ON THE EVOLUTION OF A HERMITIAN METRIC BY ITS CHERN-RICCI FORM 2 1 VALENTINO TOSATTI∗ AND BEN WEINKOVE† 0 2 Abstract. WeconsidertheevolutionofaHermitianmetriconacom- n pact complex manifold by its Chern-Ricci form. This is an evolution a equation first studied by M. Gill, and coincides with the K¨ahler-Ricci J flow if theinitial metric is K¨ahler. We findthe maximal existence time 7 for theflow in terms of the initial data. We investigate the behavior of 1 the flow on complex surfaces when the initial metric is Gauduchon, on ] complex manifolds with negative first Chern class, and on some Hopf G manifolds. Finally, we discuss a new estimate for the complex Monge- D Amp`ere equation on Hermitian manifolds. . h t a m 1. Introduction [ Let (M,J) be a compact complex manifold of complex dimension n. Let 2 v g0 be a Hermitian metric on M, that is a Riemannian metric g0 satisfying 2 g (JX,JY) = g (X,Y) for all vectors X, Y. In local complex coordinates 0 0 1 (z ,...,z ), the metric g is given by a Hermitian matrix with components 3 1 n 0 0 (g0)ij. Associated to g0 is a real (1,1) form ω0 = √−1(g0)ijdzi∧dzj, which . we will also often refer to as a Hermitian metric. 1 0 Given thesuccessofHamilton’s Ricci flow[22]inestablishingdeepresults 2 in the setting of topological, smooth and Riemannian manifolds (see e.g. 1 [5, 23, 30]), it is natural to ask whether there is a parabolic flow of metrics : v on M which starts at g , preserves the Hermitian condition and reveals 0 i X information about the structure of M as a complex manifold. In the case r when g0 is Ka¨hler (meaning dω0 = 0), the Ricci flow does precisely this. It a gives a flow of Ka¨hler metrics whose behavior is deeply intertwined with the complex and algebro-geometric properties of M (see [8, 9, 15, 31, 32, 34, 35, 36, 37, 38, 40, 41, 45, 50, 51, 52, 56], for example). However, if g is not Ka¨hler, then in general the Ricci flow does not pre- 0 serve the Hermitian condition g(JX,JY) = g(X,Y). Alternative parabolic flows on complex manifolds which do preserve the Hermitian property have been proposed by Streets-Tian [42, 43] and also Liu-Yang [28]. ∗Supported in part by NSFgrant DMS-1005457 and by a Blavatnik Award for Young Scientists. Partofthisworkwascarriedoutwhilethefirst-namedauthorwasvisitingthe Mathematical Science Center of Tsinghua University in Beijing, which he would like to thankfor its hospitality. †Supportedin part byNSFgrant DMS-1105373. 1 2 V.TOSATTIANDB.WEINKOVE This paper is concerned with another such flow, first investigated by M. Gill [19], which we will call the Chern-Ricci flow: ∂ (1.1) ω = Ric(ω), ω = ω , t=0 0 ∂t − | wherehereRic(ω)istheChern-Ricci form (sometimes called thefirst Chern form) associated to the Hermitian metric g, which in local coordinates is given by (1.2) Ric(ω) = √ 1∂∂logdetg. − − In the case when g is Ka¨hler, Ric(ω) = √ 1R dz dz , where R is the − ij i ∧ j ij usual Ricci curvature of g. Thus if g is Ka¨hler, (1.1) coincides with the 0 Ka¨hler-Ricci flow. In general, Ric(ω) does not have a simple relationship withtheRicci curvatureof g. TheBott-Chern cohomology class determined by the closed form Ric(ω) is denoted by cBC(M). We call this the first Bott- 1 Chern class of M. It is independent of the choice of Hermitian metric ω (see section 2 for more details). The following result for the Chern-Ricci flow was proved by Gill [19]: Theorem 1.1 (Gill). If cBC(M) = 0 then, for any initial metric ω , there 1 0 exists a solution ω(t) to the Chern-Ricci flow (1.1) for all time and the met- rics ω(t) converge smoothly as t to a Hermitian metric ω∞ satisfying → ∞ Ric(ω∞)= 0. Moreover, the Hermitian metric ω∞ is the unique Chern-Ricci flat metric onM oftheformω∞ = ω0+√ 1∂∂ϕforsomefunctionϕ. TheChern-Ricci − flat metrics were already known to exist [10, 54], and the estimate of [54] is usedin the proofof Theorem 1.1. If ω is Ka¨hler then Theorem 1.1 is dueto 0 Cao [8], with ω∞ being the Ricci-flat metric of Yau [60]. In Section 2 below, we discuss the work of Gill [19] further, and also explain how the Chern- Ricci flow compares with some other parabolic flows on complex manifolds studied in the literature. Our first result characterizes the maximal existence time for a solution to the Chern-Ricci flow using information from the initial Hermitian metric ω . First observe that the flow equation (1.1) may be rewritten as 0 ∂ detg(t) ω = Ric(ω )+√ 1∂∂θ(t), with θ(t)= log . 0 ∂t − − detg 0 Thus, as long as the flow exists, the solution ω(t) starting at ω must be of 0 the form ω(t) = α +√ 1∂∂Θ, for some function Θ = Θ(t), with t − (1.3) α = ω tRic(ω ). t 0 0 − Now define a number T = T(ω ) with 0< T 6 by 0 ∞ (1.4) T = sup t > 0 ψ C∞(M) with α +√ 1∂∂ψ > 0 . t { | ∃ ∈ − } By the observation above, a solution to (1.1) cannot exist beyond time T. We prove: EVOLUTION OF A HERMITIAN METRIC BY ITS CHERN-RICCI FORM 3 Theorem 1.2. There exists a unique maximal solution to the Chern-Ricci flow (1.1) on [0,T). In the special case when ω is Ka¨hler, this is already known by the result 0 of Tian-Zhang [51], who extended earlier work of Cao and Tsuji [8, 56, 57]. In the Ka¨hler case, T depends only on the cohomology class [ω ] and can be 0 written T = sup t > 0 [ω ] tc (M) > 0 . 0 1 { | − } FurthermoretheNakai-Moishezon criterion, duetoBuchdahl[7]andLamari [25] for Ka¨hler surfaces and to Demailly-P˘aun [11] for general Ka¨hler man- ifolds, implies that at time T either the volume of M goes to zero, or the volume of some proper analytic subvariety of M goes to zero (cf. the dis- cussion in [15]). Note that in the general Hermitian case, we can consider the equivalence relation of (1,1) forms on M: α α′ α= α′+√ 1∂∂ψ for some function ψ C∞(M). ∼ ⇐⇒ − ∈ Then T defined by (1.4) depends only on the equivalence class of ω . 0 In the special case when M is a complex surface (n = 2) a result of Gauduchon [16] is that every Hermitian metric is conformal to a ∂∂-closed metric ω . If ω is ∂∂-closed then so is ω(t) for t [0,T). Moreover, we 0 0 ∈ have a geometric characterization of the maximal existence time T: Theorem 1.3. Let M be a compact complex surface, ω a ∂∂-closed Her- 0 mitian metric. Then T defined by (1.4) can be written as T = sup T > 0 t [0,T ], α2 > 0, α > 0, 0 ∀ ∈ 0 t t (cid:26) (cid:12) ZM ZD (cid:12) (cid:12) for all D irreducible effective divisors with D2 < 0 , (cid:12) (cid:27) for α given by (1.3). t Note that for t [0,T), the quantity α2 = ω(t)2 is the volume of ∈ M t M M (with respect to ω(t)) and α = ω(t) is the volume of the curve D. D t DR R Thus we can restate Theorem 1.3 as: R R Corollary 1.4. Let M be a compact complex surface, ω a ∂∂-closed Her- 0 mitian metric. Then the Chern-Ricci flow (1.1) starting at ω exists until 0 either the volume of M goes to zero, or the volume of a curve of negative self-intersection goes to zero. As we remarked above, the sameresultwas known to holdfor theKa¨hler- Ricci flow thanks to the Nakai-Moishezon criterion of [7, 25]. Analogues of Theorems 1.2, 1.3 and Corollary 1.4 were conjectured by Streets-Tian [44] for their pluriclosed flow (see Section 2 below). The Ka¨hler-Ricci flow has a deep connection to the Minimal Model Pro- gram in algebraic geometry, as demonstrated by the work of Song-Tian and 4 V.TOSATTIANDB.WEINKOVE others [14, 26, 34, 35, 36, 37, 38, 39, 40, 41, 50, 51, 56, 63]. In the case of algebraic surfaces, the minimal model program is relatively simple. Indeed, a minimal surface is defined to be a surface with no ( 1)-curves (smooth − rational curves C with C2 = 1). To find the minimal model, one can just − apply a finite number of blow-downs, which are algebraic operations con- tracting the ( 1)-curves. It was shown in [38] that the Ka¨hler-Ricci flow − on an algebraic surface carries out these algebraic operations, contracting ( 1)-curves in the sense of Gromov-Hausdorff, and smoothly outside of the − curves. Moreover, the same behavior occurs on a non-algebraic Ka¨hler sur- face [40]. In all dimensions, weak solutions to the Ka¨hler-Ricci flow through singularites were constructed in [36] and a numberof conjectures were made about the metric behavior of the flow (see also [41]). We return now to the case of a complex (non-K¨ahler) surface. In this case one can also contract the ( 1) curves to arrive at a minimal surface. − We conjecture that the Chern-Ricci flow on a complex surface starting at a ∂∂-closed metric behaves in an analogous way to the Ka¨hler-Ricci flow on a Ka¨hler surface. We prove the following: Theorem 1.5. Let M be a compact complex surface with a ∂∂-closed Her- mitian metric ω , and let [0,T) be the maximal existence time of the Chern- 0 Ricci flow starting from ω . Then 0 (a) If T = then M is minimal ∞ − (b) If T < and Vol(M,ω(t)) 0 as t T , then M is either ∞ → → birational to a ruled surface or it is a surface of class VII (and in this case it cannot be an Inoue surface) − (c) If T < and Vol(M,ω(t)) stays positive as t T , then M con- ∞ → tains ( 1)-curves. − Furthermore, if M is minimal then T = unless M is CP2, a ruled surface, ∞ a Hopf surface or a surface of class VII with b > 0, in which cases (b) holds. 2 In the case that M is not minimal, and (c) occurs, we expect that the Chern-Ricci flow will contract a finite number of ( 1)-curves and can be − uniquely continued on the new manifold. Moreover, we conjecture that this process can be repeated until one obtains a minimal surface, or ends up in case (b) above. More details of this conjecture can be found in Section 6. To provide some evidence for our conjecture, we prove the following the- orem. It is an analogue of a result for the Ka¨hler-Ricci flow, whose proof is essentially contained in [51] (for a recent exposition, see Chapter 7 of [40]), and which was a key starting point for the work [36, 37, 38, 39]. We assume that the maximal existence time T is finite and, roughly speaking, that the limiting ‘class’ of the flow at time T is given by the pull-back of a Hermitian metric on a manifold N via π : M N, where π is a holomorphic map → blowing down an exceptional divisor E to a point p N. We show that the ∈ solutiontotheChern-Ricciflowwillconverge smoothlyattimeT away from E. In this way, one can obtain a Hermitian metric on the new manifold N, at least away from the point p. Our result holds in any dimension: EVOLUTION OF A HERMITIAN METRIC BY ITS CHERN-RICCI FORM 5 Theorem 1.6. Assume that there exists a holomorphic map between com- pact Hermitian manifolds π : (M,ω ) (N,ω ) blowing down the excep- 0 N → tional divisor E on M to a point p N. In addition, assume that there ∈ exists a smooth function ψ on M with (1.5) ω TRic(ω )+√ 1∂∂ψ = π∗ω , 0 0 N − − with T < given by (1.4). ∞ Then the solution ω(t) to the Chern-Ricci flow (1.1) starting at ω con- 0 ∞ verges in C on compact subsets of M E to a smooth Hermitian metric \ ω on M E. T \ There are some new obstacles to proving this in the non-K¨ahler case that we overcome using a parabolic Schwarz Lemma for volume forms, and a second order estimate for themetric which uses atrick of Phong-Sturm[33]. In the cases when the flow has a long time solution, it is natural to investigate its behavior at infinity. If the manifold has vanishing first Bott- Chern class, we have already seen by Gill’s result (Theorem 1.1) that the flow converges to a Chern-Ricci flat Hermitian metric. We now suppose that the first Chern class c (M) is negative (note that cBC(M) < 0 implies 1 1 c (M) < 0). Inthiscase,themanifoldM isKa¨hlerandafundamentalresult 1 of Aubin [1] and Yau [60] says that M admits a unique Ka¨hler-Einstein metric ω with negative scalar curvature. Cao [8] then proved that the KE Ka¨hler-Ricci flow (appropriately normalized) deforms any Ka¨hler metric in c (M) to ω . The same is true for the normalized Ka¨hler-Ricci flow 1 KE − starting at any Ka¨hler metric [51, 56]. Our next result shows that starting at any Hermitian metric on the manifold M, the (normalized) Chern-Ricci flow will converge to the Ka¨hler-Einstein metric ω . KE Theorem 1.7. Let M be a compact complex manifold with c (M) < 0 and 1 let ω be a Hermitian metric on M. Then the Chern-Ricci flow (1.1) has a 0 long-time solution ω(t), and as t goes to infinity the rescaled metrics ω(t)/t converge smoothly to the unique K¨ahler-Einstein metric ω on M. KE In particular we see that the Chern-Ricci flow on these manifolds, after normalization, deforms any Hermitian metric to a Ka¨hler one. Next we illustrate the Chern-Ricci flow with an explicit example. For α = (α ,...,α ) Cn 0 with α = = α = 1, we consider the 1 n 1 n Hopf manifold M ∈= (C\n{ }0 )/ |wh|ere ··· | | 6 α \{ } ∼ (z ,...,z ) (α z ,...,α z ). 1 n 1 1 n n ∼ This is a non-K¨ahler complex manifold of complex dimension n. If n = 2, it is an example of a class VII surface. We can write down an exact solution to the Chern-Ricci flow on M . Consider the metric ω = δij√ 1dz dz . α H r2 − i∧ j Then we have: Proposition 1.8. Themetricsω(t) := ω tRic(ω )onM giveasolution H H α − of the Chern-Ricci flow on the maximal existence interval [0,1/n). As t → 6 V.TOSATTIANDB.WEINKOVE T = 1/n, the limiting nonnegative (1,1) form ω is given by T z z ω = i j√ 1dz dz . T r4 − i ∧ j In the case of the original Hopf surface, which has α = (2,2) and is an elliptic fiberbundleover P1 via the map (z ,z ) [z ,z ], the limiting form 1 2 1 2 7→ ω is positive definite along the fibers and zero in directions orthogonal to T the fibers. Ifwestartwithanymetricω whichdiffersfromω by√ 1∂∂ψ forsome 0 H − function ψ, then we conjecture that the flow also converges as t T to a → smooth but degenerate (1,1) form on M with properties similar to ω . To α T give evidence for this conjecture, we prove an estimate: Proposition 1.9. Let ω = ω +√ 1∂∂ψ be a Hermitian metric on M , 0 H α − and let ω(t) be the solution of the Chern-Ricci flow (1.1) starting at ω on 0 M for t [0,1/n). Then there exists a uniform constant C such that α ∈ ω(t) 6 Cω , for t [0,1/n). H ∈ In particular, this result shows that we obtain convergence for the flow at the level of potential functions in C1+β for any β (0,1). For more details ∈ see Section 8. The final result in this paper concerns not the Chern-Ricci flow, but an elliptic equation: the complex Monge-Amp`ere equation (1.6) (ω+√ 1∂∂ϕ)n = eFωn, ω′ := ω+√ 1∂∂ϕ > 0, − − on a compact Hermitian manifold (M,ω), where F is a smooth function on M. We give an alternative proof of a result of [54] that ϕ is uniformly C0 k k bounded (see also [4, 12]). The result makes use of a new second order es- timate in this context: tr ω′ 6 CeA(ϕ−infMϕ) which we conjectured to hold ω in [53]. For more details see Section 9. We have included this result here because it follows easily from the argument used in Theorem 1.6 together withthemethodof [53]. Thekey newingredientisthetrick of Phong-Sturm [33] applied to this setting. 2. Preliminaries and comparison with other flows In this section, we include for the reader’s convenience some background material on local coordinate computations with Hermitian metrics. Let (M,g) be a compact Hermitian manifold of complex dimension n. We will often compute in complex coordinates z ,...,z . In this case, g 1 n is determined by the n n Hermitian matrix g = g(∂ ,∂ ), where we are × ij i j writing ∂ ,∂ for ∂ , ∂ respectively. We denote by gji the entries of the i j ∂zi ∂zj inverse matrix of (g ). ij We define the Chern connection associated to g as follows. Given a ∇ vector field X =Xi∂ and a (1,0) form a = a dz , we define X and a to i i i ∇ ∇ EVOLUTION OF A HERMITIAN METRIC BY ITS CHERN-RICCI FORM 7 be the tensors with components: Xk = ∂ Xk +ΓkXj, a = ∂ a Γka , ∇i i ij ∇i j i j − ij k where the Christoffel symbols Γk are given by ij Γk = gqk∂ g . ij i jq The tensors X and a have components Xk = ∂ Xk and a = ∂ a . i i i j i j ∇ ∇ ∇ ∇ The connection can be naturally extended to any kind of tensor, and we ∇ have g = 0. ∇k ij We write ∆ for the complex Laplacian of g, which acts on a function f by ∆f = gji∂ ∂ f = gji f. i j ∇i∇j The torsion of g is the tensor T with components Tk = Γk Γk. ij ij − ji The torsion tensor vanishes in the special case that g is Ka¨hler. We define the curvature of g to be the tensor with components p p R = ∂ Γ . kℓi − ℓ ki We will often raise and lower indices using the metric g, writing for ex- p ample R = g R . Note that R = R . We have the following kℓij pj kℓi kℓij ℓkji commutation formulae: [ , ]Xi = R iXj, [ , ]Xi = R i Xj, ∇k ∇ℓ kℓj ∇k ∇ℓ − kℓ j [ , ]a = R ia , [ , ]a = R i a , ∇k ∇ℓ j − kℓj i ∇k ∇ℓ j kℓ j i where we are writing [ , ] for . We write the Chern-Ricci ∇k ∇ℓ ∇k∇ℓ−∇ℓ∇k curvature of g as the tensor RC given by kℓ RC = gjiR = ∂ ∂ logdetg, kℓ kℓij − k ℓ so that the Chern-Ricci form is equal to Ric(ω)= √ 1RCdz dz . − kℓ k ∧ ℓ It is a closed real (1,1) form and its cohomology class in the Bott-Chern cohomology group closed real (1,1) forms H1,1(M,R) = { } , BC √ 1∂∂ψ,ψ C∞(M,R) { − ∈ } is the first Bott-Chern class of M, and is denoted by cBC(M). It is indepen- 1 dent of the choice of Hermitian metric ω. More generally, if Ω is a smooth positive volume form on M we can define locally Ric(Ω) = √ 1∂∂logΩ, − − which is a global closed real (1,1) form that represents cBC(M). For no- 1 tational convenience, we omit the factor of 2π that usually appears in the definition of cBC(M). The downside of this convention is that some factors 1 of 2π will appear later in the cohomological calculations of Section 6. 8 V.TOSATTIANDB.WEINKOVE We end this section by briefly mentioning some related parabolic equa- tions on Hermitian manifolds which have previously been studied in the literature. Streets-Tian [43] introduced the flow ∂ (2.1) g = S +Q , g = g , ∂t ij − ij ij |t=0 0 where S is given by taking ‘the other trace’ of the curvature of the Chern ij connection: S = gℓkR , ij kℓij and Q is a certain quadratic term in the torsion. If the form ω associated ij 0 to g satisfies ∂∂ω = 0, this equation becomes their pluriclosed flow 0 0 ∂ ∗ ∗ (2.2) ω = ∂∂ ω+∂∂ ω Ric(ω), ω = ω , t=0 0 ∂t − | and if g is Ka¨hler, it coincides with the Ka¨hler-Ricci flow. They analyzed 0 (2.2) in detail in [42, 44] and made a number of conjectures about it, two of whichareanaloguesofourTheorems1.2and1.3. Theyconjecturethattheir flow can be used to study the topology of class VII+ surfaces. In addition, Streets-Tian considered a family of flows of the form (2.1) with arbitrary quadratic torsion term Q and proved, among other results, a short-time existence theorem [43]. The flow (2.1) was extended to the almost complex setting by Vezzoni [58]. Liu-Yang [28] proposestudying the flow (2.1) in the case of Q = 0. In [19], Gill introduced the following parabolic complex Monge-Amp`ere equation on a compact Hermitian manifold (M,gˆ): ∂ det(gˆij +∂i∂jϕ) (2.3) ϕ= log F, gˆ +∂ ∂ ϕ > 0, ϕ =0, ∂t det(gˆ ) − ij i j |t=0 ij for a fixed smooth function F on M. He showed that the unique solution to (2.3)exists foralltimeand,afteranappropriatenormalization, converges in ∞ C to a smooth function ϕ∞ solving the complex Monge-Amp`ere equation det(gˆij +∂i∂jϕ∞) (2.4) log = F +b, det(gˆ ) ij for a constant b which is uniquely determined. The existence of solutions to the elliptic equation (2.4) on Hermitian manifolds (generalizing Yau’s Theorem [60]) was already known by the work of Cherrier [10] (if n = 2) and the authors [54] (n > 2). See also [21, 62]. In the special case where gˆ is Ka¨hler, the flow (2.3) had been considered earlier by Cao [8], who proved the analogous results. In the case when cBC(M) = 0, we can find a function F satisfying 1 ∂∂logdetgˆ= ∂∂F, and with this choice, ω(t)= ωˆ+√ 1∂∂ϕ(t) for ϕ(t) solving (2.3) is exactly − the Chern-Ricci flow starting at ωˆ. In general, the only difference between the Chern-Ricci flow and Gill’s flow (2.3) is that for the Chern-Ricci flow EVOLUTION OF A HERMITIAN METRIC BY ITS CHERN-RICCI FORM 9 we replace the fixed metric gˆ by a smoothly varying family of Hermitian metrics gˆ, and replace F by a particular function, which may also depend t ont. ManyofGill’sestimatescarryovereasilytothecaseoftheChern-Ricci flow and we will make extensive use of them here. ′ Afinalremarkaboutnotation. Inthefollowing, C,C willdenoteuniform positive constants which may vary from line to line. 3. Evolution of the trace of the metric In this section we write down a formula for the evolution of the trace of the evolving metric with respect to a fixed Hermitian metric. We will need this calculation in later sections. We carry out the computation here using tensorial quantities, following [10], rather than using a particular choice of complex coordinates as in [19, 21, 43, 53], for example. We suppose that we have three Hermitian metrics g,g and gˆ such that 0 g = g(t)satisfiestheChern-Ricciflow(1.1),andsuchthatthecorresponding forms satisfy (3.1) ω = ω +η(t), 0 for a closed (1,1) form η(t). We denote by ˆ, Tˆ, Γˆ, Rˆ the Chern connection, torsion, Christoffel sym- ∇ bols and curvature of gˆ. Denote by T the torsion tensor of g and by ∆ the 0 0 complex Laplacian associated to g = g(t). Then we have: Proposition 3.1. The evolution of logtr g is given by gˆ ∂ (3.2) ∆ logtr g = (I)+(II)+(III) gˆ ∂t − (cid:18) (cid:19) where 1 1 (I) = gjpgqigˆℓkˆ g ˆ g + gℓkˆ tr gˆ tr g tr g − ∇k ij∇ℓ pq tr g ∇k gˆ ∇ℓ gˆ gˆ (cid:20) gˆ 2Re gjigˆℓkTˆp ˆ g gjigˆℓkTˆpTˆqg − ki∇ℓ pj − ik jℓ pq (cid:21) (cid:16) (cid:17) 1 (II) = gjigˆℓk(ˆ Tˆq Rˆ gˆqp)g tr g ∇i jℓ− iℓpj kq gˆ (cid:20) (cid:21) 1 (III) = gjigˆℓk ˆ (T )p (g ) + ˆ (T )p (g ) −tr g ∇i 0 jℓ 0 kp ∇ℓ 0 ik 0 pj gˆ (cid:20) (cid:16) (cid:16) (cid:17) (cid:16) (cid:17)(cid:17) gjigˆℓkTˆq(T )p (g ) . − jℓ 0 ik 0 pq (cid:21) Moreover we have 2 (I) 6 Re gˆℓigqk(T )p (g ) ˆ tr g , (tr g)2 0 ki 0 pℓ∇q gˆ gˆ (cid:16) (cid:17) (II) 6 Ctr gˆ, g 10 V.TOSATTIANDB.WEINKOVE for a constant C that depends only on gˆ. If we are at a point where tr g > 1, gˆ then ′ (III) 6 C tr gˆ, g ′ for C depending only on g and gˆ. 0 Proof. First, ∆tr g = gjiˆ ˆ (gˆℓkg ) = gjigˆℓkˆ ˆ g . gˆ ∇i∇j kℓ ∇i∇j kℓ From the definition of covariant derivative, ˆ g = ∂ g Γˆp g , ∇j kℓ j kℓ− jℓ kp and skew-symmetrizing in j,ℓ ˆ g = ˆ g +(∂ω) Tˆpg . ∇j kℓ ∇ℓ kj jkℓ− jℓ kp But from (3.1), ∂ω = ∂ω , and we may rewrite this in terms of the torsion 0 p of g as (∂ω ) = (T ) (g ) . Thus 0 0 jkℓ 0 jℓ 0 kp ˆ ˆ g = ˆ ˆ g + ˆ (T )p (g ) (ˆ Tˆq)g Tˆq ˆ g . ∇i∇j kℓ ∇i∇ℓ kj ∇i 0 jℓ 0 kp − ∇i jℓ kq − jℓ∇i kq (cid:16) (cid:17) Switching covariant derivatives ˆ ˆ g = ˆ ˆ g Rˆ gˆqpg +Rˆ gˆqpg . ∇i∇ℓ kj ∇ℓ∇i kj − iℓkq pj iℓpj kq Arguing as above, ˆ ˆ g = ˆ ˆ g + ˆ (T )p (g ) (ˆ Tˆp)g Tˆp ˆ g . ∇ℓ∇i kj ∇ℓ∇k ij ∇ℓ 0 ik 0 pj − ∇ℓ ik pj − ik∇ℓ pj (cid:16) (cid:17) Combining all of these we have ∆tr g = gjigˆℓkˆ ˆ g +gjigˆℓk ˆ (T )p (g ) gˆ ∇ℓ∇k ij ∇i 0 jℓ 0 kp (cid:18) (cid:16) (cid:17) (3.3) + ˆ (T )p (g ) (ˆ Tˆq Rˆ gˆqp)g ∇ℓ 0 ik 0 pj − ∇i jℓ− iℓpj kq (cid:16) (cid:17) (ˆ Tˆp +Rˆ gˆqp)g Tˆq ˆ g Tˆp ˆ g . − ∇ℓ ik iℓkq pj − jℓ∇i kq − ik∇ℓ pj (cid:19) We will make a change to the second to last term using (3.4) Tˆq ˆ g = Tˆq ˆ g +Tˆq(T )p (g ) TˆpTˆqg . jℓ∇i kq jℓ∇k iq jℓ 0 ik 0 pq − ik jℓ pq On the other hand, ∂ tr g = gˆℓk∂ ∂ logdet(g) = gjigˆℓk∂ ∂ g gjpgqigˆℓk∂ g ∂ g , ∂t gˆ k ℓ k ℓ ij − k ij ℓ pq and we wish to convert the partial derivatives into covariant ones. For this, we use the relations ∂ g = ˆ g +Γˆr g k ij ∇k ij ki rj