bounded by C(S+|∇Ψ|+ 1).

• Next, using (2.10) and (2.12):

T_ip^rR_r^p_j^k=g^¯srg^qp¯ Tˆ_ip¯s

Rˆ_r¯qj^k− ∇_q¯Ψ_rj^k , so we can bound |T_ip^rRrp

jk|by C(|∇Ψ|+ 1).

• Finally, compute

∇_pRˆiqj¯k = ˆ∇_pRˆi¯qjk−ΨpirRˆr¯qjk−ΨpjrRˆi¯qrk+ ΨprkRˆi¯qjr. So |g^qp¯ ∇_pRˆ_i¯qj^k|can be bounded by C(S^1/2+ 1).

Putting this all together we arrive at the bound (3.8)

∂

∂t−∆

S ≤C(S^3/2+ 1)− 1

2(|∇Ψ|²+|∇Ψ|²).

We note here the bounds:

|∇tr_ˆgg|² ≤CS (3.9)

|∇S|²≤2S(|∇Ψ|²+|∇Ψ|²).

(3.10)

The first follows from ∇_p ˆg^¯ig_i¯

= ˆ∇_p ˆg^¯ig_i¯

= ˆg^i¯∇ˆ_pg_i¯ and the second follows from|∇S|²=

∇|Ψ|²

∇|Ψ|²

≤2|Ψ|²(|∇Ψ|²+|∇Ψ|²). Furthermore from [25, Proposition 3.1] (see also [6] in the elliptic case), we also have the following evolution equation for tr_ˆgg:

M. SHERMAN AND B. WEINKOVE

∂

∂t−∆

tr_ˆgg= (3.11)

−g^p¯g^qi¯∇ˆ_kgi¯∇ˆ^kgp¯q−2 Re

g^i¯Tˆ_ki^p∇ˆ^kgp¯

+g^i¯

∇ˆ_iTˆ¯k¯q−Rˆik¯q

¯



gk¯q−g^¯i

∇ˆ_iTˆ¯¯qq¯+ ˆ∇^kTˆik¯

+g^i¯Tˆ_¯^k¯qTˆ_ik^p(ˆg−g)_pq¯.

(Here ˆ∇^k = ˆg^¯lk∇ˆ¯l and we have raised indices on the tensor Rm using ˆd g).

This generalizes the second order evolution inequality for the K¨ahler–Ricci flow [3] (cf. [28, 1]). Hence we have the estimate

(3.12)

∂

∂t−∆

tr_ˆgg≤ − S C0

+C(S^1/2+ 1),

for a uniform positive constant C0 (in fact we can takeC0 =N).

We now would like to show that the evolution inequalities (3.8, 3.12) imply a uniform bound on S = |∇g|ˆ ² on B_r/2 ×[0, T]. Choose a smooth cutoff function ρ which is supported in B_r and is identically 1 onB_r/2. We may assume that |∇ρ|²,|∆ρ| are bounded by C/r². Let K be a large uniform constant, to be specified later, which is at least large enough so that

K

2 ≤K−tr_gˆg≤K.

Let A denote another large positive constant to be specified later. We will use a maximum principle argument with the function (cf. [5])

f =ρ² S

K−tr_gˆg+Atr_ˆgg to show thatS is bounded onB_r/2.

Suppose that the maximum off on B_r×[0, T] occurs at a point (x₀, t₀).

We assume for the moment that t₀ > 0 and that x₀ does not lie in the boundary ofBr. We wish to show that at (x0, t0),S is bounded from above by a uniform constant C. Hence we may assume without loss of generality thatS >1 at (x0, t0). In particular, we have

∂

∂t−∆

S ≤CS^3/2−1

2(|∇Ψ|²+|∇Ψ|²), (3.13)

∂

∂t−∆

tr_ˆgg≤ − S 2C0

+C.

We compute at (x0, t0),

∂

∂t−∆

f =

A ∂

∂t−∆

trˆgg+ (−∆(ρ²)) S K−tr_ˆgg +ρ² S

(K−tr_gˆg)² ∂

∂t−∆

tr_gˆg+ρ² 1 K−tr_gˆg

∂

∂t−∆

S

−4Re

ρ S

(K−tr_gˆg)²∇tr_gˆg· ∇ρ

−4Re

ρ 1

K−tr_ˆgg∇ρ· ∇S

−2Re

ρ² 1

(K−trgˆg)²∇tr_ˆgg· ∇S

− 2ρ²S

(K−trgˆg)³|∇tr_ˆgg|².

But since a maximum occurs at (x₀, t₀) we have∇f = 0 at this point, and hence

2ρ∇ρ S

K−trˆgg +ρ² ∇S

K−trgˆg +ρ² S∇tr_gˆg

(K−trgˆg)² +A∇tr_ˆgg= 0.

Then at (x₀, t₀), ∂

∂t−∆

f =

A ∂

∂t−∆

trgˆg+ (−∆(ρ²)) S

K−tr_gˆg +ρ² S (K−tr_ˆgg)²

∂

∂t−∆

trˆgg

+ρ² 1 K−tr_ˆgg

∂

∂t−∆

S−4Re

ρ 1

K−tr_gˆg∇ρ· ∇S

+ 2A|∇tr_ˆgg|² K−tr_ˆgg . Making use of (3.9, 3.10, 3.13) and Young’s inequality, we obtain at (x0, t0),

0≤ ∂

∂t−∆

f ≤

− A 2C0

S+CA

+ CS

r²K

+

− ρ² 2K²C0

S²+Cρ² K² S

+

−ρ²

2K(|∇Ψ|²+|∇Ψ|²) + ρ²

4K²C₀S²+Cρ²S

+ ρ²

4K(|∇Ψ|²+|∇Ψ|²) + C Kr²S

+CA

K S

≤ − A 2C0

S+CA+ C⁰

r²S+CA K S.

Now pick K≥4C₀C so that at (x₀, t₀), 0≤ − A

4C₀S+CA+C⁰ r²S.

Then choose A= ^8C_r⁰2^C0 so that at (x0, t0), C⁰

r²S ≤CA,

M. SHERMAN AND B. WEINKOVE

giving a uniform upper bound forS. It follows thatf is bounded from above by Cr⁻² for a uniformC. HenceS onB_r/2 is bounded above byCr⁻².

It remains to deal with the cases whent₀ = 0 or x₀ lies on the boundary ofB_r. In either case we havef(x₀, t₀)≤Atr_ˆgg(x₀, t₀)≤Cr⁻² and the same bound holds.

Remark 3.1. Tracing through the argument, one can see that the constants only depend on uniform bounds for the torsion and curvature of ˆg, and one and two derivatives (with respect to ˆ∇ or ˆ∇) of torsion and one derivative of curvature.

4. Local curvature bound

In this section we prove part (ii) of Theorem 1.1. As in the previous section, we write C for a constant of the formCN^γ for some uniform C, γ.

We compute in the ballB_r/2 on which we already have the boundS ≤C/r². Let ∆_R = ¹₂g^qp¯ (∇_p∇_q¯+∇_q¯∇_p). First we need an evolution equation for the curvature tensor. We begin with

∂

∂tR_i¯k^l = ∂

∂t

−∂_¯Γ^l_ik

=−∂_¯∂

∂t

Γ^l_ik

=−∂_¯(−∇_i(R^C)_k^l) =∇_¯∇_iR_k^lpp

and therefore,

(4.1) ∂

∂tR_i¯k¯l=−R_q¯lppRi¯kq+∇_¯∇_iR_k¯lpp.

Now, computing in coordinates whereg is the identity, we find

∆_RR_i¯k¯l=1

2(∇_p∇_p¯+∇_p¯∇_p)R_i¯k¯l

=∇_p∇_p¯R_i¯k¯l+1

2(R_p¯pi¯qR_q¯k¯l−R_ppq¯¯R_i¯qk¯l+R_{ppk¯¯ q}R_i¯q¯l−R_p¯pq¯lR_i¯kq¯)

=∇_p(∇_¯R_ipk¯ ¯l−Tp¯¯qR_i¯qk¯l) +1

2(Rppi¯¯qR_q¯k¯l−Rppq¯¯R_i¯qk¯l+Rp¯pk¯qR_i¯q¯l−R_p¯pq¯lRi¯kq¯)

=∇_¯∇_pR_i¯pk¯l−R_p¯i¯qR_qpk¯ ¯l+R_p¯qp¯R_i¯qk¯l

−R_p¯kq¯R_ipq¯¯l+R_p¯q¯lR_i¯pk¯q− ∇_p(Tp¯¯qR_iqk¯¯l) +1

2(R_ppi¯¯qR_q¯k¯l−R_ppq¯¯R_i¯qk¯l+R_p¯pk¯qR_i¯q¯l−R_p¯pq¯lR_i¯kq¯)

=∇_¯(∇_iR_ppk¯ ¯l−T_pi¯qR_q¯pk¯l)−R_p¯i¯qR_q¯pk¯l+R_p¯qp¯R_i¯qk¯l

−Rp¯kq¯R_ipq¯¯l+R_p¯q¯lRi¯pk¯q− ∇_p(Tp¯¯qR_iqk¯¯l) +1

2(R_ppi¯¯qR_q¯k¯l−R_ppq¯¯R_i¯qk¯l+R_p¯pk¯qR_i¯q¯l−R_p¯pq¯lR_i¯kq¯)

=∇_¯∇_i(R_k¯lpp¯− ∇_pT_p¯¯lk− ∇¯lT_pk¯p)− ∇_¯(T_pi¯qR_q¯pk¯l)−R_p¯i¯qR_qpk¯ ¯l

+R_p¯qp¯R_i¯qk¯l−R_p¯k¯qR_i¯pq¯l+R_p¯q¯lR_i¯pkq¯− ∇_p(T_p¯¯qR_i¯qk¯l) +1

2(R_ppi¯¯qR_q¯k¯l−R_ppq¯¯R_i¯qk¯l+R_p¯pk¯qR_i¯q¯l−R_p¯pq¯lR_i¯kq¯).

Hence ∂

∂t−∆_R

R_i¯k¯l= (4.2)

−R_q¯lp¯pR_i¯kq¯+R_p¯i¯qR_q¯pk¯l−R_p¯qp¯R_iqk¯ ¯l+R_p¯kq¯R_ipq¯¯l−R_p¯q¯lR_i¯pk¯q

−1

2(R_ppi¯¯qR_q¯k¯l−R_ppq¯¯R_iqk¯¯l+R_p¯pk¯qR_i¯q¯l−R_ppq¯¯lR_i¯kq¯) +∇_p( ˆT_p¯¯qR_i¯qk¯l) +∇_¯( ˆT_pi¯qR_q¯pk¯l) +∇_¯∇_i(∇_pT_p¯¯lk+∇¯lT_pkp¯).

To estimate this, we first compute

∇_p( ˆTp¯qR_iqkl) = ( ˆ∇_pTˆp¯q−ΨpqrTˆp¯r)R_iqkl+ ˆTp¯q∇_pR_iqkl, and this is bounded by C(|Rm|/r+|∇Rm|). Using the fact that

Ri¯kl=−∇_¯Ψikl+ ˆRi¯kl

we have

(4.3) |Rm| ≤ |∇Ψ|+C,

and hence

(4.4) |∇_p( ˆT_p¯qR_iqkl)| ≤C

|∇Rm|+|∇Ψ|

r +1 r

. Similarly for the term∇_¯( ˆTpi¯qR_q¯pk¯l).

The last two terms of (4.2) involve three derivatives of torsion. We claim that

(4.5) |∇∇∇T|, |∇∇∇T| ≤C

|∇Rm|+ |∇Ψ|+|∇Ψ|

r + 1

r³

. Indeed, applying ∇_c to (3.6), we have

∇_c∇_a∇_bT_¯ı¯^k=g^kl

∇ˆ_c∇ˆ_a∇ˆ_bTˆ_¯ı¯l−Ψ_c¯ı^q∇ˆ_a∇ˆ_bTˆ_q¯l−Ψ_c¯^q∇ˆ_a∇ˆ_bTˆ_¯ıql (4.6)

− ∇_c(Ψabr∇ˆ_rTˆ¯ı¯l)− ∇_c(Ψalr∇ˆ_bTˆ¯ı¯r)− ∇_c(Ψblr∇ˆ_aTˆ¯ı¯r)

−∇_c(∇_aΨ_bl^rTˆ_¯ı¯r) +∇_c(Ψ_bl^rΨ_ar^sTˆ_¯ı¯s) . The first three terms on the right-hand side are bounded byC(√

S+ 1) and hence byC/r. Next compute

∇_c(Ψabr∇ˆ_rTˆ¯ı¯l) = (∇_cΨabr) ˆ∇_rTˆ¯ı¯l+ Ψabr∇ˆ_c∇ˆ_rTˆ¯ı¯l−ΨabrΨc¯ıq∇ˆ_rTˆq¯l

(4.7)

−Ψ_ab^rΨ_c¯^q∇ˆ_rTˆ_¯ıql, which is bounded by C|∇Ψ|+C√

S+CS and hence by C(|∇Ψ|+ 1/r²).

The same bound holds for the other two terms on the second line of (4.6).

For the third line, compute

∇_c(∇_aΨ_bl^rTˆ_¯ı¯r) = (∇_a∇_cΨ_bl^r+R_acb^pΨ_pl^r+R_acl^pΨ_bp^r−R_acp^rΨ_bl^p) ˆT_¯ı¯r + (∇_aΨblr)

∇ˆ_cTˆ¯ı¯r−Ψc¯ıqTˆq¯r−Ψc¯qTˆ¯ıqr

,

M. SHERMAN AND B. WEINKOVE

and using the fact that ∇_cΨblr =−R_bcl^r+ ˆRbclr we obtain

∇_c(∇_aΨblrTˆ¯ı¯r) = − ∇_aRbclr+ ˆ∇_aRˆbclr−ΨabpRˆpclr−ΨalpRˆbcpr

+ Ψ_ap^rRˆ_bcl^p+R_acb^pΨ_pl^r+R_acl^pΨ_bp^r−R_acp^rΨ_bl^pTˆ_¯ı¯r + (∇_aΨ_bl^r)

∇ˆ_cTˆ¯ı¯r−Ψc¯ıqTˆq¯r−Ψc¯qTˆ¯ıqr

. It follows that

(4.8) |∇_c(∇_aΨ_bl^rTˆ_¯ı¯r)| ≤C

|∇Rm|+|Rm|

r +|∇Ψ|

r +1 r

. Finally,

∇_c(Ψ_bl^rΨ_ar^sTˆ_¯ı¯s) = (−R_bcl^r+ ˆR_bcl^r)Ψ_ar^sTˆ_¯ı¯s+ Ψ_bl^r(−R_acr^s+ ˆR_acr^s) ˆT_¯ı¯s + ΨblrΨars( ˆ∇_cTˆ¯ı¯s−Ψc¯ıqTˆq¯s−Ψc¯qTˆ¯ıqs),

giving

(4.9) |∇_c(Ψ_bl^rΨ_ar^sTˆ_¯ı¯s)| ≤C

|Rm|

r + 1 r³

.

Putting together (4.6, 4.7, 4.8, 4.9), and making use of (4.3), we obtain

|∇∇∇T| ≤C

|∇Rm|+|∇Ψ|+|∇Ψ|

r + 1

r³

,

and the bound for |∇∇∇T| follows similarly. This completes the proof of the claim (4.5).

From (4.4) and the claim we just proved, since the second and third lines of (4.2) are of the order |Rm|², we have the bound

(4.10)

∂

∂t−∆_R

Rm

≤C

|Rm|²+|∇Rm|+ |∇Ψ|+|∇Ψ|

r + 1

r³

. Now

∂

∂t−∆

|Rm|² =g^b¯g^¯ckg^¯ld(R^C)^ai¯R_i¯k¯lR_a¯bcd¯

(4.11)

+g^ai¯g^¯ckg^¯ld(R^C)^b¯R_i¯k¯lR_a¯bcd¯

+g^ai¯g^¯bg^¯ld(R^C)^¯ckR_i¯k¯lR_a¯bcd¯

+g^ai¯g^¯bg^¯ck(R^C)^¯ldR_i¯k¯lR_a¯bcd¯

+ 2Re

g^¯aig^b¯g^¯ckg^¯ld

(∂

∂t−∆_R)R_i¯k¯l

R_a¯bcd¯

−2|∇Rm|². This together with (4.10) and (4.3) implies