New Geometric Flows

National Science Review REVIEW 5: 534–541, 2018 doi: 10.1093/nsr/nww053 Advance access publication 12 September 2016

MATHEMATICS New geometric ﬂows Gang Tian1,2

ABSTRACT Downloaded from https://academic.oup.com/nsr/article/5/4/534/2615238 by guest on 27 September 2021 This is an expository paper on geometric flows. I will start with a brief tour on Ricci flow, thenIwill discuss two geometric flows introduced in my joint work with J. Streets and some recent workson them.

Keywords: Ricci flow, metrics, AMMP, Kahler,¨ Hermitian, symplectic

RICCI FLOW AND ITS APPLICATION First, we recall that a metric g is Einstein if Ric(g) = λ g, where λ is a constant, often nor- We will assume that M is a closed manifold of dimen- malized to be −(n − 1), 0 or n − 1. Note sion n and g is fixed. The Ricci flow was introduced 0 that Ric(g) = (R ) denotes the Ricci curvature by Hamilton in [1]: ij of g and measures the deviation of volume form from the Euclidean one. In dimension 2, an Ein- ∂g stein metric is simply a metric of constant Gauss ij =−2R , g (0) = a given metric. ∂t ij curvature. (1) Now we assume that M is a closed 3-manifold. It Hamilton and later DeTurck proved that for any ini- is known that any Einstein metric on a 3-manifold tial metric there is a unique solution g(t)onM × [0, is of constant sectional curvature. Therefore, it fol- T)forsomeT > 0. Hamilton also established an an- lows from the classical uniformization theorem that alytic theory for Ricci flow. It provides a way of con- if M admits an Einstein metric, then its universal cov- structing Einstein metrics as one has done for 2D or ering is of the form either round sphere S3 or R3 or 3D manifolds. the hyperbolic 3-ball H3. Thus, if we can construct an If M is a 2-manifold, it is proved in [2,3] that given Einstein metric, then we have a similar picture for 3- 1School of any initial g0 on M, (1) has a global solution g(t) after manifolds as we have for surfaces. However, not ev- Mathematical normalization and g(t) converges to a metric g∞ on ery 3-manifold admits an Einstein metric. One can Sciences, Peking 1 M. One can show that g∞ is of constant Gauss curva- easily construct such examples, such as × S for University, Beijing any surface of genus > 1. This is because its funda- 100871, China and ture. The proof is trivial if M has non-positive Euler 2 number and contained in [4]ifM has positive Eu- mental group is the product of a surface group with Department of Z Mathematics, ler number. Thus, combined with the classical uni- which is neither an Abelian group nor the funda- Princeton University, formization theorem on surfaces, we can use Ricci mental group of any hyperbolic compact 3-manifold Princeton, NJ 08544, flow to prove the following. The universal covering (cf. [5]). USA of M is isomorphic to either the 2D round sphere S2 It was known [6] that any closed 3-manifold can or the Euclidean plane R2 or the hyperbolic disk D. be decomposed along embedded 2-spheres into ir- E-mail: Of course, this conclusion can be also obtained by reducible 3-manifolds; moreover, such a decompo- [email protected] the elliptic method or the Riemann mapping theo- sition is essentially unique. Thurston’s Geometriza- rem. However, the method of Ricci flow has much tion Conjecture claims (cf. [5]) that any irreducible Received 18 July more advantages since it can be generalized to higher 3-manifold can be decomposed along incompress- 2016; Revised 15 dimensions. ible tori into finitely many complete Einstein 3- August 2016; A celebrated example is Perelman’s resolution for manifolds plus some Graph manifolds. The famous Accepted 15 August Thurston’s Geometrization Conjecture, which im- Poincare´ Conjecture is a special case of this Ge- 2016 plies the famous PoincareConjecture.´ ometrization Conjecture.

The Geometrization Conjecture was solved by If M is simply connected or its fundamental group Perelman (cf. [7,8]) using the Ricci flow. is not too big, Perelman showed us in [9]whyMt has If the Ricci flow has a solution g(t), then we can to become empty for t sufficiently large, i.e. the Ricci choose a scaling λ(t) > 0 and a reparametrization flow with surgery becomes extinct in finite time. In t = t(s) with λ(0) = 1 and t(0) = 0suchthat particular, the Poincare´ Conjecture follows. One can g˜(s ) = λ(s )g (t(s )) has fixed volume and satisfies find detailed proof of this finite extinction10 in[ ] the normalized Ricci flow: and [11]. To solve Thurston’s Geometrization Con- ∂g˜ R˜(s ) ij =−2 R˜ − g˜ , (2) jecture, one needs to study asymptotic behavior ∂s ij 3 ij of the above Ricci flow with surgery (in 2012, Bamler found and fixed a gap in section 6.4 of Downloaded from https://academic.oup.com/nsr/article/5/4/534/2615238 by guest on 27 September 2021 where R˜ij denotes the Ricci curvature of g˜(s ), and [9] that had been missed) and prove a result R˜(s ) denotes the average of the scalar curvature of on collapsing 3-manifolds with curvature locally g˜(s ). If the normalized Ricci flow (2) has a global bounded from below and geodesically convex solution g˜(s )foralls ≥ 0 and g˜(s ) converges to a boundary. smooth metric g∞ as s goes to ∞, then its limiting Perelman did not prove that there are only metric g∞ is an Einstein metric, so the universal cov- finitely many surgeries. It was accomplished by Bam- ering of M is standard. The Geometrization Conjec- ler in [12]. Moreover, he proved that after finitely ture follows. many surgeries, the normalized Ricci flow g(t)has In 1982, Hamilton proved that for any 3- bounded curvature and consequently, converges to manifold with a metric g0 with positive Ricci curva- limiting spaces (possibly collapsed) that fall into one ture, (2) with initial metric g0 has a global solution of the eight geometries classified by Thurston. This that converges smoothly to a metric of constant pos- limit should be unique, but a complete proof is still itive curvature. It follows that M is a quotient of S3 missing though it seems to be doable. Thus, if we by a finite group. could further prove that finite-time surgery is canon- However, in general, (1) or (2) develops singu- ical, then one would have a refined version of the larity at finite time, so one has to introduce amore Geometrization Conjecture: given (M, g0), there is general evolution process called ‘Ricci flow with a unique Ricci flow with surgery (Mt, g(t)) with ini- surgery’. This evolution process is still givenM by( t, tial value (M, g0) such that either Mt becomes ex- g(t)) satisfying: (i) there is a discrete set of times t0 tinct when t is sufficiently large or after normaliza- = 0 < t1 < ... < ti < ... without accumulation point tion, (Mt, g(t)) converges to a unique limit in the such that in each (ti, ti+1), Mt is a fixed smooth mani- eight geometries. fold and g(t) is the usual Ricci flow on Mt; (ii) at each For higher dimensions, we do not know much ti for i ≥ 1, the manifolds and metrics undergo topo- about singularity formation of Ricci flow unless the logical and metric discontinuities (surgeries), in par- underlying manifolds have certain structures. If the ticular, the topology type of Mt for T < ti changes initial metric g0 has either positive curvature op- to that of Mt for t > ti as t moves across ti.Inthis erator or positive bisectional curvature or positive definition, we did not specify how g(t) moves across isotropic curvature, so does g(t) along the Ricci flow. ti. We expect there is a canonical way for g(t)togo In particular, Brendle and Schoen solved a long- through the surgery time. This is still an open prob- standing problem: if a closed manifold admits a met- lem and has applications though it is not needed in ric with curvature 1/4-pinched, then it is diffeomor- Perelman’s resolution of the Geometrization Con- phic to a sphere. jecture. If the initial metric g0 is Kahler,¨ so does g(t) One of Perelman’s crucial advances is that one along the Ricci flow, thus we can restrict the Ricci needs to do surgeries only along two spheres; con- flow to Kahler¨ metrics and call it Kahler-Ricci¨ flow. sequently, he understood completely the change in The analytic minimal model program, abbreviated topology. More precisely, he proved: ‘There is a Ricci as AMMP (it is also referred as the Song-Tian pro- flow with surgery (Mt, g(t)) defined for all t ∈ [0, gram), is to use the Kahler-Ricci¨ flow to deform ∞) with any given initial 3-manifold (M, g0). The any Kahler¨ metric towards a canonical model metric, topological change in the 3-manifold as one crosses possibly through finitely many finite-time surgeries. a surgery time is a connected sum decomposition This gives a promising approach to classify all closed together with removal of connected components, Kahler¨ manifolds birationally. Let us briefly describe each of which is diffeomorphic to one of S2 × S1, the AMMP. 3 3 RP RP , or a manifold admitting a metric of con- Assume that g0 is a Kahler¨ metric on M, then in stant positive curvature’. local complex coordinates z1,... , zn, g0 is given by 536 Natl Sci Rev, 2018, Vol. 5, No. 4 REVIEW

{ } a positive Hermitain matrix-valued function g 0i j¯ . The first difficulty isthat MT may be singular so that Set one cannot continue (1) directly on MT. We need to n √ find a resolution π : M → MT such that the Ricci ω = − ∧ . 0 1 g 0i j¯ dzi dz¯ j flow can be made sense on M . The following conjec- i, j =1 ture specifies properties of M. Then it is a well-defined (1,1)-form on M, called the Kahler¨ form of g ; moreover, ω is closed 0 0 Conjecture 1.2. There is a Q-factorial variety M with and gives rise to a cohomology class [ω0]in , a holomorphic map π : M → M satisfying the H 1 1(M, C) ∩ H 2(M, R). It is proved in [13]that T T following. (1) has a maximal solution g(t)fort ∈ [0, T), where = { | ω − π > }. (i) M has only log-terminal singularities. T sup t [ 0] t 4 c 1(M) 0 ∗ Downloaded from https://academic.oup.com/nsr/article/5/4/534/2615238 by guest on 27 September 2021 (ii) For >0 sufficiently small,π [ ω¯ T ] + Note that [ω0] − t 4πc1(M)istheKahler¨ class K > 0. of g(t) since it is cohomologous to the Kahler¨ M form ω(t)ofg(t). For example, if M contains a In view of what we know from algebraic geome- > holomorphic sphere C with c1(M)([C]) 0, then try, there should be two cases for π T: M → MT: divi- ω − π [ 0] t4 c1(M) stops as a Kahler¨ class for some sorial contraction or small contraction. If π T: M → ≤ ω / π < ∞ = T [ 0]([C]) 4 c1(M)([C]) ,so(1)devel- MT is divisorial, we suppose to choose MT MT ops singularity at T. What can one do at a finite-time and continue the flow on MT.Ifπ T is small, i.e. the singularity? complex codimension of the exceptional locus is big- First, it was shown by Z. Zhang that as t tends ger than 1, then MT may not be the same as MT and to T,theKahler¨ form ω(t)ofg(t) converges to a can be referred as a flip of M. ω 1 unique positive (1,1)-current T in the L -topology If Conjecture 1.2 is affirmed, we get a flip M of on Kahler¨ potentials. It is natural to ask how regular M. The next question is to extend (1) to M.Thishas ω T is and what are its geometric properties? The fol- been done by Song and myself. We have proved that lowing conjecture in AMMP concerns these. the Ricci flow can be defined on M from Conjecture 1.2 and there is a unique solution ω(t)ofthisflow π ∗ω Conjecture 1.1. Let (M, g0) be a compact Kahler¨ man- with initial data ¯ T in the sense of distribution, ifold with T < ∞, then (M, g(t)) converges to a met- where T ≤ t < T + T1. Thus, we extend Ricci flow ω × × ric space (MT, dT) in the Gromov−Hausdorff topol- (t)onM [0, T) across T to a Ricci flow on M ogy. [T, T + T1). So, we have a solution ω(t) with surgery for t ∈ [0, T + T ). As usual, we call T a surgery time. (i) MT is a compact Kahler¨ space and there is a 1 Repeating this process, we can construct a Kahler-¨ holomorphic map π T: M → MT. Ricci flow with surgery on M with initial metric ω (ii) The current ωT descends to a current ω¯ T 0 ω = π ∗ ω for all t ≥ 0. This means that there is a sequence of on MT, i.e. T T ¯ T , which is a smooth 0 ⊂ Ricci flows ωi(t)onMi × [Ti−1, Ti), where i = 0, 1, Kahler¨ metric on MT MT and induces the 0 ... (possibly only finitely many), satisfying: distance function dT, where MT denotes the set of regular values of π . T ST1. T−1 = 0 < T1 < ... and for any T < ∞,there (iii) (M, ω(t)) converges to (M, ω )intheC∞- T are only finitely many Ti’s which are

either empty or a Q-factorial Kahler¨ variety Mmin The Hermitian curvature flows introduced by ≥ with nef canonical bundle K Mmin 0. Streets and me are given by If MN =∅, we say that the Ricci flow (1) becomes ∂ extinct at T = T − . I have conjectured g N 1 =−S + Qˆ (T), (3) ∂t Conjecture 1.4. The Ricci flow onaKahler¨ manifold ˆ M becomes extinct at finite time if and only if M is where Q(T) is a quadratic function of torsion T birational to a Fano-like manifold. and of (1,1)-type. The flows in (3) coincide with the Partial progress on this conjecture has been made Ricci flow if the initial metric isKahler.¨ It is proved in [18] that for any given Hermitian metric g0,(3) by Song and his collaborators. Furthermore, for a = similar reason, we expect that the Ricci flow (1) col- has a unique solution g(t)on[0,T) with g(0) g0 > Downloaded from https://academic.oup.com/nsr/article/5/4/534/2615238 by guest on 27 September 2021 lapses at finite time if and only if M is birational to a for some T 0. Streets and I also developed an ana- uniruled manifold. lytic theory for these Hermitian curvature flows analogous to that by Hamilton-Shi for the Ricci flow. If MN is not empty, then it has nef K MN ,soit follows from the sharp local existence theorem of There are two important cases of (3) which Iwill Song and myself that (1) has a global g(t)fort > discuss in more detail. ∗ The first is the pluri-closed flow. A Hermitian − π ω¯ − TN 1 with initial value N N 1. Next, we examine ω the limit of g(t)ast →∞. This has been done in manifold (M, g) is pluriclosed if the Kahler¨ form of g satisfies (we√ can associate to g a canonical form a rather satisfactory way: under mild assumptions ω − ∧ which is equal to 1 g i j¯dzi dz¯ j in any lo- on MN, either g(t) converges to a Calabi-Yau metric with possible singularity or t−1g(t) converges to cal coordinates z1,... ,zn. As usual in Hermitian geometry, we refer ω as the Kahler¨ form of g even if g aKahler-Einstein¨ metric with negative scalar curva- ∂∂ω¯ = ture or a generalized Kahler-Einstein¨ metric intro- is not Kahler.)¨ 0. This condition is weaker duced by Song and Tian in [14]. We refer the readers than the Kahler¨ condition, but there are many pluri- to [14,15] and [16] for details. closed manifolds, e.g. generalized Kahler¨ manifolds There are strong supporting evidences for valid- first studied by physicists and complex-symplectic ity of Conjecture 1.1 and Conjecture 1.2. In [17], manifolds named by Streets and me. Also, it follows using deep results in algebraic geometry, we have es- from Gauduchon that every complex surface admits tablished these two conjectures for M being a projec- pluri-closed metrics. = kl¯ pq¯ ˆ Putting Qi j¯ g g Tkqi¯ Tlp¯ j¯ and replacing Q tive manifold of general type and ω0 having rational Kahler¨ class. by Q in (3), we get the pluri-closed flow: ∂ g i j¯ kl¯ pq¯ =−S ¯ + g g Tkqi¯ T¯ ¯, g (0) = g 0. HERMITIAN CURVATURE FLOW ∂t i j lpj (4) In this section, we consider Hermitian manifolds of It is proved in [19] that (4) preserves the pluri- complex dimension n, i.e. those complex manifolds closed property, i.e. g(t) stays pluri closed whenever M equipped with Hermitian metrics g. The Ricci flow g0 is pluri closed. (1) does not preserve the Hermitian structures, that In [20], (4) is reformulated by using the Bismut is, if g(t) is a solution of (1) and g(0) is Hermi- connection. Recall that the Bismut connection ∇ is tian, g(t) may not be Hermitian in general. So we defined by: need new curvature flows in order to study Hermi- tian manifolds. In [18], Streets and I introduced a g (∇X Y, Z) = g (D X Y, Z) family of Hermitian curvature flows. First, let us re- 1 call some facts. In local coordinates z1,... ,zn,aHer- + dω(JX, JY, JZ), (5) mitian metric g is given by a Hermitian matrix-valued 2 { ¯} ¨ function g i j . Its Kahler form is given as in last sec- where D denotes the Levi-Civita connection and J is tion, but it may not be closed. The ‘Ricci’ curvature the complex structure of M. Equivalently, the Bismut of the Chern connection of g is given by: connection can be characterized as the unique con- 2 ∂ ¯ ∂ ∂g ¯ =− i j¯ g kl + pq¯ g kq¯ pl . nection compatible with both g and J such that the Skl¯ g g ∂zi ∂z¯ j ∂zi ∂z¯ j torsion is a 3-form. In terms of the Bismut connec- ∂ω tion, this torsion 3-form is closed if and only if g is The torsion T of g is defined by ,i.e. ∇ ∂ ∂ pluri closed. Let P be the ‘Ricci’ curvature of ,that g i j¯ g k j¯ pq = − , = . is, Pij = ω ijpq, where denotes the curvature of Ti jk¯ ∂ ∂ Tji¯ l¯ Tj¯il zk zi ∇. Unlike S from Chern connection, P may not be of Clearly, g is Kahler¨ if and only if T = 0. type (1,1). If we denote by P1,1 the (1,1)-component 538 Natl Sci Rev, 2018, Vol. 5, No. 4 REVIEW

of P, then we can rewrite (4) as Define the space P∂+∂¯ to be the cone of the classes H1,1 in ∂+∂¯ which contain pluri-closed metrics. ∂ω =− 1,1. ∂ P (6) t Conjecture 2.1. Let (M, g0) be a pluri-closed manifold with Kahler¨ form ω0.Define An application of (6) is that the pluri-closed flow τ ∗ = { | ω − π ∈ P }. can be related to the B-field renormalization group : sup t [ 0] t4 c 1(M) ∂+∂¯ t≥0 flow in the string theory (cf.21 [ ]): Then (4) has a maximal solution with initial met- τ ∗ ∂ ric g0 on [0, ). g ij =− + 1 pq 2Rcij H H jpq A positive resolution of Conjecture 2.1 would ∂t 2 i

have geometric consequences for non-Kahler¨ sur- Downloaded from https://academic.oup.com/nsr/article/5/4/534/2615238 by guest on 27 September 2021 ∂ H faces. This is because one can characterize the cone and = LBH, (7) ∂t P∂+∂¯ on non-Kahler¨ surfaces in a rather easy way. Combining this with the fact that every complex

where H is a 3-form and LB denotes the Laplace- surface is pluri-closed, we can use (4) to study geom- Beltrami operator of the time-dependent metric. etry and topology of complex surfaces, particularly, More precisely, let g(t) be a solution to (3) with still largely mysterious Class VII+ surfaces. Kahler¨ form ω(t) and the torsion T(t) of the Bismut Recently, Streets has made substantial progress ∗ ∗ connections associated to g(t), then (φ g, φ T)isa on Conjecture 2.1. We refer the readers to [22,23] solution of (7) (cf. [20]), where φ(t) is the integral and their references for details. ∗ curve of the time-dependent field dual to − 1 Jd ω Another special case of Hermitian curvature ∗ 2 (note that complex structures φ(t) J vary in t,soit flows is the Hermitian-Chern flow: is not a prior clear how a solution of (7) goes back ∂g ¯ to that of the pluri-closed flow since (7) does not tell i j =− − 1 kl¯ pq¯ , = . Si j¯ g g Tk jp¯ Tli¯ q¯ g (0) g 0 how complex structures vary). ∂t 2 It is proved in [21] that (7) is the gradient flow of (10) the following Perelman-type functional: Ustinovskiy has proved that this flow has very nice properties. Let me summarize briefly his 1 work [24]. λ(g , H) = inf R − |H|2 + f 2 We say that a Hermitain metric g is Griffiths non- 12 M negative (resp. Griffiths positive) at x ∈ M if for any ξ,η ∈ 1,0 Tx M,wehave: × − f − f = . e dV e dV 1 | ξ,ξ,η,¯ η ≥ .> , M x ( ¯) 0 (resp 0) (8) where denotes the curvature of the Chern connection. We say that g is Griffiths non-negative (resp. Hence, λ is monotonic along the pluri-closed flow. positive) if it does at every x ∈ M. Another application of (6) is that the B-field flow Ustinovskiy proved that if the initial metric g0 preserves the generalized Kahler¨ structures. This is is Griffiths non-negative, so does g(t) along the proved by Streets and me through the correspon- Hermitian-Chern flow (6). Moreover, if the Chern dence between our pluri-closed and the B-field flow. curvature at t = 0 is Griffiths-positive at some x ∈ M, Therefore, our pluri-closed flow can be also usedto then for any t > 0, g is Griffths-positive. studying generalized Kahler¨ manifolds. Ustinovskiy also proved a strong Maximum prin- We can propose a program on classification of ciple analogous to that of Hamilton for the Ricci pluri-closed manifolds as well as generalized Kahler¨ flow. Assume that g0 is Griffiths non-negative. Then manifolds similar to our AMMP for Kahler¨ mani- for any t > 0, the set folds. For example, we can state a set of conjectures ={ξ,η | ξ,η ∈ 1,0 , analogous to Conjecture 1.1, 1.2 etc. in an identical Z ( ) T M way. The first step towards this desirable program is ξ,ξ,η,¯ η = } to prove a sharp local existence theorem similar to ( ¯) 0 thatofZ.ZhangandmyselfforKahler-Ricciflow.Let¨ is invariant under torsion-twisted parallel transport us state it precisely. A pluri-closed metric defines a ∇1 ξ = γ ,ξ , ∇2 η =− η, γ , · ∗. class in the Aeppli cohomology group: γ T( ) γ g ( T( ))

1,1 2,2 , Ker (∂∂¯): R → R The work of Ustinovskiy leads to many inter- H1 1 = . (9) ∂+∂¯ {∂γ + ∂¯γ¯ | γ ∈ 0,1} esting problems on the Hermitian-Chern flow and REVIEW Tian 539

spaces of Griffiths non-negative curvature. One of where the index on the Ricci tensor has been raised these problems is related to the generalized Frenkel with respect to the associated metric. This R is actu- conjecture affirmed by N. Mok. The conjecture ally the (2,0)+(0,2) part of the Ricci curvature of the claims that any closed complex manifold that associated metric. admits a Kahler¨ metric with non-negative bi- The symplectic curvature flow26 in[ ]isgivenby sectional curvature must be biholomorphic to a dω locally symmetric space. It is desirable to generalize =−P and this to cover homogeneous manifolds. One can dt check that any homogeneous manifold admits a dJ − , + , Griffiths non-negative Hermitian metric. One may =−g 1[P (2 0) (0 2)] + R. (13) propose dt Downloaded from https://academic.oup.com/nsr/article/5/4/534/2615238 by guest on 27 September 2021 A direct computation shows: Conjecture 2.2. Any closed complex manifold which − , + , ∗ admits a Griffiths non-negative Hermitian met- g 1[P (2 0) (0 2)] = [∇ ∇ J − N ] , (14) ric must be biholomorphic to a homogeneous space. where In view of Utinovskiy’s work, we have an ap- N a = ij a ∇ p ∇ c . proach through the Hermitian-Chern flow. b g J c i J b j J p (15)

It follows that (13) can be written as SYMPLECTIC GEOMETRIC FLOW dω dJ ∗ This section is essentially taken from25 [ ], but we =−P and =−∇∇ J + N + R. dt dt discuss general symplectic manifolds instead of just (16) 4-manifolds. Since Kahler¨ manifolds are special sym- Clearly, (13) is invariant under diffeomorphisms plectic manifolds, it is desirable to extend what we of M and called symplectic curvature flow or sym- have about the Ricci flow on Kahler¨ manifolds to plectic geometric flow, or simply referred as ST-flow. symplectic manifolds. But the Ricci flow does not Similar to the Ricci flow, the ST-flow is parabolic preserve the symplectic structure in general, so we only modulo the group of diffeomorphism. More need a new flow for studying symplecic manifolds. precisely, we have Fortunately, Streets and I found a new curvature flow for symplectic manifolds,26 see[ ]. Theorem 3.1. Let (M, ω , J ) be an almost-Kahler¨ Let (M, ω) be a symplectic manifold. An almost- 0 0 manifold. There exists a unique solution to (13) with Kahler¨ metric g on M is a Riemannian metric such initial condition (ω , J )on[0,T)forsomeT > 0. that: 0 0 The ST-flow has resemblance with the Ricci flow. g (u,v) = ω(u, J v),ω(Ju, J v) = ω(u,v), In terms of associated metrics g(t), the flow be- where J is an almost complex structure. Such a triple comes (M, ω, J) is called an almost Kahler.¨ ∂g The Chern connectionM of( , ω, J)isgivenby: =−2 Ric(g ) + B 1 + B 2, (17) ∂t 1 D Y =∇ Y − J (∇ J )(Y ). X X 2 X where This connection D preserves the metric g and al- 1 = kl ∇ p ∇ q most complex structure J, but it may have torsion. Bij g g pq i J k j J l The Chern connection D induces a Hermitian 2 = kl ∇ p ∇ q . connection on the anticanonical bundle, and we de- and Bij g g pq k J i l J j note the curvature form of this connection by P.If = { ijkl} denotes the curvature of D,wehave In [26], we derive induced evolution equations on the curvature Rm(g)ofg and its derivatives as well as kl Pij = ω ijkl, (11) derivatives of J. As consequences, by using the Max- imum principle, we prove that whenever the curva- where {ωij} is the inverse of ω. By the general ture of g(t) is bounded, so do all the derivatives of Chern-Weil theory, P is a closed form and represents the curvature and J(t), in particular, the deviation of 4πc1(M). J from being parallel or integrable is controlled by the Define an endomorphism curvature. We expect that all finite-time singularities of (13) R j = k j − k j i J i Rick Rici J k (12) are modeled on Kahler¨ spaces. There are many open 540 Natl Sci Rev, 2018, Vol. 5, No. 4 REVIEW

problems, for instance, we do not know if the ST- This is analogous to Thurston’s Geometrization flow (13) is a gradient flow or behaves like a gradient for 3-manifolds. flow; in particular, is there a non-trivial breather for (13)? My guess is no. Also we would like to have a FUNDING sharp local existence theorem for (13). Define This work was supported partially by grants from National Sci- ence Foundation and National Natural Science Foundation of T = sup{t | [ω0] − t4πc 1(M) China. has a symplectic representive}. (18) REFERENCES ω This may depend on [ 0]. We conjecture 1. Hamilton R. Three-manifolds with positive Ricci curvature. JDif- ferential Geom 1982; 17: 255–306. Downloaded from https://academic.oup.com/nsr/article/5/4/534/2615238 by guest on 27 September 2021 Conjecture 3.2. Given (ω0, J0), (13) has a maximal so- 2. Hamilton R. The Ricci flow on surfaces. In: Isenberg J (ed.). lution on [0, T) with initial value (ω0, J0). 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