Collapsing of the Chern-Ricci flow on elliptic surfaces
aa r X i v : . [ m a t h . DG ] D ec COLLAPSING OF THE CHERN-RICCI FLOW ONELLIPTIC SURFACES ∗ VALENTINO TOSATTI, BEN WEINKOVE, AND XIAOKUI YANG
Abstract.
We investigate the Chern-Ricci flow, an evolution equationof Hermitian metrics generalizing the K¨ahler-Ricci flow, on elliptic bun-dles over a Riemann surface of genus greater than one. We show that,starting at any Gauduchon metric, the flow collapses the elliptic fibersand the metrics converge to the pullback of a K¨ahler-Einstein metricfrom the base. Some of our estimates are new even for the K¨ahler-Ricciflow. A consequence of our result is that, on every minimal non-K¨ahlersurface of Kodaira dimension one, the Chern-Ricci flow converges in thesense of Gromov-Hausdorff to an orbifold K¨ahler-Einstein metric on aRiemann surface. Introduction
The Chern-Ricci flow is an evolution equation for Hermitian metrics oncomplex manifolds. Given a starting Hermitian metric g , which we repre-sent as a real (1 ,
1) form ω = √− g ) ij dz i ∧ dz j , the Chern-Ricci flow isgiven by(1.1) ∂∂t ω = − Ric( ω ) , ω | t =0 = ω , where Ric( ω ) := −√− ∂∂ log det g is the Chern-Ricci form of ω . In the casewhen g is K¨ahler, namely dω = 0, (1.1) coincides with the K¨ahler-Ricciflow.The Chern-Ricci flow was first introduced by Gill [11] in the setting ofmanifolds with c BC1 ( M ) = 0, where c BC1 ( M ) is the first Bott-Chern classgiven by c BC1 ( M ) = [Ric( ω )] ∈ H , ( M, R ) = { closed real (1 , }√− ∂∂ ( C ∞ ( M )) , for any Hermitian metric ω . Making use of an estimate for the complexMonge-Amp`ere equation [6, 35], Gill showed that solutions to (1.1) on man-ifolds with c BC1 ( M ) = 0 exist for all time and converge to Hermitian metricswith vanishing Chern-Ricci form. Gill’s theorem generalizes the convergenceresult of Cao [4] for the K¨ahler-Ricci flow (which made use of estimates ofYau [39]). ∗ Research supported in part by NSF grants DMS-1105373 and DMS-1236969. Thefirst-named author is supported in part by a Sloan Research Fellowship.
The first and second named authors investigated the Chern-Ricci flow onmore general manifolds [36, 37] and proved a number of further results. Itwas shown in particular that the maximal existence time for the flow canbe determined from the initial metric; that the Chern-Ricci flow on man-ifolds with negative first Chern class smoothly deforms Hermitian metricsto K¨ahler-Einstein metrics; that when starting on a complex surface withGauduchon initial metric ω (meaning ∂∂ω = 0), the Chern-Ricci flow ex-ists until the volume of the manifold or a curve of negative self-intersectiongoes to zero; and that on surfaces with nonnegative Kodaira dimension theChern-Ricci flow contracts an exceptional curve when one exists. There areanalogues of all of these results for the K¨ahler-Ricci flow [4, 7, 32, 27].For the purpose of this discussion it will be useful to make reference tothe following condition:( ∗ ) M is a minimal non-K¨ahler complex surface and ω is Gauduchon . Surfaces which satisfy ( ∗ ) are of significant interest as they are not yetcompletely classified. Recall that a surface is minimal if it contains no ( − M satisfying ( ∗ ) fall into one of the following groups: • Kod( M ) = 1. Minimal non-K¨ahler properly elliptic surfaces. • Kod( M ) = 0. Kodaira surfaces. • Kod( M ) = −∞ . Class VII surfaces which have either: ⋄ b ( M ) = 0. Hopf surfaces or Inoue surfaces by [15, 16, 30]. ⋄ b ( M ) = 1. These are classified by [18, 31]. ⋄ b ( M ) >
1. Still unclassified.Here Kod( M ) is the Kodaira dimension of M . The result of Gill [11]shows that when Kod( M ) = 0 the Chern-Ricci flow exists for all time andconverges to a Chern-Ricci flat metric.In [37], explicit examples of solutions to the Chern-Ricci flow were foundon all M with Kod( M ) = 1, for all Inoue surfaces and for a large class ofHopf surfaces. In particular, it was shown that for any M with Kod( M ) = 1there exists an explicit solution ω ( t ) of the Chern-Ricci flow for t ∈ [0 , ∞ )with the property that as t → ∞ the normalized metrics ω ( t ) /t converge inthe sense of Gromov-Hausdorff to ( C, d KE ) where C is a Riemann surfaceand d KE is the distance function induced by an orbifold K¨ahler-Einsteinmetric on C .The main result of this paper is to show that this collapsing behavior onsurfaces of Kodaira dimension one in the examples of [37] actually occurs OLLAPSING OF THE CHERN-RICCI FLOW ON ELLIPTIC SURFACES 3 for every choice of initial starting Gauduchon metric ω . Combined withGill’s theorem, our results mean that the only remaining case (presumablythe most difficult!) under assumption ( ∗ ) is to understand the behavior ofthe Chern-Ricci flow on surfaces of negative Kodaira dimension. We believethat the results of this paper add to the growing body of evidence thatthe Chern-Ricci flow is a natural geometric evolution equation on complexsurfaces, whose behavior reflects the underlying geometry of the manifold.We first consider the case of elliptic bundles over a Riemann surface.Later we will see that this is sufficient to understand the behavior of theflow on all M satisfying ( ∗ ) with Kod( M ) = 1. Suppose that π : M → S is now an elliptic bundle over a compact Riemann surface S of genus atleast 2, with fiber an elliptic curve E . We will denote by E y = π − ( y ) thefiber over a point y ∈ S and by ω flat ,y the unique flat metric on E y in theK¨ahler class [ ω | E y ]. Let ω S be the unique K¨ahler-Einstein metric on S withRic( ω S ) = − ω S and let ω be a Gauduchon metric on M .We consider the normalized Chern-Ricci flow (1.2) ∂∂t ω = − Ric( ω ) − ω, ω | t =0 = ω , starting at ω . With this normalized flow we will see that the volume ofthe base Riemann surface S remains positive and bounded while the ellipticfibers collapse. One could equally well study the unnormalized flow (1.1) on M (so that our main collapsing result would apply to ω ( t ) /t as in [37]) butwe choose this normalization to stay in keeping with the literature on theK¨ahler-Ricci flow [24]. From [36] we know that a smooth solution to (1.2)exists for all time (see Section 2 below for more details). In this paper weprove the following convergence result as t → ∞ . Theorem 1.1.
Let π : M → S be an elliptic bundle over a Riemann surface S of genus at least 2. Let ω ( t ) be a solution of the normalized Chern-Ricciflow (1.2) on M starting at a Gauduchon metric ω . Then as t → ∞ , ω ( t ) → π ∗ ω S , exponentially fast in the C ( M, g ) topology, where ω S is the unique K¨ahler-Einstein metric on S . In particular, the diameter of each elliptic fiber tendsto zero uniformly exponentially fast and ( M, ω ( t )) converges to ( S, ω S ) inthe Gromov-Hausdorff topology.Furthermore, with the notation above, e t ω ( t ) | E y converges to the metric ω flat ,y exponentially fast in the C ( E y , g ) topology, uniformly in y ∈ S . Note that in Theorem 1.1 we do not need to assume that M is non-K¨ahler.On the other hand, we do assume that M is an elliptic bundle, so that thefibers are all isomorphic as elliptic curves. General elliptic surfaces mayhave singular fibers and in such cases, the complex structure of the smoothfibers may vary. However, we will see shortly that this does not arise for thenon-K¨ahler surfaces that are of interest to us. V. TOSATTI, B. WEINKOVE, AND X. YANG
In the case that M is K¨ahler and ω is K¨ahler, then ω ( t ) is a solutionof the normalized K¨ahler-Ricci flow. There are already a number of resultson this, which we now briefly discuss. On a general minimal K¨ahler ellipticsurface, and its higher dimensional analogue, the K¨ahler-Ricci flow was firstinvestigated by Song-Tian [24, 25]. They showed that the flow convergesat the level of potentials to a generalized K¨ahler-Einstein metric on thebase Riemann surface. The generalized K¨ahler-Einstein equation involvesthe Weil-Petersson metric and singular currents. These terms arise because,unlike in our case, the fibration structure on a K¨ahler elliptic surface is notin general locally trivial and may have singular fibers. When the K¨ahlersurface is a genuine elliptic bundle over a Riemann surface of genus largerthan one, the results of Song-Tian give C collapsing of the fibers alongthe K¨ahler-Ricci flow, as well as a uniform scalar curvature bound [26].These convergence results were strengthened by Song-Weinkove [28] andGill [12] in the special case of a product E × S , giving C ∞ convergence ofthe metrics to the pull-back of a K¨ahler-Einstein metric on the base. Fong-Zhang [9], adapting a technique of Gross-Tosatti-Zhang [13] on Calabi-Yaudegenerations, established smooth convergence for the K¨ahler-Ricci flow onmore general elliptic bundles. In particular, the statement of Theorem 1.1 isknown if the initial metric ω is K¨ahler (with the exception of the assertionthat the convergence ω ( t ) → π ∗ ω S is exponential - as far as we know, thisresult is new even in the K¨ahler-Ricci flow case).Of course, we are much more interested in manifolds which do not ad-mit K¨ahler metrics. For non-K¨ahler elliptic surfaces, we make use of thefollowing key fact : Every minimal non-K¨ahler properly elliptic surface is an elliptic bundleor has a finite cover which is an elliptic bundle.
This is well-known from the Kodaira classification (see for example [3,Lemmas 1, 2] or [38, Theorem 7.4]). Then an immediate consequence of ourTheorem 1.1 is that we can identify the Gromov-Hausdorff behavior of theChern-Ricci flow on all minimal non-K¨ahler surfaces of Kodaira dimensionone.
Corollary 1.2.
Let π : M → S be any minimal non-K¨ahler properly ellipticsurface and let ω ( t ) be the solution of the normalized Chern-Ricci flow (1.2)starting at a Gauduchon metric ω . Then ( M, ω ( t )) converges to ( S, d S ) inthe Gromov-Hausdorff topology.Here d S is the distance function induced by an orbifold K¨ahler-Einsteinmetric ω S on S , whose set Z of orbifold points is precisely the image of themultiple fibers of π . Furthermore, ω ( t ) converges to π ∗ ω S in the C ( M, g ) topology, and for any y ∈ S \ Z the metrics e t ω ( t ) | E y converge exponentiallyfast in the C ( E y , g ) topology (and uniformly as y varies in a compact setof S \ Z ) to the flat K¨ahler metric on E y cohomologous to [ ω | E y ] . OLLAPSING OF THE CHERN-RICCI FLOW ON ELLIPTIC SURFACES 5
As mentioned above, explicit examples exhibiting the behavior of Corol-lary 1.2 were given in [37].We now outline the steps we need to establish Theorem 1.1, and pointout some of the difficulties that arise from the non-K¨ahlerity of the metrics.The first parts of the proof follow quite closely the arguments used bySong-Tian [24] for the K¨ahler-Ricci flow. In Section 2, we show that theChern-Ricci flow can be written as parabolic complex Monge-Amp`ere equa-tion(1.3) ∂∂t ϕ = log e t (˜ ω + √− ∂∂ϕ ) Ω − ϕ, ˜ ω + √− ∂∂ϕ > , ϕ (0) = − ρ, where ˜ ω = ˜ ω ( t ) is a family of reference forms (which are metrics for t large)given by ˜ ω = e − t ω flat + (1 − e − t ) ω S , with ω flat = ω + √− ∂∂ρ, where ρ is chosen so that ω flat restricted to the fiber E y is exactly themetric ω flat ,y discussed above. Here Ω is a particular fixed volume formon M with the property that √− ∂∂ log Ω = ω S . If ϕ satisfies (1.3) then ω ( t ) = ˜ ω + √− ∂∂ϕ satisfies the Chern-Ricci flow (1.2).In Section 3 we establish uniform bounds for ϕ and ˙ ϕ , which imply thatthe volume form of the evolving metric ω ( t ) is uniformly equivalent to thevolume form of the reference metric. These follow in the same way as inthe case of the K¨ahler-Ricci flow [24]. In addition, we prove a crucial decayestimate for ϕ ,(1.4) | ϕ | C (1 + t ) e − t , using the argument of [28]. This estimate makes use of the Gauduchonassumption on ω , and in fact is the only place where we use this condition.So far, the torsion terms of ω ( t ) and ˜ ω have not entered the picture.They show up in the next step of obtaining uniform bounds for the metrics ω ( t ). The evolution equation for tr ˜ ω ω , essentially already computed in [36],contains terms involving the torsion and curvature of the reference metrics˜ ω . In Section 4 we prove a technical lemma giving bounds for the torsionand curvature of these metrics. In particular we show: | ˜ T | ˜ g C, | ∂ ˜ T | ˜ g + | ˜ ∇ ˜ T | ˜ g + | g Rm | ˜ g Ce t/ . To deal with these bounds of order e t/ , our idea is to exploit the strongdecay estimate (1.4) on ϕ to control these terms.In Section 5, we evolve the quantity Q = log tr ˜ ω ω − Ae t/ ϕ + 1˜ C + e t/ ϕ , noting that e t/ ϕ is bounded by (1.4). The third term of Q is the “Phong-Sturm term” [20], which was used in [36] to control some torsion termsalong the Chern-Ricci flow. Using the good positive terms arising from theLaplacian landing on e t/ ϕ we can control the bad terms of order e t/ coming V. TOSATTI, B. WEINKOVE, AND X. YANG from the torsion and curvature of ˜ ω . We obtain a uniform bound on Q whichgives the estimate(1.5) C − ˜ ω ω C ˜ ω, namely, that the solution ω is uniformly equivalent to the reference metric˜ ω . We point out that our argument here differs substantially from that ofSong-Tian [24] where they prove first a parabolic Schwarz Lemma, namelyan estimate of the type ω > C − ω S for a uniform C >
0. We were unableto prove this by a similar direct maximum principle argument, because oftroublesome torsion terms arising in the evolution of tr ω ω S . However, westill obtain the estimate ω > C − ω S once we have (1.5).The next step is to improve the bound (1.5) to the stronger exponentialconvergence result(1.6) (1 − Ce − εt )˜ ω ω (1 + Ce − εt )˜ ω, for ε >
0. To our knowledge, this estimate is new even for the K¨ahler-Ricciflow on elliptic bundles. The idea is to evolve the quantity Q = e εt (tr ω ˜ ω − − e δt ϕ, for a carefully chosen δ > / ε and again exploit the decay estimate(1.4). Showing that Q is bounded from above then gives the estimatetr ω ˜ ω − Ce − εt , and a similar argument gives the same inequality with tr ω ˜ ω replaced bytr ˜ ω ω . Combining these two estimates gives (1.6).However, in order to apply the maximum principle to Q we first requirean exponential decay estimate for ˙ ϕ . To prove this, we observe that theevolution equation for ˙ ϕ is ∂∂t ˙ ϕ = − R − − ˙ ϕ, where R is the Chern scalar curvature of g . If we had a uniform boundfor the Chern scalar curvature, an exponential decay estimate for ˙ ϕ wouldfollow from this evolution equation and the decay estimate for ϕ . However,we are only able to prove the weaker estimate(1.7) − C R Ce t/ . Nevertheless, this suffices since the coefficient of t in the exponent is strictlyless than 1. The bound (1.7) is the content of Section 6. The factor e t/ arises from the bounds on the curvature and torsion of the reference metricswe obtained in Section 4.The idea for bounding the Chern scalar curvature from above (the lowerbound is easy) is to consider the quantity u = ϕ + ˙ ϕ and bound from above − ∆ u = R + tr ω ω S > R . Using an idea that goes back to Cheng-Yau [5],and is used in the context of the K¨ahler-Ricci flow on Fano manifolds byPerelman (see Sesum-Tian [22]) and on elliptic surfaces by Song-Tian [24], OLLAPSING OF THE CHERN-RICCI FLOW ON ELLIPTIC SURFACES 7 we first bound the gradient of u by considering the quantity |∇ u | g / ( A − u ) fora fixed large A . We then evolve the quantity − ∆ u + 6 |∇ u | g , which is almostenough to obtain the estimate we need. There are some bad terms which wecan control by adding large multiples of tr ω ω S and tr ˜ ω ω . We already knowthat these terms are bounded from (1.5). Putting this together gives theupper bound on scalar curvature and the exponential decay estimate for ˙ ϕ that we require.Now that we have this exponential decay estimate on ˙ ϕ , we carry out inSection 7 the argument mentioned above for the exponential convergence ofthe metrics (1.6).In Section 8, we prove a local Calabi type estimate(1.8) | ˆ ∇ g | g Ce t/ , where ˆ ∇ is the connection associated to a local semi-flat product K¨ahlermetric defined in a neighborhood U . Note that if we had the better estimate | ˆ ∇ g | g C (as in [28, 12, 9] for example) then we could immediately concludethe global convergence of the metrics ω ( t ) to π ∗ ω S from the estimates (1.4)and (1.5) and the Ascoli-Arzel`a Theorem. We do not know whether thisstronger estimate | ˆ ∇ g | g C holds or not.To establish (1.8), we use some arguments and calculations similar to thelocal Calabi estimate in [23]. However, a key difference here is that themetrics are collapsing in the fiber directions and we need to take accountof the error terms that arise in this way. The local Calabi estimate is thenused to establish the last part of Theorem 1.1 that e t ω ( t ) | E y converges to ω flat ,y exponentially fast in the C ( E y , g ) topology, uniformly in y ∈ S .In Section 9 we complete the proofs of Theorem 1.1 (this essentially followsimmediately) and Corollary 1.2.2. Preliminaries
Hermitian geometry and notation.
We begin with a brief recapof Hermitian geometry and the Chern connection (for more details see forexample [36]).Given a Hermitian metric g , we write ω = √− g ij dz i ∧ dz j for its asso-ciated (1 ,
1) form, which we will also refer to as a metric. Write ∇ for itsChern connection, with respect to which g and the complex structure are co-variantly constant. The Christoffel symbols of ∇ are given by Γ kij = g kq ∂ i g jq .For example, if X = X i ∂ i is a vector field then its covariant derivative hascomponents ∇ i X ℓ = ∂ i X ℓ + Γ ℓij X j .The torsion tensor of g has components T kij = Γ kij − Γ kji . We will oftenlower an index using the metric g , writing T ijℓ = g kℓ T kij = ∂ i g jℓ − ∂ j g iℓ . Note that T ijℓ = T ′ ijℓ if g and g ′ are Hermitian metrics whose (1 ,
1) forms ω and ω ′ differ by a closed form. The Chern curvature of g is defined to be V. TOSATTI, B. WEINKOVE, AND X. YANG R pkℓi = − ∂ ℓ Γ pki , and we will raise and lower indices using the metric g . Wehave the usual commutation formulae involving the curvature, such as[ ∇ k , ∇ ℓ ] X i = R ikℓj X j . Define the Chern-Ricci curvature of g to be R kℓ = g ij R kℓij = − ∂ k ∂ ℓ log det g ,and we write Ric( ω ) = √− R kℓ dz k ∧ dz ℓ for the associated Chern-Ricci form, a real closed (1,1) form. Write R = g kℓ R kℓ for the Chern scalar curvature.We write ∆ for the complex Laplacian of g , which acts on a function f by ∆ f = g ij ∂ i ∂ j f . For functions f , f , we define h∇ f , ∇ f i g = g ij ∂ i f ∂ j f and |∇ f | g = h∇ f, ∇ f i g . If α = √− α ij dz i ∧ dz j is a real (1 ,
1) form and ω a Hermitian metric we write tr ω α for g ij α ij .A final remark about notation: we will write C, C ′ , C , . . . etc. for auniform constant, which may differ from line to line.2.2. Elliptic bundles and semi-flat metrics.
We now specialize to thesetting of Theorem 1.1. Let π : M → S be an elliptic bundle over a compactRiemann surface S of genus at least 2, with fiber an elliptic curve E . Clearly π : M → S is relatively minimal, because there is no ( − E y = π − ( y ) the fiber over a point y ∈ S .Let ω S be the unique K¨ahler metric on S with Ric( ω S ) = − ω S , let ω be aGauduchon metric on M .Since each fiber E y = π − ( y ) is a torus, we can find a function ρ y on E y with ω | E y + √− ∂∂ρ y = ω flat ,y , the unique flat metric on E y in the K¨ahler class [ ω | E y ]. Furthermore we cannormalize the functions ρ y by R E y ρ y ω = 0, so that they vary smoothly in y (in general this follows from Yau’s estimates [39], although in this simplecase it can also be proved directly, see also [8, Lemma 2.1]), and they definea smooth function ρ on M . We then let(2.1) ω flat = ω + √− ∂∂ρ.ω flat is a semi-flat form, in the sense that it restricts to a flat metric oneach fiber E y , but in general it is not positive definite on M . But note that ω flat ∧ π ∗ ω S is a strictly positive smooth volume form on M .2.3. The canonical bundle and long time existence for the flow.
In the same setting as above, we claim that K M = π ∗ K S . To see this,start from Kodaira’s canonical bundle formula for relatively minimal ellipticsurfaces without singular fibers [1, Theorem V.12.1] K M = π ∗ ( K S ⊗ L ) , OLLAPSING OF THE CHERN-RICCI FLOW ON ELLIPTIC SURFACES 9 where L is the dual of R π ∗ O M . But since M is an elliptic bundle, it followsthat the line bundle R π ∗ O M is trivial (see e.g. [2, Proposition 2.1]), andthe claim follows.Therefore c ( M ) = π ∗ c ( S ) (an alternative more direct proof of this factis contained in Lemma 3.2), and so there exists a unique volume form Ωwith(2.2) Ric(Ω) = − ω S and Z M Ω = 2 Z M ω ∧ ω S . Here and henceforth, we are abbreviating π ∗ ω S by ω S , and for any smoothpositive volume form Ω we write Ric(Ω) for the globally defined real (1 , −√− ∂∂ log Ω.It follows that the Bott-Chern class of the canonical bundle K M , whichequals c BC1 ( K M ) = − c BC1 ( M ) , is nef . In general this means that given any ε > f ε on M such that − Ric( ω ) + √− ∂∂f ε > − εω . Equivalently, this can be phrased by saying that forany ε > h ε on the fibers of K M withcurvature form bigger than − εω . The maximal existence theorem for theChern-Ricci flow [36, Theorem 1.2] has the following immediate corollary:
Theorem 2.1.
Let ( M, ω ) be any compact Hermitian manifold. Then theChern-Ricci flow (2.3) ∂∂t ω = − Ric( ω ) , ω | t =0 = ω , has a smooth solution defined for all t > if and only if the first Bott-Chernclass c BC1 ( K M ) is nef. The exact same statement holds for the normalizedChern-Ricci flow (1.2) . Since this theorem was not stated explicitly in [36], we provide the simpleproof.
Proof.
An elementary space-time scaling argument [14] allows one to trans-form a solution of (1.2) into a solution of (2.3) and vice versa, and oneexists for all positive time if and only if the other one does, so it is enoughto consider (2.3).In this case, we know from [36] that as long as a solution ω ( t ) exists, it isof the form ω ( t ) = ω − t Ric( ω ) + √− ∂∂ϕ ( t ) , and therefore − Ric( ω ) + √− ∂∂ (cid:16) ϕt (cid:17) > − t ω , so if the solution exists for all t >
0, then we see that c BC1 ( K M ) is nef.Conversely, if c BC1 ( K M ) is nef, then for every given t > f t with − Ric( ω ) + √− ∂∂f t > − t ω , which is equivalent to ω − t Ric( ω ) + √− ∂∂ ( tf t ) > , and so the flow exists at least on [0 , t ) by [36, Theorem 1.2]. (cid:3) Applying this to the setting of Theorem 1.1, we obtain a smooth solution ω ( t ) to the normalized Chern-Ricci flow (1.2) for t ∈ [0 , ∞ ).2.4. The parabolic complex Monge-Amp`ere equation.
From now on,until we get to Section 9, we assume we are in the setting of Theorem 1.1.We will rewrite the normalized Chern-Ricci flow (1.2) as a parabolic complexMonge-Amp`ere equation. Define reference (1 , ω = ˜ ω ( t ) by˜ ω = e − t ω flat + (1 − e − t ) ω S , where we recall that ω flat is defined by (2.1). Note that ˜ ω may not necessarilybe positive definite for all t , but there exists a time T I such that ˜ ω > t > T I . (On the other hand, observe that ˜ ω − e − t √− ∂∂ρ is positivedefinite for all t > T I now once and for all. By thelong time existence result of [36], we have uniform C ∞ estimates on ω ( t )for t ∈ [0 , T I ]. Our goal is to obtain estimates on ω ( t ) for t > T I which areindependent of t .Define a function ϕ ( t ) by ∂∂t ϕ = log e t ω ( t ) Ω − ϕ, ϕ (0) = − ρ, where we recall that ω flat = ω + √− ∂∂ρ and Ω is given by (2.2). We claimthat ω ( t ) = ˜ ω + √− ∂∂ϕ ( t ) holds. Indeed, ∂∂t ˜ ω = ω S − ˜ ω = − Ric(Ω) − ˜ ω, and so, ∂∂t ( e t ( ω − ˜ ω − √− ∂∂ϕ )) = 0 , ( e t ( ω − ˜ ω − √− ∂∂ϕ )) | t =0 = 0 , which implies that indeed ω = ˜ ω + √− ∂∂ϕ. Therefore ϕ also satisfies thePDE(2.4) ∂∂t ϕ = log e t (˜ ω + √− ∂∂ϕ ) Ω − ϕ, ˜ ω + √− ∂∂ϕ > , ϕ (0) = − ρ, and conversely every solution of (2.4) gives rise to a solution ω = ˜ ω + √− ∂∂ϕ of the normalized Chern-Ricci flow (1.2).3. Estimates on the potential and its time derivative
We now begin the proof of Theorem 1.1. We assume π : M → S isan elliptic bundle over a Riemann surface S of genus at least 2 and ω ( t )is a solution of the normalized Chern-Ricci flow (1.2) on M starting at aGauduchon metric ω . OLLAPSING OF THE CHERN-RICCI FLOW ON ELLIPTIC SURFACES 11
In this section we collect some estimates on the potential function ϕ solv-ing (2.4), and its time derivative ˙ ϕ := ∂ϕ/∂t . The proofs of these resultsare almost identical to the corresponding results for the K¨ahler-Ricci flow[24] (see also [28]). For the reader’s convenience we include here the briefarguments. We also point out the one place where we use the Gauduchoncondition. Lemma 3.1.
There exists a uniform positive constant C such that on M , (i) | ϕ ( t ) | C for all t > . (ii) | ˙ ϕ ( t ) | C for all t > . (iii) 1 C ˜ ω ω C ˜ ω for all t > T I .Proof. We follow the exposition in [28]. Since e t ˜ ω = e − t ω + 2(1 − e − t ) ω flat ∧ ω S , we have for t > T I ,(3.1) 1 C Ω e t ˜ ω C Ω . If ϕ attains a minimum at a point ( x , t ) with t > T I , then at that point0 > ∂∂t ϕ = log e t (˜ ω + √− ∂∂ϕ ) Ω − ϕ > log e t ˜ ω Ω − ϕ > − log C − ϕ, giving ϕ > − log C and hence a uniform lower bound for ϕ . The upperbound for ϕ is similar. This gives (i).For (ii), we first compute:(3.2) (cid:18) ∂∂t − ∆ (cid:19) ˙ ϕ = tr ω ( ω S − ˜ ω ) + 1 − ˙ ϕ. On the other hand, there exists a uniform constant C > C ˜ ω >ω S for t > T I . We apply the maximum principle to Q = ˙ ϕ − ( C − ϕ .Calculate for t > T I , (cid:18) ∂∂t − ∆ (cid:19) Q = tr ω ( ω S − ˜ ω ) + 1 − C ˙ ϕ + ( C − ω ( ω − ˜ ω ) − C ˙ ϕ + 2( C − , and the maximum principle shows that Q is bounded from above uniformly.This gives the upper bound for ˙ ϕ .Next consider Q = ˙ ϕ + 2 ϕ and compute (cid:18) ∂∂t − ∆ (cid:19) Q = tr ω ( ω S − ˜ ω ) + 1 + ˙ ϕ − ω ( ω − ˜ ω ) > tr ω ˜ ω + ˙ ϕ − . By the geometric-arithmetic means inequality, we have for t > T I ,(3.3) e − ˙ ϕ + ϕ = (cid:18) Ω e t ω (cid:19) C (cid:18) ˜ ω ω (cid:19) C ω ˜ ω. Then at a point ( x , t ) with t > T I where Q attains a minimum, tr ω ˜ ω − ˙ ϕ and so e − ˙ ϕ + ϕ C (3 − ˙ ϕ ) , which gives a uniform lower bound for ˙ ϕ .This completes the proof of (ii).Part (iii) follows from (i) and (ii) and the equations (2.4) and (3.1). (cid:3) Our next result is an exponential decay estimate for ϕ . We first need alemma. Recall that the volume form Ω is defined by (2.2). This lemma is theonly place in the paper where we make use of the Gauduchon assumptionon ω . Lemma 3.2.
We have that (3.4) Ω = 2 ω flat ∧ ω S . Proof.
Since 2 R M ω flat ∧ ω S = 2 R M ω ∧ ω S = R M Ω, it is enough to showthat(3.5) √− ∂∂ log Ω ω flat ∧ ω S = 0 . Recalling that M is an elliptic bundle with fiber E , we can fix a small ball B ⊂ S over which π is holomorphically trivial, so π − ( B ) ∼ = B × E . If weidentify E = C / Λ, for some lattice Λ ⊂ C , and call z the coordinate on C , then dz descends to a never vanishing holomorphic 1-form on E . If wecall α its pullback to B × E , then √− α ∧ α is a smooth semi-flat form on π − ( U ). Then there is a function u ( y ) defined on B such that for any y ∈ B we have ω flat | E y = u ( y ) √− α ∧ α. This is because both ω flat | E y and √− α ∧ α are flat volume forms on E y ,and so their ratio is a constant on E y . Integrating this equality over E y weget Z E y ω flat = u ( y ) Z E y √− α ∧ α. But on the one hand the integral R E y √− α ∧ α is independent of y bydefinition, and on the other hand the function y R E y ω flat is also constantin y , because it equals the pushforward π ∗ ω flat and we have ∂∂π ∗ ω flat = π ∗ ∂∂ω flat = π ∗ ∂∂ω = 0 . Note that the last equality uses the Gauduchon condition. This impliesthat π ∗ ω flat is constant by the strong maximum principle. Therefore u is aconstant.Fix now a point x ∈ M , call y = π ( x ), and choose local bundle coordinatesnear x and y , so that in these coordinates the projection π is given by π ( z , z ) = z . Then write locally ω S = √− g ( z ) dz ∧ dz , Ω = G ( z , z )( √− dz ∧ dz ∧ dz ∧ dz , OLLAPSING OF THE CHERN-RICCI FLOW ON ELLIPTIC SURFACES 13 and compute(3.6) F := Ω ω flat ∧ ω S = Ω u √− α ∧ α ∧ ω S = G ( ug ) − , and so locally on S we have(3.7) √− ∂∂ log F = ω S + Ric( ω S ) = 0 , because Ric(Ω) = − ω S = Ric( ω S ). This proves (3.5).Incidentally the same calculation proves that the volume form ω flat ∧ ω S satisfies Ric( ω flat ∧ ω S ) = − ω S , which gives another proof of the fact that c ( M ) = π ∗ c ( S ). (cid:3) Very similar arguments can be found in the paper of Song-Tian [24], inthe K¨ahler case (see also [25, 34]).
Remark 3.3.
Lemma 3.2 fails if we drop the assumption that ∂∂ω = 0.Indeed, consider the case when M = S × E , and let ω E be a flat K¨ahlermetric on E (while ω S is as before). If F : S → R is any nonconstantpositive function then ω = F ω E + ω S , is a Hermitian metric on M with ∂∂ω = 0. We have that ω = ω flat ,because ω is already semi-flat. On the other hand, Ω = c · ω E ∧ ω S , where c is the constant given by c = 2 R M ω ∧ ω S R M ω E ∧ ω S . Therefore we have Ω = 2 ω flat ∧ ω S = F ω E ∧ ω S . We can now prove a decay estimate for ϕ . The analogous estimate wasproved for the K¨ahler-Ricci flow on a product surface in [28, Lemma 6.7].The proof in this case is almost identical, given Lemma 3.2. Lemma 3.4.
There exists a uniform constant
C > such that on M × [0 , ∞ ) , | ϕ | C (1 + t ) e − t . Proof.
First, we claim that for t > T I ,(3.8) (cid:12)(cid:12)(cid:12)(cid:12) e t log e t ˜ ω Ω (cid:12)(cid:12)(cid:12)(cid:12) C. This follows from the argument in [28, Lemma 6.7]. Indeed, using Lemma3.2, we see that e t log e t ˜ ω Ω = e t log 2 ω S ∧ ω flat + e − t ( ω − ω flat ∧ ω S )2 ω S ∧ ω flat = e t log(1 + O ( e − t )) , which is bounded.Define now Q = e t ϕ + At , for A a large positive constant to be determined.Then(3.9) ∂Q∂t = e t log (cid:18) e t (˜ ω + √− ∂∂ϕ ) Ω (cid:19) + A. We wish to bound Q from below. Suppose that ( x , t ) is a point with t > T I at which Q achieves a minimum. At this point we have0 > ∂Q∂t > e t log e t ˜ ω Ω + A > − C ′ + A, for a uniform C ′ , thanks to (3.8). Choosing A > C ′ gives contradiction.Hence Q is bounded from below and it follows that ϕ > − C (1 + t ) e − t for auniform C . The upper bound for ϕ is similar. (cid:3) Torsion and curvature of the reference metrics
This section is devoted to proving a technical lemma on estimates forthe torsion and curvature of the reference metrics. These estimates will beneeded later in Sections 5 and 8.Recall that the reference forms ˜ ω = ˜ ω ( t ) are given by˜ ω = e − t ω flat + (1 − e − t ) ω S . For t > T I , this defines a Hermitian metric which we denote by ˜ g . We willuse a tilde to denote quantities with respect to ˜ g , such as ˜ T kij for the torsiontensor, ˜ ∇ for the Chern connection and g Rm for the Chern curvature tensor.We will write ∂ ˜ T for the tensor ∂ ℓ ˜ T ijk = ˜ ∇ ℓ ˜ T ijk .Denote by ( T ) kij the torsion tensor of the initial metric g , and T ijℓ =( T ) kij ( g ) kℓ . Then since d ˜ ω = e − t dω , we have ˜ T ijℓ = e − t T ijℓ . Lemma 4.1.
There exists a uniform constant C such that for t > T I , (i) | ˜ T | ˜ g C . (ii) | ∂ ˜ T | ˜ g + | ˜ ∇ ˜ T | ˜ g + | g Rm | ˜ g Ce t/ . (iii) | ˜ ∇ ∂ ˜ T | g + | ˜ ∇ ∂ ˜ T | g Ce t .Proof. We may choose local product holomorphic coordinates z , z , inde-pendent of t , with z in the fiber direction and z in the base direction. Since g flat is flat in the z direction, we may assume that derivatives of ( g flat ) in the z direction vanish. Now with respect to these coordinates, we maywrite ˜ g = e − t ( g flat ) , ˜ g = e − t ( g flat ) (4.1) ˜ g = e − t ( g flat ) , ˜ g = e − t ( g flat ) + (1 − e − t )( g S ) , (4.2) OLLAPSING OF THE CHERN-RICCI FLOW ON ELLIPTIC SURFACES 15 where we are writing g S for π ∗ g S . Then a straightforward computationshows that there exists a uniform constant C > e − t C ˜ g Ce − t , e t C ˜ g Ce t C ˜ g C, C ˜ g C | ˜ g | Ce − t | ˜ g | C. Then | ˜ T | g = ˜ g ij ˜ g kℓ ˜ g pq ˜ T ikq ˜ T jℓp = e − t ˜ g ij ˜ g kℓ ˜ g pq T ikq T jℓp . This is uniformly bounded since the only unbounded terms of type ˜ g ij arethe terms ˜ g , but by the skew-symmetry of T ijℓ in i, j , there can be at mosttwo such terms in the above expression, and each is bounded above by Ce t (since the components T ijk are all uniformly bounded in the holomorphiccoordinates z , z ). This completes the proof of (i).For (ii), note that we may choose the coordinates z , z as above with theadditional property that at a fixed point x say, the derivative ∂ g S vanishes.This implies that at x we have ∂ i ˜ g jℓ = e − t ∂ i ( g flat ) jℓ for all i, j, ℓ . Notethat since our coordinates are independent of t and depend continuously onthe point x ∈ M , we may allow our constants to depend on this choice ofcoordinate system.We first claim that at x ,(4.3) | ˜Γ pik | g := ˜ g ij ˜ g kℓ ˜ g pq ˜Γ pik ˜Γ qjℓ Ce t , where ˜Γ pik are the Christoffel symbols of the Chern connection of ˜ g . Note thatsince ˜Γ pik is not a tensor, this quantity depends on our choice of coordinates.For (4.3) compute, | ˜Γ pik | g = ˜ g ij ˜ g kℓ ˜ g pq ∂ i ˜ g kq ∂ j ˜ g pℓ = e − t ˜ g ij ˜ g kℓ ˜ g pq ∂ i ( g flat ) kq ∂ j ( g flat ) pℓ . But this is bounded from above by Ce t since each term of type ˜ g ij is boundedfrom above by Ce t .Next note that, at x ,(4.4) | ∂ ℓ T ijk | g Ce t . Indeed this follows from the skew symmetry of ∂ ℓ T ijk (again, not a tensor)in the indices i and j . Then at x , | ∂ ˜ T | g = e − t | ˜ ∇ ℓ T ijk | g = e − t | ∂ ℓ T ijk − ˜Γ rℓk T ijr | g e − t | ∂ ℓ T ijk | g + 2 e − t | ˜Γ rℓk | g | T ijr | g Ce t (4.5)where the last inequality follows from (4.3), (4.4) and the fact that | T ijr | g Ce t . The bound on | ˜ ∇ ˜ T | ˜ g is completely analogous (again we compute at x ): | ˜ ∇ ˜ T | g = e − t | ˜ ∇ ℓ T ijk | g = e − t | ∂ ℓ T ijk − ˜Γ rℓi T rjk − ˜Γ rℓj T irk | g e − t | ∂ ℓ T ijk | g + 4 e − t | ˜Γ rℓk | g | T ijk | g Ce t . (4.6)For the bound on the curvature ˜ R ijkℓ of ˜ g , we first compute in our coor-dinates,(4.7) | ˜ R | Ce − t , where by | · | we mean the absolute value as a complex (or real) number.Recall that the Chern curvature of ˜ g is given by˜ R ijkℓ = − ∂ i ∂ j ˜ g kℓ + ˜ g pq ∂ i ˜ g kq ∂ j ˜ g pℓ . Hence from (4.1) and (4.2) and the fact that ∂ ∂ ( g flat ) and ∂ ( g flat ) vanish in our coordinate system,˜ R = − ∂ ∂ ˜ g + ˜ g pq ∂ ˜ g q ∂ ˜ g p = e − t X ( p,q ) =(1 , ˜ g pq ∂ ( g flat ) q ∂ ( g flat ) p . but since each term ˜ g pq for ( p, q ) = (1 ,
1) is uniformly bounded, this gives(4.7).Next we show that(4.8) | ˜ R | C. For this note that | ∂ ∂ ˜ g | C and X p,q ˜ g pq ∂ ˜ g q ∂ ˜ g p = ˜ g ∂ ˜ g ∂ ˜ g + X ( p,q ) =(1 , ˜ g pq ∂ ˜ g q ∂ ˜ g p , but the first term is of order O ( e − t ) and the second is uniformly bounded.This proves (4.8).Finally, we show that(4.9) | ˜ R ijkℓ | Ce − t , for ( i, j, k, ℓ ) not all equal . To see this observe that, for i, j, k, ℓ not all equal, we have(4.10) | ∂ i ∂ j ˜ g kℓ | Ce − t . Indeed, this follows immediately from (4.1) and (4.2) unless k = ℓ = 2. Butthen one of i or j must equal 1 and we use the fact that ∂ ( g S ) = 0.Moreover, for i, j, k, ℓ not all equal, we claim:(4.11) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X p,q ˜ g pq ∂ i ˜ g kq ∂ j ˜ g pℓ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Ce − t . OLLAPSING OF THE CHERN-RICCI FLOW ON ELLIPTIC SURFACES 17
Indeed, first assume that neither k nor ℓ is equal to 2. Then | ˜ g pq | Ce t , | ∂ i ˜ g kq | Ce − t and | ∂ j ˜ g pℓ | Ce − t , and so | ˜ g pq ∂ i ˜ g kq ∂ j ˜ g pℓ | Ce − t . Next suppose that k = 2 and ℓ = 1. Then X p,q ˜ g pq ∂ i ˜ g q ∂ j ˜ g p = X p ˜ g p ∂ i ˜ g ∂ j ˜ g p + X p ˜ g p ∂ i ˜ g ∂ j ˜ g p . The first term on the right hand side is of order O ( e − t ) by the same argumentas above, and the second is of order O ( e − t ) since | ˜ g p | C , | ∂ i ˜ g | C and | ∂ j ˜ g p | Ce − t . This proves (4.11) if k = 2 and ℓ = 1 and the case k = 1, ℓ = 2 is similar.Finally we deal with the case k = ℓ = 2. We wish to bound X p,q ˜ g pq ∂ i ˜ g q ∂ j ˜ g p . If one of p or q is equal to 1 then the summand is of order O ( e − t ) by similararguments to the ones given above. Otherwise the summand is˜ g ∂ i ˜ g ∂ j ˜ g , and then we use that fact that one of i or j must be 1, since we are assumingthat i, j, k, ℓ are not all equal. But | ∂ ˜ g | Ce − t and so the summand isof order O ( e − t ). This completes the proof of (4.11).Combining (4.10) and (4.11) gives (4.9).To complete the proof of (ii), we note that | g Rm | g = ˜ g iq ˜ g pj ˜ g ks ˜ g rℓ ˜ R ijkℓ ˜ R qpsr . Recall that ˜ g ab is bounded by C if ( a, b ) = (1 ,
1) and by Ce t if a = b = 1.When i = j = k = ℓ = p = q = r = s = 1, then we apply (4.7) to see thatthe summand is bounded by C . We get the same bound if i = j = k = ℓ = 2or p = q = r = s = 2 by applying (4.7), (4.8) and (4.9). Otherwise, bothof the terms ˜ R ijkℓ and ˜ R qpsr are bounded by Ce − t (or better) by (4.7) and(4.9) (because the term ˜ R does not appear). Moreover, at least one ofthe metric terms ˜ g − is bounded by a uniform constant C , while the otherthree are each bounded from above by Ce t . Thus in every case we obtain | g Rm | g Ce t . Combining this with (4.5) and (4.6) gives (ii).For (iii), compute˜ ∇ p ˜ ∇ q ˜ T ijk = ˜ ∇ p ( ∂ q ˜ T ijk − ˜Γ ℓqk ˜ T ijℓ )= ∂ p ∂ q ˜ T ijk − ˜ T ijℓ ∂ p ˜Γ ℓqk − ˜Γ ℓqk ∂ p ˜ T ijℓ − ˜Γ ℓpq ∂ ℓ ˜ T ijk − ˜Γ ℓpk ∂ q ˜ T ijℓ + ˜Γ rpq ˜Γ ℓrk ˜ T ijℓ + ˜Γ rpk ˜Γ ℓqr ˜ T ijℓ . For the first term we observe that(4.12) | ∂ p ∂ q ˜ T ijk | g = e − t | ∂ p ∂ q T ijk | g Ce t , because of the skew symmetry in i, j . And, as in the proof of (ii),(4.13) | ∂ p ˜ T ijℓ | g = e − t | ∂ p T ijk | g Ce t . We claim that(4.14) | ∂ p ˜Γ ijk | g := ˜ g pq ˜ g ja ˜ g kb ˜ g ic ∂ p ˜Γ ijk ∂ q ˜Γ cab Ce t . To see this note that at x we have ∂ p ˜Γ ijk = ˜ g iℓ ∂ p ∂ j ˜ g kℓ − ˜ g is ˜ g rℓ ∂ p ˜ g rs ∂ j ˜ g kℓ = e − t ˜ g iℓ ∂ p ∂ j ( g flat ) kℓ + (1 − e − t )˜ g iℓ ∂ p ∂ j ( g S ) kℓ − e − t ˜ g is ˜ g rℓ ∂ p ( g flat ) rs ∂ j ( g flat ) kℓ , and so (with the obvious notation) | ∂ p ˜Γ ijk | g C (cid:0) e − t | ∂ p ∂ j ( g flat ) kℓ | g + | ∂ p ∂ j ( g S ) kℓ | g + e − t | ∂ p ( g flat ) rs ∂ j ( g flat ) kℓ | g (cid:1) . But the second term equals C | ∂ ∂ ( g S ) | | ˜ g | , and so is bounded by C ,while the other two terms are bounded by Ce t because each term of type˜ g ij is bounded above by Ce t . This establishes the claim (4.14).Combining (4.3), (4.12), (4.13), (4.14) and parts (i) and (ii) we haveproved the bound | ˜ ∇ ∂ ˜ T | g Ce t .Finally, calculate˜ ∇ p ˜ ∇ q ˜ T ijk = ˜ ∇ p ( ∂ q ˜ T ijk − ˜Γ ℓqk ˜ T ijℓ )= ∂ p ∂ q ˜ T ijk − ˜ T ijℓ ∂ p ˜Γ ℓqk − ˜Γ ℓqk ∂ p ˜ T ijℓ − ˜Γ ℓpi ∂ q ˜ T ℓjk − ˜Γ ℓpj ∂ q ˜ T iℓk + ˜Γ rpi ˜Γ ℓqk ˜ T rjℓ + ˜Γ rpj ˜Γ ℓqk ˜ T irℓ . Arguing as above we can bound the | · | ˜ g of all these terms by Ce t . Indeed,the only terms which are different are the one involving ∂ p ˜Γ ℓqk = − ˜ R qpkℓ ,which can be bounded using part (ii) and the one involving ∂ p ˜ T ijℓ which canbe bounded by the same argument as in (4.13). This finishes the proof of(iii). (cid:3) Evolution of the trace of the metric
Let ω = ω ( t ) solve the normalized Chern-Ricci flow (1.2) in the setting ofTheorem 1.1. The main theorem we prove in this section is: Theorem 5.1.
There exists a uniform constant
C > such that for t > T I , tr ˜ ω ω C. Hence the metrics ω and ˜ ω are uniformly equivalent for t > T I . OLLAPSING OF THE CHERN-RICCI FLOW ON ELLIPTIC SURFACES 19
For the last assertion, note that we have(5.1) tr ω ˜ ω = ˜ ω ω tr ˜ ω ω, and the uniform equivalence of the volume forms ω and ˜ ω (Lemma 3.1).Then an upper bound for tr ˜ ω ω is equivalent to an upper bound for tr ω ˜ ω andhence also the uniform equivalence of ω and ˜ ω .In [28] a similar estimate is proved using a direct maximum principleargument and a bound for the potential ϕ . Here, as discussed in the Intro-duction, there are new unbounded terms arising from the torsion. We willcontrol these terms using the exponential decay estimate for ϕ (Lemma 3.4).First we need the following lemma. Lemma 5.2.
For t > T I , the following evolution inequality holds: (5.2) (cid:18) ∂∂t − ∆ (cid:19) log tr ˜ ω ω ˜ ω ω ) Re (cid:16) ˜ g iℓ g kq ˜ T kiℓ ∂ q tr ˜ ω ω (cid:17) + Ce t/ tr ω ˜ ω. Proof.
From [36, Proposition 3.1] we have (cid:18) ∂∂t − ∆ (cid:19) log tr ˜ ω ω = 1tr ˜ ω ω (cid:18) − g pj g iq ˜ g kℓ ˜ ∇ k g ij ˜ ∇ ℓ g pq + 1tr ˜ ω ω g kℓ ∂ k tr ˜ ω ω∂ ℓ tr ˜ ω ω − (cid:16) g ij ˜ g kℓ ˜ T pki ˜ ∇ ℓ g pj (cid:17) − g ij ˜ g kℓ ˜ T pik ˜ T qjℓ g pq + g ij ˜ g kℓ ( ˜ ∇ i ˜ T qjℓ − ˜ R iℓpj ˜ g pq ) g kq − g ij ˜ ∇ i ˜ T ℓjℓ − g ij ˜ g kℓ ˜ g pj ˜ ∇ ℓ ˜ T pik + g ij ˜ g kℓ ˜ T pik ˜ T qjℓ ˜ g pq − tr ˜ ω ω − ˜ g iℓ ˜ g kj g ij ∂∂t ˜ g kℓ (cid:19) . Note that there are some differences from the computation in [36] since herewe are evolving ω by the normalized Chern-Ricci flow, and our referencemetrics ˜ ω depend on time. In particular here we have T ijk = ˜ T ijk (insteadof T ijk = ( T ) ijk in [36]). Also, the last two terms above are new: the firstarising from the − ω term on the right hand side of (1.2) and the secondfrom the time derivative of ˜ ω . Fortunately, the contribution of these twoterms has a good sign. Indeed, observe that ∂∂t ˜ g = g S − ˜ g > − ˜ g and hence − tr ˜ ω ω − ˜ g iℓ ˜ g kj g ij ∂∂t ˜ g kℓ . Again from Proposition 3.1 in [36], we have1tr ˜ ω ω (cid:18) − g pj g iq ˜ g kℓ ˜ ∇ k g ij ˜ ∇ ℓ g pq + 1tr ˜ ω ω g kℓ ∂ k tr ˜ ω ω∂ ℓ tr ˜ ω ω − (cid:16) g ij ˜ g kℓ ˜ T pki ˜ ∇ ℓ g pj (cid:17) − g ij ˜ g kℓ ˜ T pik ˜ T qjℓ g pq (cid:19) ˜ ω ω ) Re (cid:16) ˜ g iℓ g kq ˜ T kiℓ ∂ q tr ˜ ω ω (cid:17) . Hence to complete the proof of the lemma it remains to show that for t > T I we have 1tr ˜ ω ω (cid:16) g ij ˜ g kℓ ( ˜ ∇ i ˜ T qjℓ − ˜ R iℓpj ˜ g pq ) g kq − g ij ˜ ∇ i ˜ T ℓjℓ − g ij ˜ g kℓ ˜ g pj ˜ ∇ ℓ ˜ T pik + g ij ˜ g kℓ ˜ T pik ˜ T qjℓ ˜ g pq (cid:17) C (tr ω ˜ ω ) e t/ . But this follows easily from Lemma 4.1, the fact that the quantities tr ω ˜ ω and tr ˜ ω ω are uniformly equivalent and the inequality tr ˜ ω ω > C − > C (the geometric-arithmetic means inequality). Indeed,1tr ˜ ω ω | g ij ˜ g kℓ ˜ ∇ i ˜ T qjℓ g kq | ˜ ω ω | g − | ˜ g | ˜ g − | ˜ g | ˜ ∇ ˜ T | ˜ g | g | ˜ g C (tr ω ˜ ω ) e t/ , ˜ ω ω | g ij ˜ g kℓ ˜ g pq g kq ˜ R iℓpj | ˜ ω ω | g − | ˜ g | ˜ g − | g | g | ˜ g | g Rm | ˜ g C (tr ω ˜ ω ) e t/ , ˜ ω ω | g ij ˜ ∇ i ˜ T ℓjℓ | ˜ ω ω | g − | ˜ g | ˜ ∇ ˜ T | ˜ g Ce t/ , ˜ ω ω | g ij ˜ g kℓ ˜ g pj ˜ ∇ ℓ ˜ T pik | ˜ ω ω | g − | ˜ g | ˜ g − | ˜ g | ˜ g | ˜ g | ˜ ∇ ˜ T | ˜ g Ce t/ , ˜ ω ω | g ij ˜ g kℓ ˜ T pik ˜ T qjℓ ˜ g pq | ˜ ω ω | g − | ˜ g | ˜ g − | ˜ g | ˜ g | ˜ g | ˜ T | g C, as required. (cid:3) We can now prove Theorem 5.1, making use of the decay estimate of ϕ (Lemma 3.4) and the bound on ˙ ϕ (Lemma 3.1). Proof of Theorem 5.1.
We use the fact that e t/ ϕ is uniformly bounded, andconsider the quantity Q = log tr ˜ ω ω − Ae t/ ϕ + 1˜ C + e t/ ϕ , where ˜ C is a uniform constant chosen so that ˜ C + e t/ ϕ >
1, and A is a largeconstant to be determined later. The idea of adding an extra term, of theform of a reciprocal of a potential function, comes from Phong-Sturm [20]and was used in the context of the Chern-Ricci flow in [36]. Notice that0 C + e t/ ϕ . We will show that at a point ( x , t ) with t > T I at which Q achieves amaximum, we have a uniform upper bound of tr ˜ ω ω , and the theorem willfollow thanks to Lemma 3.4. OLLAPSING OF THE CHERN-RICCI FLOW ON ELLIPTIC SURFACES 21
First compute, using the fact that ∆ ϕ = 2 − tr ω ˜ ω and the bounds for ϕ and ˙ ϕ from Lemma 3.1, (cid:18) ∂∂t − ∆ (cid:19) (cid:18) − Ae t/ ϕ + 1˜ C + e t/ ϕ (cid:19) = − (cid:18) A + 1( ˜ C + e t/ ϕ ) (cid:19) (cid:18) e t/ ˙ ϕ + 12 e t/ ϕ (cid:19) + (cid:18) A + 1( ˜ C + e t/ ϕ ) (cid:19) ∆( e t/ ϕ ) − | ∂ ( e t/ ϕ ) | g ( ˜ C + e t/ ϕ ) CAe t/ − Ae t/ tr ω ˜ ω − | ∂ ( e t/ ϕ ) | g ( ˜ C + e t/ ϕ ) . (5.3)At the point ( x , t ), we have ∂ q Q = 0, which implies that ∂ q tr ˜ ω ω tr ˜ ω ω = (cid:18) A + 1( ˜ C + e t/ ϕ ) (cid:19) e t/ ∂ q ϕ. Then at this point,2(tr ˜ ω ω ) Re (cid:16) ˜ g iℓ g kq ˜ T kiℓ ∂ q tr ˜ ω ω (cid:17) = 2tr ˜ ω ω Re (cid:18) ˜ g iℓ g kq ˜ T kiℓ (cid:18) A + 1( ˜ C + e t/ ϕ ) (cid:19) e t/ ∂ q ϕ (cid:19) CA (tr ˜ ω ω ) ( ˜ C + e t/ ϕ ) g kq ˜ g iℓ ˜ T kiℓ ˜ g mj ˜ T qjm + | ∂ ( e t/ ϕ ) | g ( ˜ C + e t/ ϕ ) CA tr ˜ ω ω + | ∂ ( e t/ ϕ ) | g ( ˜ C + e t/ ϕ ) , (5.4)where for the last step we have used Lemma 3.4, part (i) of Lemma 4.1, andthe fact that tr ˜ ω ω and tr ω ˜ ω are uniformly equivalent.Combining (5.2), (5.3) and (5.4), we have, at a point at which Q achievesa maximum, for a uniform C > (cid:18) ∂∂t − ∆ (cid:19) Q CA + Ce t/ tr ω ˜ ω + CAe t/ − Ae t/ tr ω ˜ ω where we are assuming, without loss of generality, that at this maximumpoint of Q we have tr ˜ ω ω >
1. Choose a uniform A large enough so that A > C + 1 . Then we obtain at the maximum of Q , e t/ tr ω ˜ ω CA + CAe t/ , which implies that tr ω ˜ ω and hence tr ˜ ω ω is uniformly bounded from above atthe maximum of Q . This establishes the estimate tr ˜ ω ω C and completesthe proof of the theorem. (cid:3) A bound on the Chern scalar curvature
In this section, we establish the following estimate for the Chern scalarcurvature:
Theorem 6.1.
There exists a uniform constant C such that along the nor-malized Chern-Ricci flow (1.2) we have − C R Ce t/ , for all t > . First note that the lower bound for the Chern scalar curvature followsfrom the same argument as in the K¨ahler-Ricci flow (see for example Theo-rem 2.2 in [28]). Indeed, from (1.2), we have g kℓ ∂∂t g kℓ = − R − . But R = − g ij ∂ i ∂ j log det g and hence ∂R∂t = − g ij ∂ i ∂ j (cid:18) g kℓ ∂∂t g kℓ (cid:19) − (cid:18) ∂∂t g ij (cid:19) ∂ i ∂ j log det g = ∆ R + | Ric | + R > ∆ R + 12 R + R and then the lower bound for R follows.We now establish the upper bound of the Chern scalar curvature. Beforewe start the main argument, we need a few preliminary calculations. Lemma 6.2.
There exists a uniform constant
C > such that for t > T I ,we have (6.1) (cid:18) ∂∂t − ∆ (cid:19) tr ˜ ω ω − C − | ˜ ∇ g | g + Ce t/ . and (6.2) (cid:18) ∂∂t − ∆ (cid:19) tr ω ω S | ˜ ∇ g | g − C − |∇ tr ω ω S | g + Ce t/ . As a consequence, there are uniform positive constants C , C such that for t > T I , (6.3) (cid:18) ∂∂t − ∆ (cid:19) (tr ω ω S + C tr ˜ ω ω ) −| ˜ ∇ g | g − C − |∇ tr ω ω S | g + C e t/ . OLLAPSING OF THE CHERN-RICCI FLOW ON ELLIPTIC SURFACES 23
Proof.
For (6.1), we compute the evolution of tr ˜ ω ω . As in Lemma 5.2 wemay modify [36, Proposition 3.1] to obtain, for t > T I , (cid:18) ∂∂t − ∆ (cid:19) tr ˜ ω ω = − g pj g iq ˜ g kℓ ˜ ∇ k g ij ˜ ∇ ℓ g pq − (cid:16) g ij ˜ g kℓ ˜ T pki ˜ ∇ ℓ g pj (cid:17) − g ij ˜ g kℓ ˜ T pik ˜ T qjℓ g pq + g ij ˜ g kℓ ( ˜ ∇ i ˜ T qjℓ − ˜ R iℓpj ˜ g pq ) g kq − g ij ˜ ∇ i ˜ T ℓjℓ − g ij ˜ g kℓ ˜ g pj ˜ ∇ ℓ ˜ T pik + g ij ˜ g kℓ ˜ T pik ˜ T qjℓ ˜ g pq − tr ˜ ω ω − ˜ g iℓ ˜ g kj g ij (( g S ) kℓ − ˜ g kℓ ) . Using Theorem 5.1, Lemma 4.1, and the Cauchy-Schwarz inequality in thesecond term, we obtain (6.1).The inequality (6.2) is a parabolic Schwarz Lemma calculation for themap π : M → S [24, 40]. Note that we already know that tr ω ω S C sincethe metrics ω and ˜ ω are uniformly equivalent.The computation for (6.2) is similar to that of Song-Tian [24], except thatof course here we need to control the extra torsion terms. Given any point x ∈ M we choose local coordinates { z i } on M centered at x such that g is theidentity at x , and a coordinate w on S near π ( x ) ∈ N , which we can assumeis normal for the metric g S . In these coordinates we can represent the map π as a local holomorphic function f . We will use subscripts like f i , f ij , ... toindicate covariant derivatives of f with respect to g . For example we have f i = ∂ i f, f ij = ∂ j f i − (cid:16) g kq ∂ j g iq (cid:17) f k , f ij = f ji = 0 . We will also write g S for the coefficient of the metric g S in the coordinate w , so g S ( x ) = 1 and the pullback of the metric g S to M is given by f i f j g S .We use the shorthand h ij = g iℓ g kj f k f ℓ g S , where h ij is semipositive definiteand satisfies | h | g := h ij h kℓ g iℓ g kj C . Then we have (cf. [33])∆tr ω ω S = g ij ∂ i ∂ j (cid:16) g kℓ f k f ℓ g S (cid:17) = g ij g kℓ f ki f ℓj g S + g ij h pq R ijpq − g ij g kℓ f k f ℓ f i f j R S , for R S the scalar curvature of g S , and so (cid:18) ∂∂t − ∆ (cid:19) tr ω ω S =tr ω ω S − g ij g kℓ f ki f ℓj g S + g ij h pq ( R pqij − R ijpq )+ g ij g kℓ f k f ℓ f i f j R S . (6.4) The last term can be dropped since R S <
0. Now at x we have ∂ i (tr ω ω S ) = P k f ki f k , and using the Cauchy-Schwarz inequality we have |∇ tr ω ω S | g = X i,k,p f ki f pi f p f k X k,p | f k || f p | X i | f ki | ! / X j | f pj | / = X k | f k | X i | f ki | ! / X ℓ | f ℓ | ! X i,k | f ki | =(tr ω ω S ) g ij g kℓ f ki f ℓj g S Cg ij g kℓ f ki f ℓj g S . (6.5)Next we claim that given any constant C we can find a constant C suchthat(6.6) | g ij h pq ( R pqij − R ijpq ) | Ce t/ + 12 C | ˜ ∇ g | g . In fact, we only need the case C = 1 here, but the general case will beuseful later. To prove this claim, we first calculate (see also [23, (2.6)]), R ijpq = − g rq ∂ j Γ rip = − g rq ∂ j Γ rpi + g rq ∂ j T rpi = R pjiq + g rq ∂ j T rpi = R jpqi + g rq ∂ j T rpi = R qpji + g is ∂ p T sqj + g rq ∂ j T rpi = R pqij + g is ∂ p T sqj + g rq ∂ j T rpi , (6.7)and therefore(6.8) g ij h pq ( R pqij − R ijpq ) = − h pq ∂ p T jqj − g ij h pq g rq ∂ j T rpi . Recall that T ijℓ = ˜ T ijℓ . Differentiating this gives(6.9) ∂ p T jqj = ˜ g rs g rj ∂ p ˜ T sqj − ˜ g rs g ru g tj ˜ T sqj ˜ ∇ p g tu = g rj ˜ ∇ p ˜ T qjr − g ru g tj ˜ T qjr ˜ ∇ p g tu , (6.10) ∂ j T rpi = ˜ g sb g rb ∂ j ˜ T spi − ˜ g sb g ru g tb ˜ T spi ˜ ∇ j g tu = g rb ˜ ∇ j ˜ T pib − g ru g tb ˜ T pib ˜ ∇ j g tu . Putting together (6.8), (6.9), (6.10) we get g ij h pq ( R pqij − R ijpq ) = − h pq g rj ˜ ∇ p ˜ T qjr + h pq g ru g tj ˜ T qjr ˜ ∇ p g tu − h pq g ij ˜ ∇ j ˜ T piq + h pq g ij g tr ˜ T pir ˜ ∇ j g tq . (6.11)Using again that the metrics g and ˜ g are equivalent and | h | g C , we canthen bound this by | g ij h pq ( R pqij − R ijpq ) | C | ∂ ˜ T | ˜ g + C | ˜ T | ˜ g | ˜ ∇ g | g . But from Lemma 4.1 we have that | ˜ T | ˜ g C and | ∂ ˜ T | ˜ g Ce t/ , and so wehave (6.6) as required. OLLAPSING OF THE CHERN-RICCI FLOW ON ELLIPTIC SURFACES 25
Then (6.2) follows from (6.4), (6.5) and (6.6). (6.3) follows immediatelyfrom (6.1) and (6.2). (cid:3)
We can now start the main argument for the proof of Theorem 6.1. Notethat since many of our inequalities require ˜ ω to be a metric, we will oftenassume (without saying it explicitly) that t > T I , which is not a problemsince R is bounded on [0 , T I ]. As in [24], we consider the quantity u = ϕ + ˙ ϕ =log e t ω Ω . We know that | u | C , and we have that − ∆ u = R + tr ω ω S > R ,so our goal is to get an upper bound for − ∆ u . First compute from (3.2),(6.12) (cid:18) ∂∂t − ∆ (cid:19) u = tr ω ω S − , and (cid:18) ∂∂t − ∆ (cid:19) ∆ u = R ij u ij + ∆ u + ∆tr ω ω S . But on the other hand R ij = − u ij − ( g S ) ij , and so (cid:18) ∂∂t − ∆ (cid:19) ∆ u = −|∇∇ u | g − h g S , ∇∇ u i g + ∆ u + ∆tr ω ω S > − |∇∇ u | g + ∆ u + ∆tr ω ω S − C, using that | g S | g C . From (6.2) we have − ∆tr ω ω S Ce t/ + | ˜ ∇ g | g − C − |∇ tr ω ω S | g − h ij ( R ij + g ij ) Ce t/ + | ˜ ∇ g | g − C − |∇ tr ω ω S | g + h ij ( u ij + ( g S ) ij ) Ce t/ + | ˜ ∇ g | g − C − |∇ tr ω ω S | g + 12 |∇∇ u | g , where we have used the fact that ∂∂t tr ω ω S = − h ij ∂∂t g ij , with h ij as in theproof of Lemma 6.2. Therefore (cid:18) ∂∂t − ∆ (cid:19) ( − ∆ u ) |∇∇ u | g − ∆ u + Ce t/ + | ˜ ∇ g | g − C − |∇ tr ω ω S | g . (6.13)We need a quantity whose evolution can kill the bad term 2 |∇∇ u | g , andthis quantity is |∇ u | g . Before we compute its evolution, we need formulaeto commute two covariant derivatives of the same type. For any function ψ and (0 ,
1) form a = a k dz k , a short calculation gives[ ∇ j , ∇ ℓ ] ψ = − T kjℓ ∇ k ψ, [ ∇ i , ∇ j ] a k = − T ℓij ∇ ℓ a k . We will also use the familiar formulae[ ∇ i , ∇ j ] a k = g pℓ R ijpk a ℓ , R iℓpj = R pℓij + g rj ∂ ℓ T rpi , where the second equation is contained in (6.7). We then compute:∆ |∇ u | g = g ij g kℓ (cid:18) ∇ i ∇ j ∇ k u ∇ ℓ u + ∇ k u ∇ i ∇ j ∇ ℓ u + ∇ i ∇ k u ∇ j ∇ ℓ u + ∇ i ∇ ℓ u ∇ j ∇ k u (cid:19) = |∇∇ u | g + |∇∇ u | g + g ij g kℓ (cid:18) ∇ i ∇ k ∇ j u ∇ ℓ u + ∇ k u ∇ i ∇ ℓ ∇ j u − ∇ k u ∇ i ( T pjℓ ∇ p u ) (cid:19) = |∇∇ u | g + |∇∇ u | g + 2Re h∇ ∆ u, ∇ u i g − g ij g kℓ T pik ∇ p ∇ j u ∇ ℓ u + g ij g kℓ g pq R iℓpj ∇ q u ∇ k u − g ij g kℓ T pjℓ ∇ k u ∇ i ∇ p u − g ij g kℓ ∂ i T pjℓ ∇ k u ∇ p u = |∇∇ u | g + |∇∇ u | g + 2Re h∇ ∆ u, ∇ u i g + g kℓ g pq R pℓ ∇ q u ∇ k u − (cid:16) g ij g kℓ T pik ∇ p ∇ j u ∇ ℓ u (cid:17) + g kℓ g pq ∂ ℓ T ipi ∇ q u ∇ k u − g ij g kℓ ∂ i T pjℓ ∇ k u ∇ p u. We use again (6.9) and (6.10) ∂ i T pjℓ = g qp ˜ ∇ i ˜ T jℓq − g rp g qs ˜ T jℓq ˜ ∇ i g rs , ∂ ℓ T ipi = g ij ˜ ∇ ℓ ˜ T pij − g is g rj ˜ T pij ˜ ∇ ℓ g rs , and Lemma 4.1 to conclude that∆ |∇ u | g > |∇∇ u | g + |∇∇ u | g + 2Re h∇ ∆ u, ∇ u i g + g kℓ g pq R pℓ ∇ q u ∇ k u − C |∇∇ u | g |∇ u | g − Ce t/ |∇ u | g − C | ˜ ∇ g | g |∇ u | g > |∇∇ u | g + |∇∇ u | g + 2Re h∇ ∆ u, ∇ u i g + g kℓ g pq R pℓ ∇ q u ∇ k u − Ce t/ |∇ u | g − C | ˜ ∇ g | g |∇ u | g . Next, using (6.12), we have ∂∂t |∇ u | g = g kℓ g pq R pℓ ∇ k u ∇ q u + |∇ u | g +2Re h∇ ∆ u, ∇ u i g +2Re h∇ tr ω ω S , ∇ u i g and hence (cid:18) ∂∂t − ∆ (cid:19) |∇ u | g − |∇∇ u | g − |∇∇ u | g + 2Re h∇ tr ω ω S , ∇ u i g + Ce t/ |∇ u | g + C | ˜ ∇ g | g |∇ u | g . (6.14)We will use this evolution inequality to bound |∇ u | g . Proposition 6.3.
There is a constant C such that |∇ u | g Ce t/ . OLLAPSING OF THE CHERN-RICCI FLOW ON ELLIPTIC SURFACES 27
Proof.
We use the method of Cheng-Yau [5], see also [22, 24]. We fix aconstant A such that | u | A − (cid:18) ∂∂t − ∆ (cid:19) |∇ u | g A − u ! = 1 A − u (cid:18) ∂∂t − ∆ (cid:19) |∇ u | g + |∇ u | g ( A − u ) (cid:18) ∂∂t − ∆ (cid:19) u − h∇|∇ u | g , ∇ u i g ( A − u ) − |∇ u | g ( A − u ) A − u (cid:18) − |∇∇ u | g − |∇∇ u | g + 2Re h∇ tr ω ω S , ∇ u i g + Ce t/ |∇ u | g + C | ˜ ∇ g | g |∇ u | g (cid:19) + (tr ω ω S − |∇ u | g ( A − u ) − h∇|∇ u | g , ∇ u i g ( A − u ) − |∇ u | g ( A − u ) . (6.15)Note that the term (tr ω ω S − |∇ u | g ( A − u ) can be absorbed in the term Ce t/ |∇ u | g A − u .For ε > h∇|∇ u | g , ∇ u i g ( A − u ) = ε h∇|∇ u | g , ∇ u i g ( A − u ) + 2(1 − ε ) A − u Re * ∇ |∇ u | g A − u ! , ∇ u + g − − ε ) |∇ u | g ( A − u ) , (6.16)and use the Cauchy-Schwarz inequality to bound − ε h∇|∇ u | g , ∇ u i g ( A − u ) ε √ |∇ u | g ( |∇∇ u | g + |∇∇ u | g ) / ( A − u ) ε |∇ u | g ( A − u ) + 4 ε |∇∇ u | g + |∇∇ u | g ( A − u ) ε |∇ u | g ( A − u ) + 12 |∇∇ u | g + |∇∇ u | g A − u , (6.17)provided ε / ε ). We also bound(6.18) C | ˜ ∇ g | g |∇ u | g A − u C | ˜ ∇ g | g + ε |∇ u | g ( A − u ) . Putting (6.15), (6.16), (6.17), (6.18) together, we get (cid:18) ∂∂t − ∆ (cid:19) |∇ u | g A − u ! A − u (cid:18) h∇ tr ω ω S , ∇ u i g + Ce t/ |∇ u | g (cid:19) + C | ˜ ∇ g | g − ε |∇ u | g ( A − u ) − − ε ) A − u Re * ∇ |∇ u | g A − u ! , ∇ u + g . Call now Q = |∇ u | g A − u + C (tr ω ω S + C tr ˜ ω ω ) , where C is as in (6.3), and C is a large uniform constant to be fixed soon.We can use (6.3) to get (cid:18) ∂∂t − ∆ (cid:19) Q A − u (cid:18) h∇ tr ω ω S , ∇ u i g + Ce t/ |∇ u | g (cid:19) − ε |∇ u | g ( A − u ) − − ε ) A − u Re * ∇ |∇ u | g A − u ! , ∇ u + g − C | ˜ ∇ g | g − |∇ tr ω ω S | g + Ce t/ , (6.19)provided C is large enough. We now observe that |∇ tr ˜ ω ω | g | ˜ ∇ g | g . Indeed, if we choose local coordinates such that ˜ g is the identity at a point,then at that point we have(6.20) |∇ tr ˜ ω ω | g = X k | X i ˜ ∇ k g ii | X i,k | ˜ ∇ k g ii | X i,j,k | ˜ ∇ k g ij | = 2 | ˜ ∇ g | g . Since ˜ g and g are uniformly equivalent, we conclude that |∇ tr ˜ ω ω | g C | ˜ ∇ g | g . We can then assume that C was large enough (this fixes C ) so that(6.21) − C | ˜ ∇ g | g −|∇ tr ˜ ω ω | g . Furthermore, we can bound(6.22)
C e t/ |∇ u | g A − u ε |∇ u | g ( A − u ) + Ce t , (6.23) 2 Re h∇ tr ω ω S , ∇ u i g A − u |∇ tr ω ω S | g + ε |∇ u | g ( A − u ) + C, so combining (6.19), (6.21), (6.22), (6.23) we get (cid:18) ∂∂t − ∆ (cid:19) Q − ε |∇ u | g ( A − u ) − − ε ) A − u Re * ∇ |∇ u | g A − u ! , ∇ u + g − |∇ tr ˜ ω ω | g − |∇ tr ω ω S | g + Ce t . OLLAPSING OF THE CHERN-RICCI FLOW ON ELLIPTIC SURFACES 29
We can write this as (cid:18) ∂∂t − ∆ (cid:19) Q − ε |∇ u | g ( A − u ) − − ε ) A − u Re h∇ Q , ∇ u i g + 2(1 − ε ) C A − u Re h∇ tr ω ω S , ∇ u i g + 2(1 − ε ) C C A − u Re h∇ tr ˜ ω ω, ∇ u i g − |∇ tr ˜ ω ω | g − |∇ tr ω ω S | g + Ce t , which together with the bounds2(1 − ε ) C A − u Re h∇ tr ω ω S , ∇ u i g |∇ tr ω ω S | g + ε |∇ u | g ( A − u ) + C, − ε ) C C A − u Re h∇ tr ˜ ω ω, ∇ u i g |∇ tr ˜ ω ω | g + ε |∇ u | g ( A − u ) + C, gives us (cid:18) ∂∂t − ∆ (cid:19) Q − ε |∇ u | g ( A − u ) − − ε ) A − u Re h∇ Q , ∇ u i g + Ce t . Now define Q = e − t/ Q , which satisfies (cid:18) ∂∂t − ∆ (cid:19) Q − C − e − t/ |∇ u | g − − ε ) A − u Re h∇ Q , ∇ u i g + Ce t/ . At a maximum of Q (occurring at a time t > T I ) we conclude that |∇ u | g Ce t , which implies that Q C everywhere. This proves the proposition. (cid:3) Now that we know that |∇ u | g Ce t/ , we can go back to (6.14) and get (cid:18) ∂∂t − ∆ (cid:19) |∇ u | g − |∇∇ u | g − |∇∇ u | g + |∇ tr ω ω S | g + | ˜ ∇ g | g + Ce t . (6.24)Finally we can put everything together to prove Theorem 6.1. Proof of Theorem 6.1.
We will show that there is a constant C such that − ∆ u Ce t/ . and this will give R Ce t/ .From (6.13) and (6.24), we see that (cid:18) ∂∂t − ∆ (cid:19) ( − ∆ u + 6 |∇ u | g ) − |∇∇ u | g − ∆ u + C |∇ tr ω ω S | g + C | ˜ ∇ g | g + Ce t . Using (6.3) we get (cid:18) ∂∂t − ∆ (cid:19) (cid:0) − ∆ u + 6 |∇ u | g + C (tr ω ω S + C tr ˜ ω ω ) (cid:1) −|∇∇ u | g − ∆ u + Ce t , provided C is large enough. Define now Q = e − t/ (cid:0) − ∆ u + 6 |∇ u | g + C (tr ω ω S + C tr ˜ ω ω ) (cid:1) . Note that Q > − Ce − t/ , because − ∆ u > R > − C . Then, (cid:18) ∂∂t − ∆ (cid:19) Q − e − t/ |∇∇ u | g − e − t/ ∆ u + Ce t/ , where we absorbed a term like e − t/ into e t/ . From the Cauchy-Schwarzinequality we also have that( − ∆ u ) |∇∇ u | g . It follows that at a maximum of Q (occurring at t > T I ) we have that( − ∆ u ) Ce t , which implies that Q C everywhere. This completes theproof of the theorem. (cid:3) We end this section by applying our Chern scalar curvature bound toobtain an exponential decay estimate for ˙ ϕ . Lemma 6.4.
For any η with < η < / and any σ with < σ < / ,there exists a constant C such that − Ce − ηt ˙ ϕ Ce − σt . Proof.
We first prove the lower bound by refining an argument in [28]. Wehave(6.25) ∂∂t ˙ ϕ = − R − − ˙ ϕ and hence, by Theorem 6.1 and the fact that | ˙ ϕ | is bounded,(6.26) ∂∂t ˙ ϕ ( t ) C , for a uniform C . Suppose for a contradiction that we do not have the bound˙ ϕ > − Ce − ηt for any C . Then there exists a sequence ( x k , t k ) ∈ M × [0 , ∞ )with t k → ∞ as k → ∞ such that˙ ϕ ( x k , t k ) − ke − ηt k . Put γ k = k C e − ηt k . From now on we work at the point x k . Then by (6.26),we have that ˙ ϕ − k e − ηt k on [ t k , t k + γ k ] . Indeed, ˙ ϕ ( t k + a ) − ˙ ϕ ( t k ) = Z t k + at k ∂∂t ˙ ϕ dt C γ k , for a ∈ [0 , γ k ] , and hence ˙ ϕ ( t k + a ) ˙ ϕ ( t k ) + C γ k − ke − ηt k + k e − ηt k . OLLAPSING OF THE CHERN-RICCI FLOW ON ELLIPTIC SURFACES 31
Then, using Lemma 3.4, − C (1+ t k ) e − t k ϕ ( t k + γ k ) − ϕ ( t k ) = Z t k + γ k t k ˙ ϕdt − γ k k e − ηt k = − k C e − ηt k . But if 2 η < k → ∞ and we are done.For the upper bound of ϕ we use the upper bound R Ce t/ of Theorem6.1. From (6.25), we have(6.27) ∂∂t ˙ ϕ ( t ) > − C e t/ , for a uniform C . Note that we may assume, by increasing C , that we have(6.28) ∂ ∂t ϕ ( t ) > − C e t ′ / for t ∈ [ t ′ , t ′ + 1] , for any time t ′ . Suppose for a contradiction that we do not have the bound˙ ϕ Ce − σt for any C . Then there exists a sequence ( x k , t k ) ∈ M × [0 , ∞ )with t k → ∞ as k → ∞ such that˙ ϕ ( x k , t k ) > ke − σt k . Put γ k = k C e − ( σ +1 / t k which we assume for the moment satisfies γ k x k . Then by (6.28), we have that˙ ϕ > k e − σt k on [ t k , t k + γ k ] . Indeed, this follows from (6.28) since˙ ϕ ( t k + a ) − ˙ ϕ ( t k ) = Z t k + at k ∂∂t ˙ ϕ dt > − γ k C e t k / , for a ∈ [0 , γ k ] , and hence ˙ ϕ ( t k + a ) > ˙ ϕ ( t k ) − γ k C e t k / > ke − σt k − k e − σt k .Then from Lemma 3.4, C (1+ t k ) e − t k > ϕ ( t k + γ k ) − ϕ ( t k ) = Z t k + γ k t k ˙ ϕdt > γ k k e − σt k = k C e − (2 σ +1 / t k . But since 2 σ + 1 / < k → ∞ and we aredone.It remains to check the case when γ k >
1. But then we have ˙ ϕ > k e − σt k on [ t k , t k + 1] since ˙ ϕ ( t k + a ) − ˙ ϕ ( t k ) > − γ k C e t k / for a ∈ [0 ,
1] and so weget ˙ ϕ ( t k + a ) > k e − σt k for a ∈ [0 , C (1 + t k ) e − t k > ϕ ( t k + 1) − ϕ ( t k ) = Z t k +1 t k ˙ ϕdt > k e − σt k and we get a contradiction since σ < (cid:3) Exponential decay estimates for the metric
In this section we establish the key estimates which show that ω ( t ) and ˜ ω approach each other exponentially fast as t → ∞ . More precisely we prove: Theorem 7.1.
For any ε with < ε < / , there exists C such that for t > T I (1 − Ce − εt )˜ ω ω ( t ) (1 + Ce − εt )˜ ω. In this section we will always assume t > T I , without necessarily mention-ing it explicitly. First, we have the following evolution inequality for tr ω ˜ ω which has the same form as the inequality for tr ˜ ω ω given by (6.1). We willmake use of both of these inequalities to prove Theorem 7.1. Lemma 7.2.
We have, for a uniform
C > , (cid:18) ∂∂t − ∆ (cid:19) tr ω ˜ ω Ce t/ − C − | ˜ ∇ g | g . Proof.
We will use the shorthand h ij = g iℓ g kj ˜ g kℓ . Note that we know alreadythat h, g and ˜ g are all uniformly equivalent to each other.We start by computing a formula for the evolution of tr ω ˜ ω . First of all,we have∆tr ω ˜ ω = g ij ˜ ∇ i ˜ ∇ j ( g kℓ ˜ g kℓ ) = − g ij ˜ ∇ i ( g kq g pℓ ˜ g kℓ ˜ ∇ j g pq )= g ij g ks g rq g pℓ ˜ g kℓ ˜ ∇ i g rs ˜ ∇ j g pq + g ij g ps g rℓ g kq ˜ g kℓ ˜ ∇ i g rs ˜ ∇ j g pq − g ij g kq g pℓ ˜ g kℓ ˜ ∇ i ˜ ∇ j g pq . But˜ ∇ i ˜ ∇ j g pq = ˜ ∇ i (cid:16) ∂ j g pq − ˜Γ sjq g ps (cid:17) = ∂ i ∂ j g pq − ˜Γ rip ∂ j g rq − g ps ∂ i ˜Γ sjq − ˜Γ sjq ∂ i g ps + ˜Γ rip ˜Γ sjq g rs = ˜ R ijrq ˜ g rs g ps − R ijpq + g rs ˜ ∇ i g ps ˜ ∇ j g rq , and so∆tr ω ˜ ω = g ij g ks g rq g pℓ ˜ g kℓ ˜ ∇ i g rs ˜ ∇ j g pq + g ij g kq g pℓ ˜ g kℓ R ijpq − g ij g pq ˜ R ijpq . On the other hand ∂∂t tr ω ˜ ω = tr ω ˜ ω + tr ω ( ω S − ˜ ω ) + g ij h pq R pqij . Therefore, (cid:18) ∂∂t − ∆ (cid:19) tr ω ˜ ω =tr ω ω S + g ij g kℓ ˜ R ijkℓ + g ij h pq ( R pqij − R ijpq ) − g ij g kℓ h pq ˜ ∇ i g kq ˜ ∇ j g pℓ . OLLAPSING OF THE CHERN-RICCI FLOW ON ELLIPTIC SURFACES 33
From Lemma 4.1, we conclude that(7.1) (cid:18) ∂∂t − ∆ (cid:19) tr ω ˜ ω Ce t/ + g ij h pq ( R pqij − R ijpq ) − g ij g kℓ h pq ˜ ∇ i g kq ˜ ∇ j g pℓ . We also use the equivalence of g and h to bound(7.2) − g ij g kℓ h pq ˜ ∇ i g kq ˜ ∇ j g pℓ − C − | ˜ ∇ g | g . Next we have(7.3) | g ij h pq ( R pqij − R ijpq ) | Ce t/ + 12 C | ˜ ∇ g | g . Indeed, this follows from the same argument as in the proof of (6.6), eventhough the tensor h there is different. Combining (7.1), (7.2), (7.3), we getthe desired inequality. (cid:3) Next we use the exponential decay of ϕ (Lemma 3.4) and ˙ ϕ (Lemma 6.4)to obtain an exponential decay bound from above for tr ω ˜ ω − ˜ ω ω − Proposition 7.3.
For any < ε < / there is a constant C > such thatfor t > T I , (7.4) tr ω ˜ ω − Ce − εt and (7.5) tr ˜ ω ω − Ce − εt . Proof.
Given 0 < ε < /
4, choose η > ε such that ε + 1 / η < < η < / δ satisfying 2 ε + 1 / < δ < ε + 1 / η (in particular 0 < δ < ε < η . For (7.4), wecompute the evolution of Q = e εt (tr ω ˜ ω − − e δt ϕ. From Lemma 3.4, it suffices to obtain a uniform upper bound for Q . Com-pute using Lemma 7.2 and the fact that ∆ ϕ = 2 − tr ω ˜ ω , (cid:18) ∂∂t − ∆ (cid:19) Q Ce ( ε +1 / t + εe εt (tr ω ˜ ω − − δe δt ϕ − e δt ˙ ϕ − e δt (tr ω ˜ ω − Ce ( ε +1 / t + Ce ( δ − η ) t − e δt (tr ω ˜ ω − , (7.6)where in the last line we have used the lower bound ˙ ϕ > − Ce − ηt fromLemma 6.4 (since 0 < η < / ω ˜ ω C .But we have δ − η < ε + 1 / Q ,0 Ce ( ε +1 / t − e δt (tr ω ˜ ω − , and hence e εt (tr ω ˜ ω − Ce (2 ε − δ +1 / t C. since we chose δ so that 2 ε − δ + 1 / <
0. This implies that Q is boundedfrom above at any maximum point, and completes the proof of (7.4).The proof of (7.5) is slightly more complicated. First recall that (see(6.1)) for t > T I , (cid:18) ∂∂t − ∆ (cid:19) tr ˜ ω ω Ce t/ . Fix σ with 0 < ε < σ < /
4. From Lemma 6.4 we have(7.7) − Ce − σt ˙ ϕ Ce − σt . Now choose δ with 1 / ε < δ < / ε + σ , which we can do because ε < σ . Then set Q = e εt (tr ˜ ω ω − − e δt ϕ. Compute (cid:18) ∂∂t − ∆ (cid:19) Q Ce ( ε +1 / t − e δt ˙ ϕ − e δt (tr ω ˜ ω − Ce ( ε +1 / t − e δt (tr ω ˜ ω − , (7.8)using (7.7) and the fact that δ − σ < / ε . We now wish to replace theterm tr ω ˜ ω by the sum of tr ˜ ω ω and a small error term.(7.9) tr ω ˜ ω = ˜ ω ω tr ˜ ω ω = tr ˜ ω ω + (cid:18) ˜ ω ω − (cid:19) tr ˜ ω ω. Then from (2.4), (3.8) and Lemma 3.4,˙ ϕ = log ω ˜ ω + O ( e − σt ) , and so using (7.7) again (cid:12)(cid:12)(cid:12)(cid:12) ˜ ω ω − (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) e O ( e − σt ) − (cid:12)(cid:12)(cid:12) Ce − σt , which implies that, since tr ˜ ω ω is uniformly bounded,(7.10) (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) ˜ ω ω − (cid:19) tr ˜ ω ω (cid:12)(cid:12)(cid:12)(cid:12) Ce − σt . Then combining (7.8), (7.9), (7.10) and again using the fact that e ( δ − σ ) t e ( ε +1 / t we obtain for t > T I , (cid:18) ∂∂t − ∆ (cid:19) Q Ce ( ε +1 / t − e δt (tr ˜ ω ω − − e δt (cid:18) ˜ ω ω − (cid:19) tr ˜ ω ω Ce ( ε +1 / t − e δt (tr ˜ ω ω − . Then at the maximum point of Q (occurring at a time t > T I ) we have e εt (tr ˜ ω ω − Ce (2 ε +1 / − δ ) t C ′ , since we chose δ > / ε . This shows that Q is bounded from above andcompletes the proof. (cid:3) OLLAPSING OF THE CHERN-RICCI FLOW ON ELLIPTIC SURFACES 35
To show that ω and ˜ ω approach each other exponentially fast we use anelementary lemma: Lemma 7.4.
Let ε > be small. Suppose that tr ω ˜ ω − ε and tr ˜ ω ω − ε. Then (1 − √ ε )˜ ω ω (1 + 2 √ ε )˜ ω. Proof.
We may work at a point at which ˜ g is the identity and g is diagonalwith eigenvalues λ , λ . Then the lemma amounts to proving that if λ , λ > λ + λ ε, λ + 1 λ ε, then 1 − √ ε λ i √ ε, for i = 1 , . By symmetry, we only have to prove the estimate for λ . We have(7.11) λ ε − λ , λ (2 + ε ) λ − λ , which implies in particular that (2 + ε ) λ − >
0. This last inequalityimplies that − λ − λ (2 + ε ) λ − . Then in (7.11), λ ε − λ (2 + ε ) λ − . Multiplying this by (2 + ε ) λ − > λ − (2 + ε ) λ + 1 . Completing the square, we obtain( λ − (1 + ε/ ε + ε / . Then, assuming ε > − √ ε λ √ ε, as required. (cid:3) Finally, we complete the proof of Theorem 7.1:
Proof of Theorem 7.1.
Combine Proposition 7.3 with Lemma 7.4. (cid:3) A third order estimate
In this section we prove a “Calabi-type” estimate for the first derivative ofthe evolving metric. One might guess that the natural quantity to consideris | ˜ ∇ g | g , following the computation in [28], say. However, we encountereddifficulties in obtaining a good bound for this quantity because of the non-K¨ahlerity of the reference metrics ˜ ω . Our idea then is to take a K¨ahler reference metric. Of course, in general M may not admit a global K¨ahlermetric, so we work locally on an open set where the bundle M is trivial.We obtain a Calabi estimate on this open set, using a cut-off function anda local reference K¨ahler metric. Our computations are based on those in [23].However, the situation here is complicated by the fact that the metrics arecollapsing in the fiber directions, and we will need to make careful use ofthe bounds from Lemma 4.1.Fix a point y ∈ S and neighborhood B of y over which π is trivial, so U = π − ( B ) ∼ = B × E . Over U we have ω E = iα ∧ α a d -closed semi-flat(1 , ω = ω E + ω S is a semi-flat product K¨ahler metric on U . From now on we work exclusivelyon U , where we define S = | ˆ ∇ g | g . Fix a smaller open set V ⊂⊂ U . Theorem 8.1. On V we have S Ce t/ , for all t > . By compactness, we obtain the same bound in any such neighborhood V .Recall that ω flat ,y denotes the unique flat metric on the fiber E y = π − ( y )in the K¨ahler class [ ω | E y ]. Exactly as in [28, Lemma 6.9] (see also [12], [13,Theorem 1.1], [9, Proposition 5.8]) we have that: Corollary 8.2.
For any y ∈ S , we have on E y , e t ω ( t ) | E y → ω flat ,y exponentially fast in the C ( E y , g ) topology. Moreover, the convergence isuniform in y ∈ S .Proof. We use an idea from [34, p. 440]. Since g | E y is uniformly equivalentto e − t ˆ g | E y = e − t g E , we conclude that |∇ g E ( e t g | E y ) | g E = e − t |∇ g E ( g | E y ) | e − t g E Ce − t |∇ ˆ g | Ey ( g | E y ) | g Ce − t S Ce − t/ , using Theorem 8.1. But on E y , g flat ,y is a constant multiple of g E , and so |∇ g E ( e t g | E y − g flat ,y ) | g E Ce − t/ . The rest of the proof follows easily, and exactly as in [28, Lemma 6.9], since e t g | E y and g flat ,y lie in the same K¨ahler class on E y . (cid:3) OLLAPSING OF THE CHERN-RICCI FLOW ON ELLIPTIC SURFACES 37
Before we start the proof of Theorem 8.1, we need some preliminarycalculations. Denote by Ψ ijk = Γ ijk − ˆΓ ijk , the difference of the Christoffelsymbols of g and ˆ g . It is a tensor which satisfies | Ψ | g = S . The evolutionof S is computed in [23, (3.4)], generalizing calculations of [39, 4, 19], whichgives (cid:18) ∂∂t − ∆ (cid:19) S = S − |∇ Ψ | g − |∇ Ψ | g + g ij g rs g ab (cid:16) ∇ r T bja + ∇ b T arj (cid:17) Ψ kip Ψ ℓsq g pq g kℓ + g ij g rs g ab (cid:16) ∇ r T bja + ∇ b T arj (cid:17) Ψ kpi Ψ ℓqs g pq g kℓ − g ab (cid:0) ∇ k T bsa + ∇ b T aks (cid:1) Ψ kip Ψ sjq g ij g pq − (cid:20) g rs (cid:0) ∇ i ∇ p T sℓr + ∇ i ∇ s T rpℓ − T air R aspℓ + g kℓ ∇ r ˆ R ispk (cid:1) Ψ ℓjq g ij g pq (cid:21) , (8.1)where ˆ R ispk is the curvature tensor of ˆ g . We are going to bound each ofthese terms separately. The key difference from the calculation in [23] isthat in our case the torsion T ijℓ of g does not equal the torsion of ˆ g (whichhere is zero), but rather the torsion ˜ T ijℓ of ˜ g .Therefore we let ˜Ψ ijk = Γ ijk − ˜Γ ijk and H ijk = ˜Γ ijk − ˆΓ ijk = Ψ ijk − ˜Ψ ijk . Lemma 8.3.
For all t > T I , we have that (8.2) | H | g Ce t/ . (8.3) | ∂H | g Ce t/ . (8.4) | ˜ ∇ H | g Ce t . Proof.
At any given point x ∈ M we can choose local bundle coordinates( z , z ) as before. Then in this coordinate system (4.3) gives | ˜Γ ijk | g Ce t , where we are using the fact that g and ˜ g are uniformly equivalent. Since ˆ g is a semi-flat product K¨ahler metric, ˆΓ ijk is zero except when i = j = k = 2,and so | ˆΓ ijk | g C. The bound (8.2) follows immediately from these bounds.Next, note that ∂ ℓ H ijk = − ˜ R jℓki + ˆ R jℓki . From Lemma 4.1, part (ii), we have that | ˜ R jℓki | g Ce t/ , while the factthat ˆ g is a semi-flat product K¨ahler metric implies that the only nonzerocomponent of ˆ R is ˆ R , and so(8.5) | ˆ R jℓki | g C, from which (8.3) follows.We have that˜ ∇ p H ijk = ∂ p H ijk − ˜Γ ℓpj H iℓk − ˜Γ ℓpk H ijℓ + ˜Γ ipℓ H ℓjk . Thanks to (4.3) and (8.2), in these coordinates we can bound the | · | g normof the last three terms by Ce t . As for the first term, we have ∂ p H ijk = ∂ p ˜Γ ijk − ∂ p ˆΓ ijk , and ∂ p ˆΓ ijk = ˆ g iℓ ∂ p ∂ j ˆ g kℓ − ˆ g is ˆ g rℓ ∂ p ˆ g rs ∂ j ˆ g kℓ , which is zero except when i = j = k = p = 2, and so | ∂ p ˆΓ ijk | g C. On the other hand | ∂ p ˜Γ ijk | g Ce t , thanks to (4.14). This proves (8.4). (cid:3) We now start bounding the terms in (8.1). We have ∇ ℓ T ikj = ˜ ∇ ℓ ˜ T ikj − Ψ qℓj ˜ T ikq + H qℓj ˜ T ikq , and so thanks to Lemma 4.1 and Lemma 8.3 we have that |∇ ℓ T ikj | g Ce t/ + C S / . Therefore we can bound g ij g rs g ab (cid:16) ∇ r T bja + ∇ b T arj (cid:17) Ψ kip Ψ ℓsq g pq g kℓ + g ij g rs g ab (cid:16) ∇ r T bja + ∇ b T arj (cid:17) Ψ kpi Ψ ℓqs g pq g kℓ − g ab (cid:0) ∇ k T bsa + ∇ b T aks (cid:1) Ψ kip Ψ sjq g ij g pq C ( e t/ + S / ) S . (8.6)Next, we compute ∇ a ∇ b T ijk = ∇ a ( ˜ ∇ b ˜ T ijk − Ψ rbk ˜ T ijr + H rbk ˜ T ijr )= ˜ ∇ a ˜ ∇ b ˜ T ijk − Ψ rab ˜ ∇ r ˜ T ijk − Ψ rak ˜ ∇ b ˜ T ijr + H rab ˜ ∇ r ˜ T ijk + H rak ˜ ∇ b ˜ T ijr − ( ∇ a Ψ rbk ) ˜ T ijr − Ψ rbk ˜ ∇ a ˜ T ijr + Ψ rbk Ψ sar ˜ T ijs − Ψ rbk H sar ˜ T ijs + ( ˜ ∇ a H rbk ) ˜ T ijr + H rbk ˜ ∇ a ˜ T ijr − Ψ sab H rsk ˜ T ijr − Ψ sak H rbs ˜ T ijr + H sab H rsk ˜ T ijr + H sak H rbs ˜ T ijr . OLLAPSING OF THE CHERN-RICCI FLOW ON ELLIPTIC SURFACES 39
Using again Lemma 4.1 and Lemma 8.3 we can bound |∇ a ∇ b T ijk | g C ( e t + e t/ S / + S + |∇ Ψ | g ) , and so(8.7) − (cid:18) g rs ( ∇ i ∇ p T sℓr )Ψ ℓjq g ij g pq (cid:19) C ( e t S / + e t/ S + S / + |∇ Ψ | g S / ) . Similarly we have ∇ a ∇ b T ijk = ∇ a ( ˜ ∇ b ˜ T ijk − Ψ rbk ˜ T ijr + H rbk ˜ T ijr )= ˜ ∇ a ˜ ∇ b ˜ T ijk − Ψ rai ˜ ∇ b ˜ T rjk − Ψ raj ˜ ∇ b ˜ T irk + H rai ˜ ∇ b ˜ T rjk + H raj ˜ ∇ b ˜ T irk − ( ∇ a Ψ rbk ) ˜ T ijr − Ψ rbk ˜ ∇ a ˜ T ijr + Ψ rbk Ψ sai ˜ T sjr + Ψ rbk Ψ saj ˜ T isr − Ψ rbk H sai ˜ T sjr − Ψ rbk H saj ˜ T isr + ( ˜ ∇ a H rbk ) ˜ T ijr + H rbk ˜ ∇ a ˜ T ijr − Ψ sai H rbk ˜ T sjr − Ψ saj H rbk ˜ T isr + H sai H rbk ˜ T sjr + H saj H rbk ˜ T isr . Using again Lemma 4.1 and Lemma 8.3 we can bound |∇ a ∇ b T ijk | g C ( e t + e t/ S / + S + |∇ Ψ | g ) , and so(8.8) − (cid:18) g rs ∇ i ∇ s T rpℓ Ψ ℓjq g ij g pq (cid:19) C ( e t S / + e t/ S + S / + |∇ Ψ | g S / ) . Next, we have ∂ p Ψ ℓqk = ˆ R qpkℓ − R qpkℓ , and so we can bound(8.9)2Re (cid:18) g rs T air R aspℓ Ψ ℓjq g ij g pq (cid:19) C |∇ Ψ | g S / + C | d Rm | g S / C (1+ |∇ Ψ | g ) S / , because of (8.5).Finally, we have ∇ r ˆ R ispk = ˆ ∇ r ˆ R ispk − Ψ ℓri ˆ R ℓspk − Ψ ℓrp ˆ R isℓk + Ψ krℓ ˆ R ispℓ . But ˆ g is a product of K¨ahler-Einstein metrics on Riemann surfaces, thereforeˆ ∇ r ˆ R ispk = 0. Using (8.5) again, we conclude that(8.10) − (cid:18) g rs g kℓ ∇ r ˆ R ispk Ψ ℓjq g ij g pq (cid:19) C S . Putting together (8.1), (8.6), (8.7), (8.8), (8.9) and (8.10), we concludethat(8.11) (cid:18) ∂∂t − ∆ (cid:19) S C ( e t/ S + e t S / + S / ) − |∇ Ψ | g − |∇ Ψ | g . Next, we define K¨ahler metrics on U by ˆ ω t = e − t ω E + ω S . These areuniformly equivalent to ω independent of t thanks to Theorem 5.1. Further-more, the covariant derivative of ˆ ω t is independent of t and equal to that ofˆ ω , and we will denote it by ˆ ∇ as before. The same is true for the curvatureof ˆ ω t , which equals ˆ R ijkp . We use [36, Proposition 3.1] to compute (cid:18) ∂∂t − ∆ (cid:19) tr ˆ ω t ω = − g pj g iq ˆ g kℓt ˆ ∇ k g ij ˆ ∇ ℓ g pq − g ij ˆ g kℓt ˆ g pqt g kq ˆ R iℓpj − g ij ˆ g kℓt ˆ ∇ i ˜ T jℓk − g ij ˆ g kℓt ˆ ∇ ℓ ˜ T ikj − tr ˆ ω t ω + e − t ˆ g iℓt ˆ g kjt g ij ( g E ) kℓ . We have that ˆ ∇ i ˜ T jℓk = ˜ ∇ i ˜ T jℓk + H pik ˜ T jℓp , and so we can use Lemma 4.1 and (8.2) to bound | ˆ ∇ i ˜ T jℓk | g Ce t/ . Hence, making use of (8.5), we have(8.12) (cid:18) ∂∂t − ∆ (cid:19) tr ˆ ω t ω − C − S + Ce t/ , for a uniform C >
Proof.
Let K be a large constant such that K K − tr ˆ ω t ω K, whose value will be fixed later. Let 0 ρ B in S , which is identically 1 in asmaller neighborhood of y , and denote the pullback ρ ◦ π also by ρ . Considerthe quantity Q = ρ e − t/ S K − tr ˆ ω t ω + tr ˆ ω t ω on supp( ρ ) ⊂ U. Our goal is to obtain an upper bound for Q , giving the bound S Ce t/ on a smaller neighborhood of U , which we may assume contains V . We willapply the maximum principle to this function Q , noting that it is equal totr ˆ ω t ω , and hence is bounded, on the boundary of supp( ρ ).We start with the following observations. Since ρ is the pullback of afunction from the base S , we have from the estimate Cω > ω S ,(8.13) |∇ ρ | g C, | ∆ ρ | g C, independent of t . Furthermore, we have the simple inequalities (see [23,(3.9), (3.10)]),(8.14) |∇ tr ˆ ω t ω | g C S , |∇S| g S ( |∇ Ψ | g + |∇ Ψ | g ) , OLLAPSING OF THE CHERN-RICCI FLOW ON ELLIPTIC SURFACES 41 where the first one also follows from the argument for (6.20).We can now compute (cid:18) ∂∂t − ∆ (cid:19) Q = − ρ e − t/ S K − tr ˆ ω t ω ) + ρ e − t/ K − tr ˆ ω t ω (cid:18) ∂∂t − ∆ (cid:19) S + ρ e − t/ S ( K − tr ˆ ω t ω ) ! (cid:18) ∂∂t − ∆ (cid:19) tr ˆ ω t ω − ρ e − t/ Re h∇S , ∇ tr ˆ ω t ω i g ( K − tr ˆ ω t ω ) − ρ e − t/ S ( K − tr ˆ ω t ω ) |∇ tr ˆ ω t ω | g − ∆( ρ ) e − t/ S K − tr ˆ ω t ω ! − ρ e − t/ K − tr ˆ ω t ω Re h∇ ρ, ∇Si g − ρ e − t/ S ( K − tr ˆ ω t ω ) Re h∇ ρ, ∇ tr ˆ ω t ω i g . Let ( x , t ) be a point in supp( ρ ) at which Q achieves a maximum. We mayassume without loss of generality that t > x lies in the interior ofsupp( ρ ). We have at ( x , t ),2 ρ ∇ ρ e − t/ S K − tr ˆ ω t ω + ρ e − t/ K − tr ˆ ω t ω ∇S + ρ e − t/ S ( K − tr ˆ ω t ω ) ∇ tr ˆ ω t ω + ∇ tr ˆ ω t ω = 0 . Taking the inner product of this with ∇ tr ˆ ω t ω , we see that, at this point,0 (cid:18) ∂∂t − ∆ (cid:19) Q = − ρ e − t/ S K − tr ˆ ω t ω ) + ρ e − t/ K − tr ˆ ω t ω (cid:18) ∂∂t − ∆ (cid:19) S + ρ e − t/ S ( K − tr ˆ ω t ω ) ! (cid:18) ∂∂t − ∆ (cid:19) tr ˆ ω t ω − ∆( ρ ) e − t/ S K − tr ˆ ω t ω ! − ρ e − t/ K − tr ˆ ω t ω Re h∇ ρ, ∇Si g + 2 |∇ tr ˆ ω t ω | K − tr ˆ ω t ω . (8.15)We use (8.11), (8.12), (8.13), (8.14) to obtain at this point,0 ρ e − t/ K (cid:18) C ( e t/ S + e t S / + S / ) − |∇ Ψ | g − |∇ Ψ | g (cid:19) + ρ e − t/ S K ! (cid:18) − S C + Ce t/ (cid:19) + Ce − t/ S K + ρ e − t/ K (cid:0) |∇ Ψ | g + |∇ Ψ | g (cid:1) + C S K , where we have used the Young inequality and (8.14):4 ρ e − t/ K − tr ˆ ω t ω |h∇ ρ, ∇Si g | Ce − t/ S K + ρ e − t/ K |∇S| S Ce − t/ S K + ρ e − t/ K (cid:0) |∇ Ψ | g + |∇ Ψ | g (cid:1) . Suppose that at ( x , t ) we have e − t/ S K − >
1, and hence e t S / K − / (otherwise, e − t/ S is bounded and thus so is Q ). We may also assume that S is much larger than K , say S > K . Then at this point, S C + 4 ρ e − t/ S CK Cρ e − t/ K S / K / + S K / + S K ! + C S / K / + Cρ e − t/ S / K / + Ce − t/ S K + C S K Choosing K to be much larger than C , we see that the second term on theleft hand side of this inequality dominates all the terms involving ρ on theright hand side. This gives S C − CK / S / − Ce − t/ K − CK ! , a contradiction since we chose K to be much larger than C . It follows that Q is uniformly bounded from above at the point ( x , t ). This completesthe proof of the theorem. (cid:3) Proofs of the main results
In this section we give the proofs of Theorem 1.1 and Corollary 1.2.
Proof of Theorem 1.1.
The estimate proved in Theorem 7.1 immediately im-plies that given any 0 < ε < / C such that k ω ( t ) − ˜ ω ( t ) k C ( M,g ) Ce − εt . From the definition of ˜ ω ( t ) = e − t ω flat + (1 − e − t ) π ∗ ω S we deduce that k ω ( t ) − π ∗ ω S k C ( M,g ) Ce − εt . The Gromov-Hausdorff convergence of (
M, ω ( t )) to ( S, ω S ) follows fromLemma 9.1 below. Finally, given any y ∈ S , the exponential convergence of e t ω ( t ) | E y to ω flat ,y in the C ( E y , g ) topology (uniformly in y ), follows fromCorollary 8.2 and the compactness of M . (cid:3) We used the following elementary result, which is undoubtedly well-known(cf. [37, Theorem 8.1]).
Lemma 9.1.
Let π : M → S be a fiber bundle, where ( M, g M ) and ( S, g S ) are closed Riemannian manifolds. If g ( t ) , t > , is a family of Riemannianmetrics on M with k g ( t ) − π ∗ g S k C ( M,g M ) → as t → ∞ , then ( M, g ( t )) converges to ( S, g S ) in the Gromov-Hausdorff sense as t → ∞ . OLLAPSING OF THE CHERN-RICCI FLOW ON ELLIPTIC SURFACES 43
Proof.
For any y ∈ S we denote by E y = π − ( y ) the fiber over y . Fix ε > L t the length of a curve in M measured with respect to g ( t ), andby d t the induced distance function on M . Similarly we have L S , d S on S .Using the standard formulation of Gromov-Hausdorff convergence (see e.g.[37]), let F = π : M → S and define a map G : S → M by sending everypoint y ∈ S to some chosen point in M on the fiber E y . The map G will ingeneral be discontinuous, and it satisfies F ◦ G = Id, so(9.1) d S ( y, F ( G ( y ))) = 0 . On the other hand since g ( t ) | E y goes to zero, we have that for any t largeand for any x ∈ M (9.2) d t ( x, G ( F ( x ))) ε. Next, given two points x , x ∈ M let γ : [0 , L ] → S be a unit-speed mini-mizing geodesic in S joining F ( x ) and F ( x ). Since the bundle π is locallytrivial, we can cover the image of γ by finitely many open sets U j , j N, such that π − ( U j ) is diffeomorphic to U j × E (where E is the fiber of thebundle) and there is a subdivision 0 = t < t < · · · < t N = L of [0 , L ]such that γ ([ t j − , t j ]) ⊂ U j . Fix a point e ∈ E , and use the trivializationsto define ˜ γ j ( s ) = ( γ ( s ) , e ), for s ∈ [ t j − , t j ], which are curves in M with theproperty that | L t (˜ γ j ) − L S ( γ | [ t j − ,t j ] ) | ε/N, as long as t is sufficiently large (because g ( t ) → π ∗ g S ). The points ˜ γ j ( t j ) and˜ γ j +1 ( t j ) lie in the same fiber of π , so we can join them by a curve containedin this fiber with L t -length at most ε/ N (for t large). We also join x with ˜ γ (0) and x with ˜ γ N ( L ) in the same fashion. Concatenating these“vertical” curves and the curves ˜ γ j , we obtain a piecewise smooth curve ˜ γ in M joining x and x , with π (˜ γ ) = γ and | L t (˜ γ ) − d S ( F ( x ) , F ( x )) | ε. Therefore,(9.3) d t ( x , x ) L t (˜ γ ) d S ( F ( x ) , F ( x )) + 2 ε. Since F ◦ G = Id , we also have that for all t large and for all y , y ∈ S ,(9.4) d t ( G ( y ) , G ( y )) d S ( y , y ) + 2 ε. Given now two points x , x ∈ M , let γ be a unit-speed minimizing g ( t )-geodesic joining them. If we denote by L π ∗ g S ( γ ) the length of γ using thedegenerate metric π ∗ g S , then we have for t large,(9.5) d S ( F ( x ) , F ( x )) L S ( F ( γ )) = L π ∗ g S ( γ ) L t ( γ ) + ε = d t ( x , x ) + ε, where we used again that g ( t ) → π ∗ g S . Obviously this also implies that forall t large and for all y , y ∈ S ,(9.6) d S ( y , y ) d t ( G ( y ) , G ( y )) + ε. Combining (9.1), (9.2), (9.3), (9.4), (9.5) and (9.6) we get the requiredGromov-Hausdorff convergence. (cid:3)
Proof of Corollary 1.2.
The proof is similar to [37, Theorem 8.2]. From [3,Lemmas 1, 2] or [38, Theorem 7.4] we see that there is a finite unramifiedcovering p : M ′ → M (with deck transformation group Γ) which is alsoa minimal properly elliptic surface π ′ : M ′ → S ′ and π ′ is an elliptic fiberbundle with S ′ a compact Riemann surface of genus at least 2. Furthermore,Γ also acts on S ′ (so that π ′ is Γ-equivariant), with finitely many fixed pointswhose union Z is precisely the image of the multiple fibers of π , with quotient S = S ′ / Γ, and so that the quotient map q : S ′ → S satisfies q ◦ π ′ = π ◦ p. Denote by ω S ′ the orbifold K¨ahler-Einstein metric on S ′ with Ric( ω S ′ ) = − ω S ′ . From the description of M and M ′ as quotients of H × C ∗ , where H is the upper half plane in C (see e.g. [17], [37, Section 8]), it follows that π ′∗ ω S ′ is a smooth real (1 ,
1) form on M ′ , which also equals p ∗ π ∗ ω S . Indeed,if we let z ∈ H be the variable in the upper half plane, w ∈ C ∗ , and y = Im z ,then from the arguments in [37, Section 8] we see that the form π ∗ ω S on M is induced from the form y √− dz ∧ dz on H × C ∗ , and the exact sameformula holds on M ′ .Given any Gauduchon metric ω on M , call ω ( t ) its evolution under thenormalized Chern-Ricci flow on M , as before. Let ω ′ = p ∗ ω , which is aΓ-invariant Gauduchon metric on M ′ . If we call ω ′ ( t ) its evolution under thenormalized Chern-Ricci flow on M ′ , then ω ′ ( t ) is also Γ-invariant, and equalto p ∗ ω ( t ). Furthermore, Γ also acts by isometries of the distance function d S ′ of ω S ′ , with quotient space ( S, d S ), the distance function of the orbifoldmetric ω S .Now Theorem 1.1 applied to the elliptic bundle π ′ : M ′ → S ′ showsthat ( M ′ , ω ′ ( t )) converges to ( S ′ , ω S ′ ) in the Gromov-Hausdorff topology.But exactly as in [37, Theorem 8.2] we see that the convergence happensalso in the Γ-equivariant Gromov-Hausdorff topology, and therefore by [10,Theorem 2.1] or [21, Lemma 1.5.4] we conclude that ( M, ω ( t )) converges to( S, d S ) in the Gromov-Hausdorff topology.Now we apply Theorem 1.1 again to M ′ to see that k ω ′ ( t ) − π ′∗ ω S ′ k C ( M ′ ,p ∗ g ) Ce − εt . Fix now an open set U of M , small enough so that p − ( U ) is a disjoint unionof finitely many copies U j of U . Then p : U j → U is a biholomorphism foreach j and the Γ-action on p − ( U ) permutes the U j ’s. Therefore for each j , the map p : U j → U gives an isometry between ω ′ ( t ) | U j and ω ( t ) | U ,and also between ( π ′∗ ω S ′ ) | U j and ( π ∗ ω S ) | U . Fixing one value of j , from k ω ′ ( t ) − π ′∗ ω S ′ k C ( U j ,p ∗ g ) Ce − εt , we conclude that k ω ( t ) − π ∗ ω S k C ( U,g ) Ce − εt . Covering M by finitely many such open sets U shows that ω ( t ) converges to π ∗ ω S in the C ( M, g ) topology.Finally, fix any point y ∈ S \ Z , and let V be a small open neighborhoodof y such that π − ( V ) ∼ = V × E and q − ( V ) is a disjoint union of finitelymany copies V j of V with points y j ∈ V j mapping to y and with q : V j → V OLLAPSING OF THE CHERN-RICCI FLOW ON ELLIPTIC SURFACES 45 a biholomorphism. Then π ′− ( V ) ∼ = V × E , and under these identificationsthe biholomorphism p : π ′− ( V ) → π − ( V ) equals ( q, Id) : V × E → V × E . Under this map the fiber E ′ y := π ′− ( y ) is carried to the fiber E y .Applying Theorem 1.1 to M ′ , we see that e t ω ′ ( t ) | E ′ y converges exponentiallyfast in the C ( E ′ y , g ′ ) topology to ω ′ flat ,y , the flat K¨ahler metric on E ′ y cohomologous to [ ω ′ | E ′ y ], and the convergence is uniform when varying y .But the local biholomorphism p maps ω ′ ( t ) to ω ( t ), and ω ′ flat ,y to ω flat ,y ,and the result follows. (cid:3) Acknowledgements.
The authors thank the referee for some suggestionswhich improved the presentation.
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