Ricci-Yang-Mills flow on surfaces and pluriclosed flow on elliptic fibrations
aa r X i v : . [ m a t h . DG ] F e b RICCI-YANG-MILLS FLOW ON SURFACES AND PLURICLOSED FLOW ONELLIPTIC FIBRATIONS
JEFFREY STREETS
Abstract.
We give a complete description of the global existence and convergence for the Ricci-Yang-Mills flow on T k bundles over Riemann surfaces. These results equivalently describe solutionsto generalized Ricci flow and pluriclosed flow with symmetry. Introduction
Suppose
G → M → Σ is the total space of a principal G -bundle. Given a choice h of metricon the Lie algebra g of G , a one-parameter family of metrics g t on Σ and principal connections µ t satisfies the Ricci-Yang-Mills flow (RYM flow) [39, 43] if ∂∂t g = − g + F µ ,∂∂t µ = − d ∗ g F µ , (1.1)where F µ denotes the curvature of µ , and F µ = tr h tr g − F ⊗ F . This system of equations has arisenin mathematical physics literature in the study of renormalization group flows. Also, this flow arisesby modifying the Ricci flow of an invariant metric on a principal bundle to fix the metric on thefibers. This modification is a natural way to simplify the equation to understand the geometry ofprincipal bundles, while of course the Ricci flow of invariant metrics is also natural in understandingcollapsing limits and has been significantly studied (cf. [17]). Further results on RYM flow appearin [14, 27, 28, 42].A second origin of these equations is as a symmetry reduction of the generalized Ricci flow (cf.[9]). Given a smooth manifold M , a one-parameter family of metrics G t and closed three-forms H t is a solution of generalized Ricci flow if ∂∂t G = − G + H ,∂∂t H = ∆ d H, (1.2)where H = tr g − tr g − H ⊗ H . This system of equations also arises in the study of renormalizationgroup flows [4], coupling the Ricci flow for a metric with the natural heat flow for a closed three-form. As discussed in § g t , µ t ) over a Riemann surface,the pairs G t = π ∗ g t + tr h µ t ⊗ µ t , H t = tr h F µ ∧ µ satisfy generalized Ricci flow.A third origin of Ricci-Yang-Mills flow is complex geometry, where it arises as a special case of pluriclosed flow [36]. The pluriclosed flow is a natural evolution equation generalizing the K¨ahler-Ricci flow to complex, non-K¨ahler manifolds. Given ( M , J ) a complex surface, a one-parameter Date : February 19, 2021.We gratefully acknowledge support from the NSF via DMS-1454854. family of Hermitian, pluriclosed metrics ω t satisfies pluriclosed flow if ∂∂t ω = − ρ , B , where ρ B is the Ricci curvature of the Bismut connection. In [38] it was shown that after a certaingauge modification pluriclosed flow is in fact equivalent to generalized Ricci flow. Furthermore, itwas shown in [35] that the pluriclosed flow of T -invariant metrics on complex surfaces reduces tothe RYM flow. What is somewhat surprising is that the pluriclosed flow, defined in general withno symmetry considerations in mind, naturally fixes the metric on the Lie algebra and results inthe RYM flow in this setting.In this paper, building on prior results (cf. [28]), we give a complete description of the Ricci-Yang-Mills flow for T k -bundles over Riemann surfaces. A complete description of the Ricci flow onRiemann surfaces was achieved by Hamilton/Chow [7, 12], with other approaches coming later (cf.[3, 40]). Also, a complete description of Yang-Mills flow over Riemann surfaces was given by R˚ade[22]. We first state the main result describing the behavior of Ricci-Yang-Mills flow: Theorem 1.1.
Let T k → M → Σ denote a principal T k -bundle over a Riemann surface Σ , andfix h a choice of metric on t k . Let g denote a Riemannian metric on Σ and µ a choice ofprincipal connection. We let ( g t , µ t ) denote the solution to Ricci-Yang-Mills flow with this initialdata, and G t = π ∗ g t + tr h µ t ⊗ µ t the associated one-parameter family of invariant metrics on M .The following hold: (1) If χ (Σ) < then ( g t , µ t ) exists on [0 , ∞ ) and ( M, G t t ) converges in the Gromov-Hausdorfftopology to (Σ , g Σ ) , where g Σ denotes the canonical metric of constant curvature − . (2) If χ (Σ) = 0 then ( g t , µ t ) exists on [0 , ∞ ) and ( M, G t t ) converges in the Gromov-Hausdorfftopology to a point. (3) If χ (Σ) > and c ( M ) = 0 , then there exists T < ∞ such that ( g t , µ t ) exists on [0 , T ) , and ( M, T − t G t ) converges in the C ∞ topology to (Σ × R , g Σ × g R ) , where g Σ denotes a metricof constant curvature . (4) If χ (Σ) > , and c ( M ) = 0 , then ( g t , µ t ) exists on [0 , ∞ ) and there are constants λ i and aone-parameter family of diffeomorphisms φ t such that lim t →∞ φ ∗ t g t = λ g Σ lim t →∞ φ ∗ t F µ t = λ ω Σ , where g Σ denotes a metric of constant curvature , and ω Σ denotes the associated area form. Remark 1.2.
Note that the qualitative behavior of the metric g t in cases (1) and (2) agrees withthe case of Ricci flow, regardless of the topology of the bundle. The cases (3) and (4), where χ (Σ) >
0, are more subtle. We recall that the Ricci flow on S with arbitrary initial data contractsin finite time to a round point [7, 12]. In case the bundle is trivial, the RYM flow includes theseRicci flow lines as special cases, and case (3) shows that this behavior still holds in general, thatis, the flow still converges to a round point, with F → F = 0. Specifically, the F term in the evolution of g acts as a restoring force which doesn’t allow the sphere to collapse. Rather, along RYM flow the base willconverge to a round sphere of fixed size depending on the Chern class of the bundle and the choiceof h , without need for normalization, with F remaining fixed (note this indicates the necessity ofthe constants λ i ). Outside of the homogeneous setting one can hope that this behavior will stillhold due to the topological nontriviality of the bundle. Case (4) of Theorem 1.1 confirms thisbehavior for all initial data. Remark 1.3.
The proofs of cases (1) and (2) follow from maximum principle arguments combinedwith monotonicity of a modified Liouville energy. On the other hand, the proofs of cases (3) and (4)
ICCI-YANG-MILLS FLOW ON SURFACES AND PLURICLOSED FLOW ON ELLIPTIC FIBRATIONS 3 are significantly more intricate. For case (3) we rely on a modified Perelman-type entropy functionalto prove a κ -noncollapsing result for the metrics along the flow. Constructing a blowup limit ata singular time thus yields an ancient solution, which by maximum principle arguments can beshown to satisfy F ≡
0, so that it is in fact a solution to Ricci flow. A result of Perelman [21] yieldsthat this is then isometric to the shrinking sphere solution, yielding the claimed behavior. Case(4) requires studying a certain gauge-modified flow to allow for an application of Aubin’s improvedMoser-Trudinger inequality. We furthermore use the topological hypothesis of nontriviality of thebundle to establish an a priori lower bound for the volume along the flow.As discussed above, Ricci-Yang-Mills flow over Riemann surfaces is equivalently described bysolutions to generalized Ricci flow, thus Theorem 1.1 has an immediate corollary:
Corollary 1.4.
Let T k → M → Σ denote a principal T k bundle over a Riemann surface Σ ,with h a choice of metric on t k . Given G = π ∗ g + tr h µ ⊗ µ an invariant metric on M , let H = tr h F ∧ µ . Let ( G t , H t ) denote the unique solution to generalized Ricci flow on M withinitial data ( G , H ) . Then G t = π ∗ g t + tr h µ t ⊗ µ t , H t = tr h F t ∧ µ t , where ( g t , µ t ) is the solutionto RYM flow with initial condition ( g , µ ) . In particular, the existence and convergence propertiesare described as in Theorem 1.1 according to the topology of M . Remark 1.5.
As solutions to generalized Ricci flow with a torus symmetry, the flow lines inCorollary 1.4 are subject to the action of T -duality (cf. [8, 9, 31]). As an elementary example,homogeneous solutions on the unit tangent bundle over Σ are T -dual to solutions on the samebundle with g and µ preserved, while the length of the circles is inverted.Furthermore, due to the connection to pluriclosed flow described above, Theorem 1.1 has appli-cations to complex geometry. Complex surfaces give the first examples of non-K¨ahler manifoldsand among these, elliptic surfaces form a diverse and interesting class. Conjectures of the behav-ior of pluriclosed flow on these surfaces were described in [25], and the next result confirms theseconjectures in the case of invariant initial data on principal bundles, announced in [25]. Corollary 1.6.
Let ( M, J ) be a compact complex surface which is the total space of a holomorphic T -principal bundle over a Riemann surface Σ . (1) Suppose χ (Σ) < . Given ω an invariant pluriclosed metric on ( M, J ) , the solution topluriclosed flow with initial data ω exists on [0 , ∞ ) , and ( M, ω t t ) converges in the Gromov-Hausdorff topology to (Σ , g Σ ) , where g Σ denotes the canonical metric of constant curvature − . (2) Suppose χ (Σ) = 0 . Given ω an invariant pluriclosed metric on ( M, J ) , the solution topluriclosed flow with initial data ω exists on [0 , ∞ ) , and ( M, ω t t ) converges in the Gromov-Hausdorff topology to a point. (3) Suppose ( M, J ) ∼ = S × T . Given ω an invariant pluriclosed metric on ( M, J ) , let T = (8 π ) − Area ω ( S ) . The solution to pluriclosed flow with this initial data exists on [0 , T ) , and ( M, T − t ω t ) converges in the C ∞ topology to ( S × R , ω S × ω R ) . (4) Suppose ( M, J ) is a Hopf surface. Given ω an invariant pluriclosed metric on ( M, J ) , thesolution to pluriclosed flow with this initial data exists on [0 , ∞ ) , and ( M, ω t ) converges inthe C ∞ topology to a multiple of ω Hopf , the standard Hopf metric.
Remark 1.7. (1) Up to finite covers, all non-K¨ahler surfaces of Kodaira dimension 0 or 1 aretotal spaces of holomorphic T bundles, and admit invariant pluriclosed metrics and flowlines as described in items (1) and (2).(2) The surfaces in case (1) have Kodaira dimension 1. The general behavior of the K¨ahler-Ricci flow on K¨ahler surfaces of Kodaira dimension 1, analyzing the more subtle case ofsingular fibrations, is described in [24]. JEFFREY STREETS (3) The surfaces in case (2) have Kodaira dimension 0, and include both K¨ahler and non-K¨ahlerKodaira surfaces.(4) The surfaces in cases (3) and (4) have Kodaira dimension −∞ . In fact, the only Hopfsurfaces which are principal holomorphic T bundles are standard , described as Z -quotients S × S ∼ = C \{ } / h ( z , z ) → ( αz , βz ) i , where | α | = | β | <
1. These surfaces admit the Hermitian, pluriclosed, metric defined bythe Z -invariant K¨ahler form ω Hopf = ρ − √− ∂∂ρ , where ρ = q | z | + | z | is distance to the origin. The associated Riemannian metric is thestandard cylindrical metric on C \{ } ∼ = S × R . We denote the metric on the quotient as ω Hopf , which is Bismut-flat.Here is an outline of the rest of this paper. In § g . We also describe the symmetryreductions discussed above. Next in § § χ ≤ § χ (Σ) > χ (Σ) > Acknowledgements:
We thank Matthew Gursky for several helpful conversations.2.
Background
Setup and scalar reductions.
Fix T k → M → Σ a principal T k bundle over a compactRiemann surface Σ. An invariant metric G on M is determined by a triple ( g, µ, h ), where g is ametric on Σ, µ is a principal T k -connection, and h is a family of metrics on t k parameterized by Σ.Specifically, G = π ∗ g + tr h µ ⊗ µ . We will make the further restriction that this family h is constantover M , thus determined by a choice of inner product on t k . We will study a certain normalizationof the Ricci-Yang-Mills flow system, namely we fix λ ∈ {− , , } and consider ∂∂t g = − g + F µ − λg,∂∂t µ = − d ∗ g F µ ,∂∂t h = − λh. (2.1)where ( F µ ) ij = h IJ g kl F Iik F Jjl . We note that our normalized flow also scales the metric h . This isnatural from the point of view of RYM flow as a symmetry reduction of flows on the total space ofthe bundle, and in particular it follows that the data described by (2.1) differs from the system bya rescaling of the associated metrics G t = π ∗ g t + tr h t µ t ⊗ µ t . To begin our analysis we first showthat RYM flow over a Riemann surface can be described using a conformal factor on the base spaceand a t k -valued function determining the principal connection. ICCI-YANG-MILLS FLOW ON SURFACES AND PLURICLOSED FLOW ON ELLIPTIC FIBRATIONS 5
Lemma 2.1.
Given a solution to Ricci-Yang-Mills flow, one has g t = e u t g Σ , where ∂∂t u = e − u (∆ g Σ u − R Σ ) + | F µ | g,h − λ. (2.2) Proof.
As Σ is a surface, it follows easily thatRc g = Rg, F µ = | F µ | g,h g. It follows easily that the ansatz g t = e u t is preserved, and furthermore (cid:18) ∂∂t u (cid:19) g t = ∂∂t g t = − (cid:16) R − | F µ | g,h + λ (cid:17) g t . Using the formula R = e − u ( R Σ − ∆ g Σ u ), the result follows. (cid:3) Lemma 2.2.
Given a solution to Ricci-Yang-Mills flow , and a background connection µ thereexists a one-parameter family of t k -valued functions f t such that µ t = µ + d c f, ∂∂t f = ∆ g f + tr ω F µ . (2.3) Proof.
Since µ and µ are connections on the same bundle over a Riemann surface, using Hodgetheory and a gauge transformation we can solve for a function f such that µ = µ + d c f . Giventhe solution to Ricci-Yang-Mills flow, we can solve for f t as in the statement by the theory of linearparabolic equations, using initial data f . We then define e µ t = µ + d c f t and observe that ∂∂t e µ = d c (∆ g f + tr ω F µ )= d c tr ω ( F µ + dd c f )= d c tr ω F e µ = − d ∗ g F e µ . Thus e µ satisfies the Yang-Mills flow with respect to the time-dependent metric g , and since e µ = µ and solutions to Yang-Mills flow are unique, it follows that µ t = e µ t = µ + d c f , as required. (cid:3) Thus we have shown that the metric G t is defined equivalently in terms of a conformal factor u on Σ and a t k -valued function f , and this notation will be used throughout. Furthermore we willrefer to the principal connection as µ f := µ + d c f , where µ is some background connection.2.2. Energy functional.
We next exhibit a gradient formulation for Ricci-Yang-Mills flow in thissetting. From [40] we know that the Ricci flow on surfaces is the gradient flow of the Liouvilleenergy. A generalization of this to Ricci-Yang-Mills flow was shown in [28], and below we give aminor modification of this to account for the scaling parameter λ . Definition 2.3.
Given data ( u, f, h ) as above, let F ( u, f, h ) = ˆ Σ (cid:16) | du | + e − u (cid:12)(cid:12) F µ f (cid:12)(cid:12) g Σ ,h (cid:17) dV Σ + R Σ ˆ Σ udV Σ + λ ˆ Σ e u dV Σ . Proposition 2.4.
Given a solution to Ricci-Yang-Mills flow, one has ddt F ( u t , f t , h t ) = − ˆ Σ e u ˙ u dV Σ − λ ˆ Σ e − u (cid:12)(cid:12) F µ f (cid:12)(cid:12) g Σ ,h dV Σ − ˆ Σ e − u (cid:10) ∇ g F µ f , ∇ g F µ f (cid:11) g Σ ,h dV Σ . Proof.
First we compute ddt ˆ Σ 12 | du | dV Σ = ˆ Σ h d ˙ u, du i dV Σ = − ˆ Σ ˙ u ∆ g Σ udV Σ JEFFREY STREETS
Next ddt ˆ Σ e − u (cid:12)(cid:12) F µ f (cid:12)(cid:12) g Σ ,h dV Σ = ˆ Σ h ( − ˙ u − λ ) e − u (cid:12)(cid:12) F µ f (cid:12)(cid:12) g Σ ,h + 2 e − u (cid:10) ∆ g F µ f , F µ f (cid:11) g Σ ,h i dV Σ = ˆ Σ h ( ˙ u + λ ) (cid:16) − e − u (cid:12)(cid:12) F µ f (cid:12)(cid:12) g Σ ,h (cid:17) − e − u (cid:10) ∇ g F µ f , ∇ g F µ f (cid:11) g Σ ,h i dV Σ . Lastly ddt (cid:18) R Σ ˆ Σ udV Σ + λ ˆ Σ e u dV Σ (cid:19) = ˆ Σ ( R Σ ˙ u + λ ˙ ue u ) dV Σ . Combining these yields ddt F ( u t , f t ) = ˆ Σ ˙ u (cid:16) − ∆ g Σ u + R Σ − e − u (cid:12)(cid:12) F µ f (cid:12)(cid:12) g Σ ,h + λe u (cid:17) dV Σ − λ ˆ Σ e − u (cid:12)(cid:12) F µ f (cid:12)(cid:12) g Σ ,h dV Σ − ˆ Σ e − u (cid:10) ∇ g F µ f , ∇ g F µ f (cid:11) g Σ ,h dV Σ = − ˆ Σ e u ˙ u dV Σ − λ ˆ Σ e − u (cid:12)(cid:12) F µ f (cid:12)(cid:12) g Σ ,h dV Σ − ˆ Σ e − u (cid:10) ∇ g F µ f , ∇ g F µ f (cid:11) g Σ ,h dV Σ , as claimed. (cid:3) Higher Regularity.
One key application of the energy monotonicity of Proposition 2.4 is toobtain higher regularity estimates for the flow in the presence of certain bounds.
Proposition 2.5.
Given a solution ( g t , µ f t , h t ) to (2.1), suppose there exists a constant C > sothat for all t > , C S ( g t ) ≤ C, C − ≤ Vol( g t ) ≤ C, || du || L ≤ C, F ( u t , f t , h t ) ≤ C. (2.4) There exists ǫ, A depending on C so that if [ t , t ] is a time interval such that | t − t | + F ( u t , f t , h t ) − F ( u t , f t , h t ) ≤ ǫ, then sup [ t ,t ] (cid:18) || u || H + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e u |∇ g F | g (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L (cid:19) ≤ A (cid:18) || u ( t ) || H + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e u |∇ g F | g (cid:12)(cid:12)(cid:12) ( t ) (cid:12)(cid:12)(cid:12) L + 1 (cid:19) . Moreover, if the assumptions (2.4) hold on a finite time interval [0 , T ) , then the flow extendssmoothly past time T .Proof. This is established in ([28] § (cid:3) Symmetry Reductions.
In this subsection we record two ways in which the Ricci-Yang-Mills flow on Riemann surfaces arises via considering natural flow equations in higher dimensionswith symmetries. First, certain generalized Ricci flow lines on a three-manifold with a principal S symmetry reduce to Ricci-Yang-Mills flow on the base space. Also, invariant solutions to pluriclosedflow on complex surfaces which are principal T bundles reduce to Ricci-Yang-Mills flow on thebase space. These reductions show how to obtain Corollaries 1.4 and 1.6 from Theorem 1.1, butare also instrumental in showing the behavior of the Ricci-Yang-Mills flow in the most difficult caseof the sphere. ICCI-YANG-MILLS FLOW ON SURFACES AND PLURICLOSED FLOW ON ELLIPTIC FIBRATIONS 7 T k -invariant generalized Ricci flow over Riemann surfaces. As described in the introduction,the generalized Ricci flow is the parabolic system for a Riemannian metric G and closed three-form H defined by ∂∂t G = − G + H ,∂∂t H = ∆ d H. This is a parabolic system of equations and basic regularity and long-time existence obstructionshave been established in [26]. A Perelman-type F -functional for this was found in [19], and anexpander entropy functional was found in [26]. Some recent results in the homogeneous settinghave appeared [20], as well as a stability result near Ricci-flat metrics [23]. In [9] it was shownthat the equation reduces to Ricci-Yang-Mills flow in the case of a U (1) principal bundle over aRiemann surface. This extends to T k bundles: Proposition 2.6.
Let T k → M → Σ be a principal T k -bundle over a Riemann surface, and suppose ( g t , µ t ) is a solution of Ricci-Yang-Mills flow (1.1). Let G t = π ∗ g + tr h µ t ⊗ µ t , H t = tr h F t ∧ µ t . Then ( G t , H t ) is a solution of generalized Ricci flow.Proof. The proof is identical to (cf. [9] Prop 4.39), which is written for the case k = 1 but generalizesimmediately to the case of arbitrary k . (cid:3) T -invariant pluriclosed flow on complex surfaces. Given a complex manifold ( M n , J ), aHermitian metric g is pluriclosed if the associated K¨ahler form ω satisfies √− ∂∂ω = 0. For apluriclosed metric we define H = − d c ω = √− ∂ − ∂ ) ω , noting dH = 0. There is a Hermitianconnection on T M , the
Bismut connection , defined by ∇ B = ∇ + g − H . This has an associatedcurvature tensor Ω B , and the Bismut-Ricci form is the natural contraction ρ B = tr Ω B J. The pluriclosed flow is the equation ∂∂t ω = − ρ , B . This is a parabolic equation [36], which solves K¨ahler-Ricci flow if the initial metric is K¨ahler.Furthermore, after a gauge modification, the associated pairs ( g t , H t ) of metrics and three-formsare a solution of generalized Ricci flow [38]. Global existence and convergence results for pluriclosedflow have appeared in [15, 29, 30]. Pluriclosed flow also preserves generalized K¨ahler geometry [37],and global existence and convergence results have been shown in this setting [1, 34, 33, 32], andspecifically the results of [34] overlap partially with Corollary 1.6.Let us now restrict to the case where ( M , J ) is a complex surface which is the total space of aholomorphic principal T bundle over a base manifold Σ. Let Z, W denote canonical vertical vectorfields associated to a basis Z , W for the torus action, such that W = J Z . Let g denote an invariantHermitian metric on J , and let h , i denote a metric on t such that h Z , W i = 0. As explained in[35], a choice of invariant Hermitian metric G is equivalent to a triple ( g, µ, ψ ), where ψ = G ( Z, Z ) = G ( W, W ) ,µ ( X ) = ψ − g ( X, Z ) Z + ψ − g ( X, W ) W ,g ( X, Y ) = G ( X, Y ) − ψ h µ ( X ) , µ ( Y ) i . Furthermore, the metric is pluriclosed if and only if ψ is constant. Here ψ h , i is playing the roleof the metric h as described above. By general principle the pluriclosed flow will preserve the T symmetry, and the fact that ψ is constant. As shown in ([35] Lemma 6.2), the natural Hermitianconnection is determined by functions f and f where µ f = (cid:16) µ Z + d c f + df (cid:17) ⊗ Z + (cid:16) µ W + d c f − df (cid:17) ⊗ W. Furthermore, the symmetry reduced equations are gauge-equivalent to the Ricci-Yang-Mills flow:
Proposition 2.7. (cf. [35]
Proposition 6.3) Given ( M , J ) as above, a one-parameter family of T invariant metrics G t = π ∗ g t + tr h t µ f ⊗ µ f is a solution to normalized pluriclosed flow if andonly if ∂g∂t = − (cid:16) R − ψ (cid:12)(cid:12) F µ f (cid:12)(cid:12) + λ (cid:17) g T ,∂∂t f = tr ω F Z µ f = ∆ g f + tr ω F Z µ ,∂∂t f = tr ω F W µ f = ∆ g f + tr ω F W µ ,∂∂t ψ = − λψ. (2.5)The proof in [35] is by direct computation, but a more conceptual proof for a more general casecan be given as follows. Suppose T k → M → Σ is a principal T k bundle. By taking a product witha trivial T k bundle we obtain a T k bundle f M over Σ, which admits a natural complex structureinduced by a choice of complex structure on t k determined by the natural splitting. We can extenda choice of principal connection on M using a flat connection to determine a principal connection e µ on f M . Letting ξ i denote a basis for T k , it follows that the (1 ,
1) form ω = π ∗ ω g + k X i =1 µ ξ i ∧ µ Jξ i is positive, and moreover H = − d c ω = P ki =1 F ξ i ∧ µ ξ i . Since the solution to pluriclosed flow isequivalent to generalized Ricci flow after a gauge transformation ([38]), it follows from Proposition2.6 that, up to a gauge transformation, the solution to pluriclosed flow is the same as the associatedsolution to Ricci-Yang-Mills flow with initial data ( g, µ ) on Σ.3. A priori estimates
In this section we establish a priori estimates building towards the global existence claims ofTheorem 1.1. In all cases we will choose a background conformal metric g Σ with constant scalarcurvature. With this choice we obtain a solution u t to the flow of conformal factors (2.2) by Lemma2.1. Furthermore using Hodge theory we can choose a background connection µ such that F µ ≡ ω Σ ⊗ ζ ∈ Λ (Σ) ⊗ t k . With this choice we apply Lemma 2.2 to obtain a solution f t to the potential flow (2.3).3.1. Evolution equations.Lemma 3.1.
Given a solution to Ricci-Yang-Mills flow, we have (cid:18) ∂∂t − ∆ (cid:19) e − u = − |∇ u | g t e − u + R Σ e − u − (cid:12)(cid:12) F µ f (cid:12)(cid:12) g t ,h t e − u + λe − u . (3.1) ICCI-YANG-MILLS FLOW ON SURFACES AND PLURICLOSED FLOW ON ELLIPTIC FIBRATIONS 9
Proof.
We directly compute using Lemma 2.1 ∂∂t e − u = − e − u (cid:16) ∆ u − e − u R Σ + (cid:12)(cid:12) F µ f (cid:12)(cid:12) g t ,h t − λ (cid:17) = ∆ e − u − |∇ u | g t e − u + R Σ e − u − (cid:12)(cid:12) F µ f (cid:12)(cid:12) g t ,h t e − u + λe − u . (cid:3) Lemma 3.2.
Given a solution to Ricci-Yang-Mills flow, we have (cid:18) ∂∂t − ∆ (cid:19) |∇ f | g t ,h = − (cid:12)(cid:12) ∇ f (cid:12)(cid:12) g t ,h − (cid:12)(cid:12) F µ f (cid:12)(cid:12) g t ,h t |∇ f | g t ,h + λ |∇ f | g t ,h + 2 (cid:10) ∇ e − u ⊗ tr ω Σ F µ , ∇ f (cid:11) g t ,h . (3.2) Proof.
We directly compute using Lemma 2.1, Lemma 2.2, and the Bochner formula (cid:18) ∂∂t − ∆ (cid:19) |∇ f | g t ,h = (cid:16) R − (cid:12)(cid:12) F µ f (cid:12)(cid:12) g t ,h t + λ (cid:17) |∇ f | g t ,h + 2 h∇ (∆ f + tr ω F µ ) , ∇ f i g t ,h − h ∆ ∇ f, ∇ f i g t ,h − (cid:12)(cid:12) ∇ f (cid:12)(cid:12) g t ,h = − (cid:12)(cid:12) ∇ f (cid:12)(cid:12) g t ,h − (cid:12)(cid:12) F µ f (cid:12)(cid:12) g t ,h t |∇ f | g t ,h + λ |∇ f | g t ,h + 2 (cid:10) ∇ e − u ⊗ tr ω Σ F µ , ∇ f (cid:11) g t ,h , as claimed. (cid:3) The case χ (Σ) < . Assume χ (Σ) <
0, so that by the uniformization theorem we may choosethe background metric g Σ so that R g Σ = −
1. In this case we also set λ = 1, and these choices holdthroughout this subsection. Proposition 3.3.
Given a solution to Ricci-Yang-Mills flow with χ (Σ) < , we have sup M ×{ t } (cid:0) e − u − (cid:1) ≤ e − t sup M ×{ } (cid:0) e − u − (cid:1) . Proof.
Specializing Lemma 3.1 to the case R Σ = − λ = 1, and dropping negative terms yields (cid:18) ∂∂t − ∆ (cid:19) (cid:0) e − u − (cid:1) ≤ − ( e − u ) + e − u = − (cid:0) e − u − (cid:1) − (cid:0) e − u − (cid:1) . The result follows from the maximum principle. (cid:3)
Proposition 3.4.
Given a solution to Ricci-Yang-Mills flow with χ (Σ) < , we have sup M ×{ t } | f | h ≤ C (1 + t ) . Proof.
Using the a priori estimate of Proposition 3.3 we see by Lemma 2.2 that (cid:18) ∂∂t − ∆ (cid:19) f = tr ω F µ ≤ C. The upper bound follows by the maximum principle, and the lower bound is similar. (cid:3)
Lemma 3.5.
Given a solution to Ricci-Yang-Mills flow with χ (Σ) < , there exists a constant A > so that (cid:18) ∂∂t − ∆ (cid:19) (cid:16) A ( e − u −
1) + e − t |∇ f | g t ,h + e − t | f | h (cid:17) ≤ − (cid:12)(cid:12) F µ f (cid:12)(cid:12) g t ,h t − C − A (cid:12)(cid:12) ∇ e − u (cid:12)(cid:12) g t − e − t |∇ f | g t ,h − A e − u + 2 Ae − u . Proof.
Fix
A > A ( e − u −
1) + e − t |∇ f | g t ,h + e − t | f | h . By combining Lemmas 2.2,3.1, and 3.2 we obtain (cid:18) ∂∂t − ∆ (cid:19) Φ = A n − |∇ u | g t e − u − (cid:12)(cid:12) F µ f (cid:12)(cid:12) g t ,h t e − u − e − u + e − u o + e − t n − (cid:12)(cid:12) ∇ f (cid:12)(cid:12) g t ,h − (cid:12)(cid:12) F µ f (cid:12)(cid:12) g t ,h t |∇ f | g t ,h + 2 (cid:10) ∇ e − u ⊗ tr ω Σ F µ , ∇ f (cid:11) g t ,h o − e − t |∇ f | g t ,h + 2 e − t e − u h tr ω Σ F µ , f i h − e − t | f | h ≤ − C − A (cid:12)(cid:12) ∇ e − u (cid:12)(cid:12) g t + A (cid:0) e − u − e − u (cid:1) − e − t (cid:12)(cid:12) ∇ f (cid:12)(cid:12) g t ,h + Ce − t (cid:12)(cid:12) ∇ e − u (cid:12)(cid:12) g t |∇ f | g t ,h − e − t |∇ f | g t ,h + Ce − t e − u | f | h − e − t | f | h ≤ − e − t (cid:12)(cid:12) ∇ f (cid:12)(cid:12) g t ,h − C − A (cid:12)(cid:12) ∇ e − u (cid:12)(cid:12) g t − e − t |∇ f | g t ,h + A (cid:0) e − u − e − u (cid:1) + C (1 + t ) e − t e − u , (3.3)where the second line follows from the estimate of Proposition 3.3, and the third line followsby choosing A sufficiently large and applying the Cauchy-Schwarz inequality, and the estimateof Proposition 3.4. Next we use the formula F µ f = F µ + √− ∂∂f , together with the fact that h t = e − t h to obtain the estimate (cid:12)(cid:12) ∇ f (cid:12)(cid:12) g t ,h ≥ (cid:12)(cid:12) √− ∂∂f (cid:12)(cid:12) g t ,h = (cid:12)(cid:12) F µ f − F µ (cid:12)(cid:12) g t ,h ≥ (cid:12)(cid:12) F µ f (cid:12)(cid:12) g t ,h − | F µ | g t ,h ≥ e t (cid:12)(cid:12) F µ f (cid:12)(cid:12) g,h − Ce − u . (3.4)Inserting this into (3.3) and choosing A sufficiently large yields the inequality (cid:18) ∂∂t − ∆ (cid:19) Φ ≤ − (cid:12)(cid:12) F µ f (cid:12)(cid:12) g t ,h t − C − A (cid:12)(cid:12) ∇ e − u (cid:12)(cid:12) g t − e − t |∇ f | g t ,h + A (cid:0) e − u − e − u (cid:1) + C (1 + t ) e − t e − u + C ( e − u ) ≤ − (cid:12)(cid:12) F µ f (cid:12)(cid:12) g t ,h t − C − A (cid:12)(cid:12) ∇ e − u (cid:12)(cid:12) g t − e − t |∇ f | g t ,h − A e − u + 2 Ae − u , where the last line follows by choosing A sufficiently large. (cid:3) Proposition 3.6.
Given a solution to Ricci-Yang-Mills flow with χ (Σ) < , there exists a constant C > so that sup M ×{ t } e − t |∇ f | g t ,h ≤ C. Proof.
Defining Φ as in Lemma 3.5 and returning to line (3.3) and using the result of Proposition3.3 yields the differential inequality (cid:18) ∂∂t − ∆ (cid:19) Φ ≤ − e − t |∇ f | g t ,h + C. Using the estimates of Proposition 3.3 and 3.4, at a sufficiently large maximum for Φ it followsthat e − t |∇ f | g t ,h is also arbitrarily large, yielding a contradiction. Hence Φ has an a priori upperbound and thus so does e − t |∇ f | g t ,h . (cid:3) ICCI-YANG-MILLS FLOW ON SURFACES AND PLURICLOSED FLOW ON ELLIPTIC FIBRATIONS 11
Proposition 3.7.
Given a solution to Ricci-Yang-Mills flow with χ (Σ) < , there exists a constant C > so that sup M ×{ t } u ≤ C. Proof.
Choose
A > u + A ( e − u −
1) + e − t |∇ f | g t ,h + e − t | f | h Combining Lemma 2.1 and Lemma 3.5 we obtain, dropping some negative terms, (cid:18) ∂∂t − ∆ (cid:19) Φ ≤ (cid:12)(cid:12) F µ f (cid:12)(cid:12) g t ,h t + e − u − (cid:26) − (cid:12)(cid:12) F µ f (cid:12)(cid:12) g t ,h t − A e − u + 2 Ae − u (cid:27) ≤ Ce − u − . (3.5)Suppose there exists ( x , t ) such thatΦ( x , t ) = sup M × [0 ,t ] Φ = B. Using Propositions 3.3, 3.4, and 3.6, if B is sufficiently large, it follows that u ( x , t ) ≥ B . Rearranging this inequality we obtain e − u ( x , t ) ≤ e − B . Since ( x , t ) is a spacetime maximum for Φ, returning to (3.5) and applying the maximum principlewe obtain 0 ≤ (cid:20)(cid:18) ∂∂t − ∆ (cid:19) Φ (cid:21) ( x , t ) ≤ Ce − B − , a contradiction for B chosen sufficiently large. It follows that Φ has a uniform upper bound, andthe proposition follows. (cid:3) The case χ (Σ) = 0 . Assume χ (Σ) = 0. By the uniformization theorem we may choose thebackground metric g Σ so that R Σ = 0. In this case we again fix λ = 1, and these choices holdthroughout this subsection. Proposition 3.8.
Given a solution to Ricci-Yang-Mills flow with χ (Σ) = 0 , we have sup M ×{ t } e − u ≤ e t sup M ×{ } e − u . Proof.
Since in this setting R Σ = 0, this follows directly by applying the maximum principle to theevolution equation of Lemma 3.1. (cid:3) Proposition 3.9.
Given a solution to Ricci-Yang-Mills flow with χ (Σ) = 0 , we have sup M ×{ t } | f | h ≤ Ce t . Proof.
Using the a priori estimate of Proposition 3.8 we see by Lemma 2.2 that (cid:18) ∂∂t − ∆ (cid:19) f = tr ω F µ ≤ Ce t . The upper bound follows by the maximum principle, and the lower bound is similar. (cid:3)
Lemma 3.10.
Given a solution to Ricci-Yang-Mills flow with χ (Σ) = 0 , there exists a constant A > so that (cid:18) ∂∂t − ∆ (cid:19) (cid:16) Ae − u + e − t |∇ f | g t ,h + e − t | f | h (cid:17) ≤ − (cid:12)(cid:12) F µ f (cid:12)(cid:12) g t ,h t − C − A e − t (cid:12)(cid:12) ∇ e − u (cid:12)(cid:12) g t − e − t |∇ f | g t ,h + Ce − u . Proof.
Fix
A > Ae − u + e − t |∇ f | g t ,h + e − t | f | h . By combining Lemmas 2.2, 3.1, and3.2, using R Σ = 0, and the estimates of Propositions 3.8 and 3.9 we obtain (cid:18) ∂∂t − ∆ (cid:19) Φ = A n − |∇ u | g t e − u − (cid:12)(cid:12) F µ f (cid:12)(cid:12) g t ,h t e − u + e − u o + e − t n − (cid:12)(cid:12) ∇ f (cid:12)(cid:12) g t ,h − (cid:12)(cid:12) F µ f (cid:12)(cid:12) g t ,h t |∇ f | g t ,h + 2 (cid:10) ∇ e − u ⊗ tr ω Σ F µ , ∇ f (cid:11) g t ,h o − e − t |∇ f | g t ,h + 2 e − t e − u h tr ω Σ F µ , f i h − e − t | f | h ≤ − C − Ae − t (cid:12)(cid:12) ∇ e − u (cid:12)(cid:12) g t + Ce t − e − t (cid:12)(cid:12) ∇ f (cid:12)(cid:12) g t ,h + Ce − t (cid:12)(cid:12) ∇ e − u (cid:12)(cid:12) g t |∇ f | g t ,h − e − t |∇ f | g t ,h + Ce − t e − u | f | h − e − t | f | h ≤ − e − t (cid:12)(cid:12) ∇ f (cid:12)(cid:12) g t ,h − C − A e − t (cid:12)(cid:12) ∇ e − u (cid:12)(cid:12) g t − e − t |∇ f | g t ,h + Ce − u . (3.6)Arguing as in line (3.4) we obtain (cid:12)(cid:12) ∇ f (cid:12)(cid:12) g t ,h ≥ e t (cid:12)(cid:12) F µ f (cid:12)(cid:12) g t ,h t − Ce − u . Using this estimate in (3.6) and applying Proposition 3.8, the result follows. (cid:3)
Proposition 3.11.
Given a solution to Ricci-Yang-Mills flow with χ (Σ) = 0 , one has sup M ×{ t } u ≤ Ce t . Proof.
Choose
A > u + Ae − u + e − t |∇ f | g t ,h + e − t | f | h . Combining Lemmas 2.1 and 3.10 we obtain, dropping some negative terms and applying Proposition3.8, (cid:18) ∂∂t − ∆ (cid:19) Φ ≤ (cid:12)(cid:12) F µ f (cid:12)(cid:12) g t ,h t − n − (cid:12)(cid:12) F µ f (cid:12)(cid:12) g t ,h t + Ce − u o ≤ Ce − u − ≤ Ce t − . Applying the maximum principle gives the result. (cid:3)
The case χ (Σ) > , c ( M ) = 0 . Assume χ (Σ) >
0, so that by the uniformization theoremwe may choose the background metric g Σ so that R Σ = 1. We furthermore make the assumptionthat c ( M ) = 0 ∈ t k , and thus we may choose a background connection µ so that F µ ≡ . We also set λ = 0, noting then that h t ≡ h . These choices hold throughout this subsection. ICCI-YANG-MILLS FLOW ON SURFACES AND PLURICLOSED FLOW ON ELLIPTIC FIBRATIONS 13
Lemma 3.12.
Given a solution to Ricci-Yang-Mills flow with χ (Σ) > , c = 0 , one has (cid:18) ∂∂t − ∆ (cid:19) |∇ f | g t ,h = − (cid:12)(cid:12) F µ f (cid:12)(cid:12) g t ,h − (cid:12)(cid:12) ( ∇ f ) , , (cid:12)(cid:12) g t ,h − (cid:12)(cid:12) F µ f (cid:12)(cid:12) g t ,h |∇ f | g t ,h Proof.
In the case c = 0, Lemma 3.2 yields (cid:18) ∂∂t − ∆ (cid:19) |∇ f | g t ,h = − (cid:12)(cid:12) ∇ f (cid:12)(cid:12) g t ,h − (cid:12)(cid:12) F µ f (cid:12)(cid:12) g t ,h |∇ f | g t ,h . Since F µ ≡
0, it follows that (cid:12)(cid:12) ∇ f (cid:12)(cid:12) g t ,h = (cid:12)(cid:12) ( ∇ f ) , (cid:12)(cid:12) g t ,h + (cid:12)(cid:12) ( ∇ f ) , , (cid:12)(cid:12) g t ,h = (cid:12)(cid:12) F µ f (cid:12)(cid:12) g t ,h + (cid:12)(cid:12) ( ∇ f ) , , (cid:12)(cid:12) g t ,h , and the result folows. (cid:3) Proposition 3.13.
Given a solution to Ricci-Yang-Mills flow with χ (Σ) > , c = 0 , one has sup M ×{ t } |∇ f | g t ,h ≤ sup M ×{ } |∇ f | g ,h . (3.7) Proof.
Lemma 3.2 yields (cid:18) ∂∂t − ∆ (cid:19) |∇ f | g t ,h ≤ , and the result follows from the maximum principle. (cid:3) The case χ (Σ) > , c ( M ) = 0 .Proposition 3.14. Given a solution to Ricci-Yang-Mills flow as above with χ (Σ) > , c = 0 ,there exists a constant C > so that C − ≤ Vol( g t ) ≤ C. Proof.
To establish the lower bound we use the fact that the bundle is nontrivial. Fix X ∈ t k suchthat ´ Σ h F µ , X i 6 = 0. Then we observe that there is a uniform constant λ > λ = (cid:12)(cid:12)(cid:12)(cid:12) ˆ Σ h F µ , X i h (cid:12)(cid:12)(cid:12)(cid:12) ≤ ˆ Σ | F µ | g,h dV g ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F µ f (cid:12)(cid:12) g,h (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L Vol( g t ) . It follows that ddt
Vol( g t ) ≥ − π + λ g t ) , and the lower bound follows. Using the upper bound on the Liouville energy from Proposition 2.4,we obtain an upper bound on the volume from the upper bound on Liouville energy and the sharpform of the Moser-Trudinger inequality on S [2, 18, 41]. (cid:3) Global existence and convergence
The case χ (Σ) < .Proposition 4.1. Let T k → M → Σ denote a principal torus bundle over a compact Riemannsurface with χ (Σ) < . Given G = π ∗ g + tr h µ ⊗ µ an invariant metric on M , the solution to(2.1) with λ = 1 and initial condition ( g , µ , h ) exists on [0 , ∞ ) , and further satisfies (1) lim t →∞ g t = g Σ , the unique conformal metric on Σ of constant curvature − . (2) lim t →∞ F µ t = F µ = ω Σ ⊗ ζ for some ζ ∈ t k . (3) ( M, G t ) converges in the Gromov-Hausdorff topology to (Σ , g Σ ) . Proof.
By combining the estimates of Propositions 3.3, 3.7, and then applying Proposition 2.5, weconclude long time existence of the flow. Note that the upper and lower estimates on u are uniformin time.To address the convergence we use the energy monotonicity. Since λ = 1 it follows from Proposi-tion 2.4 that F is monotonically decreasing. As the functional F is bounded below, it follows thatthere exists a sequence { t i } → ∞ such thatlim i →∞ ddt F ( u t i , f t i ) = 0 . Choose a small ǫ > t i such that 0 ≥ ddt F ( u t i , f t i ) ≥ − ǫ . From Proposition 2.4 itfollows that for such times t i one has ˆ Σ e u ˙ u dV Σ + ˆ Σ e − u (cid:12)(cid:12) F µ f (cid:12)(cid:12) g Σ ,h dV Σ + 2 ˆ Σ e − u (cid:10) ∇ g F µ f , ∇ g F µ f (cid:11) g Σ ,h dV Σ ≤ ǫ. Using the uniform bound for u from Propositions 3.3, 3.7, it follows that ˆ Σ ˙ u dV Σ + ˆ Σ (cid:12)(cid:12) F µ f (cid:12)(cid:12) g,h dV Σ + 2 ˆ Σ (cid:12)(cid:12) ∇ g F µ f (cid:12)(cid:12) g,h dV Σ ≤ Cǫ. (4.1)We next show that the L smallness of ˙ u can be split into the scalar curvature and bundle curvaturepieces to give smallness of the H norm of u . To that end we estimate using the Sobolev inequality, ˆ Σ (cid:12)(cid:12) F µ f (cid:12)(cid:12) g,h dV Σ = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F µ f (cid:12)(cid:12) g,h (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L ≤ C (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F µ f (cid:12)(cid:12) g,h (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F µ f (cid:12)(cid:12) g,h (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) H ≤ Cǫ . It follows from the L estimate for ˙ u above that ˆ Σ (cid:0) e − u (∆ Σ u + 1) − (cid:1) dV Σ ≤ Cǫ. (4.2)As u is also uniformly bounded, we conclude a uniform H estimate for u at these times, and thusa uniform C α estimate by Sobolev embedding. Returning to (4.1), we conclude an H estimate for f − f , and thus a C α estimate as well. It follows from parabolic Schauder estimates that f hasa uniform C ,α estimate on [ t i , t i + 1]. The equation for u is now uniformly parabolic with a C α inhomogeneous term, thus we obtain a uniform C ,α estimate for u on [0 , T ). Alternating betweenthe equations for f and u we continue bootstrapping to get C ∞ estimates for u and f on [ t i , t i + 1].Since we can find a time t i satisfying (4.1) in any time interval [ T, T + 1] for sufficiently large T , itfollows that there are uniform C ∞ estimates for u and f .From the discussion above, specifically (4.2), it follows that the metrics g t converge to a conformalmetric of constant curvature −
1, which must be g Σ , and in particular u converges to zero. Withthis in place it follows from (2.3) that f t converges to a constant function, so that F µ f → F µ , asclamed. As h t = e − t h , it follows immediately that ( M, G t = π ∗ g t + h t µ t ⊗ µ t ) converges to (Σ , g Σ )as claimed. (cid:3) The case χ (Σ) = 0 . This case requires a further technical lemma to obtain the limitingbehavior, namely a bound on the Sobolev constant of the conformal metric. This lemma is alsoemployed in obtaining estimates in the case Σ = S . Lemma 4.2.
Suppose (Σ , g Σ ) is a compact Riemann surface. Fix u ∈ C ∞ (Σ) such that ˆ Σ udV Σ = 0 , || du || L ≤ A. There exists a constant C = C ( A ) such that C S ( e u g Σ ) ≤ C . ICCI-YANG-MILLS FLOW ON SURFACES AND PLURICLOSED FLOW ON ELLIPTIC FIBRATIONS 15
Proof.
Using the L gradient bound for u , it follows from the Moser-Trudinger inequality that forany p > C such that ˆ Σ e p | u | dV Σ ≤ C. We note that W , embeds into any L p space, so we choose the constant C S ( g Σ ) such that (cid:18) ˆ Σ f dV Σ (cid:19) ≤ C S ( g Σ ) (cid:18) ˆ Σ | df | g Σ dV Σ + ˆ Σ f dV Σ (cid:19) . Using the above estimates we will bound the Sobolev constant for the metric g = e u g Σ , for theembedding W , → L . The argument modifies in on obvious way to estimate other Sobolevconstants. To begin we have (cid:18) ˆ Σ f dV g (cid:19) = (cid:18) ˆ Σ f e u dV Σ (cid:19) ≤ (cid:18) ˆ Σ f dV Σ (cid:19) (cid:18) ˆ Σ e u dV Σ (cid:19) ≤ C (cid:18) ˆ Σ f dV Σ (cid:19) . Using the Sobolev inequality for g Σ and the conformal invariance of the Dirichlet energy we obtain (cid:18) ˆ Σ f dV g (cid:19) + (cid:18) ˆ Σ f dV Σ (cid:19) ≤ C ˆ Σ (cid:16) |∇ f | g Σ + f (cid:17) dV Σ = C ˆ Σ |∇ f | g dV g + C ˆ Σ f dV Σ . We furthermore choose
N > ˆ e − u ≥ N dV Σ ≤ N ˆ Σ e | u | dV Σ ≤ CN .
Using this we estimate ˆ Σ f dV Σ = ˆ e − u ≤ N f dV Σ + ˆ e − u ≥ N f dV Σ ≤ N ˆ Σ f dV g + (cid:18) ˆ Σ f (cid:19) (cid:18) ˆ e − u ≥ N dV Σ (cid:19) ≤ N ˆ Σ f dV g + CN − (cid:18) ˆ Σ f dV Σ (cid:19) . Choosing N sufficiently large and using this above gives the claim. (cid:3) Proposition 4.3.
Let T k → M → Σ denote a principal torus bundle over a compact Riemannsurface with χ (Σ) = 0 . Given G = π ∗ g + tr h µ ⊗ µ an invariant metric on M , the solutionto (2.1) with λ = 1 and initial condition ( g , µ , h ) exists on [0 , ∞ ) and ( M, G t ) converges in theGromov-Hausdorff topology to a point.Proof. By combining the estimates of Propositions 3.8, 3.11, and then applying Proposition 2.5,we conclude long time existence of the flow. In this case we cannot conclude L ∞ estimates for u which are uniform in time.To obtain the convergence, we first observe that by Proposition 2.4, since λ = 1, there is ana priori upper bound on F , which in this case immediately implies a uniform upper bound for ||∇ u || L . Furthermore, using that F is bounded below since R Σ = 0 we choose a sequence of times { t i } → ∞ such that lim i →∞ ddt F ( u t i , f t i ) = 0 . Thus we may fix ǫ > i it follows from Proposition 2.4that for t i one has ˆ Σ h e u ˙ u + e − u (cid:12)(cid:12) F µ f (cid:12)(cid:12) g Σ ,h + 2 e − u (cid:10) ∇ g F µ f , ∇ g F µ f (cid:11) g Σ ,h i dV Σ ≤ ǫ. (4.3)Note that we can decompose the first term to yield ǫ ≥ ˆ Σ e u (cid:16) − R g + | F µ | g,h − (cid:17) dV Σ = ˆ Σ e u (cid:16) − R g + | F µ | g,h (cid:17) dV Σ + ˆ Σ (cid:16) R g − | F µ | g,h (cid:17) e u dV Σ + ˆ Σ e u dV Σ ≥ ˆ Σ e u (cid:16) − R g + | F µ | g,h (cid:17) dV Σ − ǫ + ˆ Σ e u dV Σ . Thus ˆ Σ e u dV Σ + ˆ Σ e u (cid:16) e − u ∆ Σ u + | F µ | g,h (cid:17) dV Σ . ≤ ǫ. (4.4)Now let w = u t i − ´ Σ u t i dV Σ . Using Lemma 4.2 we conclude that the W , → L Sobolev constantof g = e w g Σ is bounded. It follows from a standard iteration argument that there is a uniformlower bound on the volume of balls of the form Vol( B r ( p )) ≥ cr . Since the volume is uniformlybounded above it follows from this that the diameter of the metric e w g Σ is bounded. Since ˆ Σ u t i dV Σ ≤ log (cid:18) ˆ Σ e u ti dV Σ (cid:19) ≤ log ǫ, it follows easily that the diameter of e u ti g is approaching zero as i → ∞ .We note using the monotonicity and lower bound for F that for any ǫ > T > t ≥ T , there exists e t ∈ [ t, t + 1] such that ddt F ( u e t , f e t ) ≤ ǫ . The argumentabove applies at time e t to show that diam( g e t ) = o ( ǫ ). We further claim that for all t ≥ T ,diam( g t ) = o ( ǫ ). To see this, fix a time t ≥ T and choose e t ∈ [ t, t + 1] as above. Since H (Σ) = 0,we apply the diameter lower bound (with normalization λ = 1) for Ricci-Yang-Mills flow ([27, 13])to see that there is a uniform constant c > g t ) ≤ c diam( g e t ). The claim follows, andwe conclude lim t →∞ diam( g t ) = 0. Since h t = e − t h , it follows that lim t →∞ diam( G t ) = 0, and so( M, G t ) converges in the Gromov-Hausdorff topology to a point. (cid:3) The case χ (Σ) > and trivial bundle. κ -noncollapsing. Here we sketch an argument for proving uniform κ -noncollapsing of themetrics in this setting, adapting arguments of ([39, 11, 9]), all based on the original argumentof Perelman [21]. To begin we define a modification of Perelman’s entropy formula adapted toRicci-Yang-Mills flow, which is not quite monotone, though still of some interest in studying Ricci-Yang-Mills flow more generally. Similar entropy monotonicity formulas have appeared in [16]. Inthe setting of a trivial bundle, we can make a further modification which yields a monotone quantitywhich gives the required κ -noncollapsing. ICCI-YANG-MILLS FLOW ON SURFACES AND PLURICLOSED FLOW ON ELLIPTIC FIBRATIONS 17
Definition 4.4.
Given T k → M → Σ a principal T k bundle over Σ, fix data ( g, µ, h ) as above,with F µ = dµ ∈ Λ T ∗ ⊗ t k . Fix u ∈ C ∞ ( M ) , u > τ >
0, and define f − via u = e − f − (4 πτ ) n . Define W − ( g, µ, h, f − , τ ) = ˆ Σ (cid:20) τ (cid:18) |∇ f − | + R − | F µ | g,h (cid:19) + f − − n (cid:21) udV. Lemma 4.5.
Given T k → M → Σ a principal T k bundle over Σ , fix data ( g, µ, h, u, τ ) as above,and let δg = v g , δh = v h , δµ = α, δf = φ, δτ = σ. Then δ W − ( v g , v h , α, φ, σ )= ˆ Σ (cid:20) σ (cid:18) |∇ f − | + R − | F µ | g,h (cid:19) − τ (cid:28) v g , Rc − F µ + ∇ f (cid:29) − τ h v h , tr g F µ ⊗ F µ i− τ (cid:10) α, d ∗ g F µ + ∇ f − y F µ (cid:11) + φ + (cid:20) τ (cid:18) f − − |∇ f − | + R − | F µ | g,h (cid:19) + f − n (cid:21) (cid:18) tr g v g − φ − nσ τ (cid:19)(cid:21) udV g Definition 4.6.
Given T k → M → Σ a principal T k bundle over Σ, suppose ( g t , µ t ) is a solution ofRicci-Yang-Mills flow. We say that a one-parameter family of functions f − satisfies the conjugateheat equation if ∂∂t f − = − ∆ f − + |∇ f − | − R + 12 | F | g,h + n τ . Proposition 4.7.
Let ( M n , g t , µ t , h ) be a solution to Ricci-Yang-Mills flow. Let f − denote anassociated solution of the conjugate heat equation. Then, for T > , setting τ = T − t , ddt W − ( g, µ, h, f − , τ ) = ˆ Σ " τ (cid:12)(cid:12)(cid:12)(cid:12) Rc − F µ + ∇ f − − g τ (cid:12)(cid:12)(cid:12)(cid:12) + τ (cid:12)(cid:12) d ∗ g F µ + ∇ f − y F µ (cid:12)(cid:12) − | F µ | g,h udV g . Proof.
By diffeomorphism invariance of the functional it suffices to compute for the gauge-fixedsystem: ∂∂t g = − F µ − ∇ f − ,∂∂t µ = − d ∗ g F µ − ∇ f − y F µ ,∂∂t f − = − ∆ f − − R + 12 | F µ | g,h + n τ . Observe that for this system of equations, tr g ∂∂t g − ∂∂t f − + n τ = 0. It follows from Lemma 4.5that ddt W − ( g, µ, h, f − , τ )= ˆ Σ (cid:20) τ (cid:28) Rc − F µ + ∇ f − , Rc − F µ + ∇ f − (cid:29) + τ (cid:12)(cid:12) d ∗ g F µ + ∇ f − y F µ (cid:12)(cid:12) − ∆ f − − R + 12 | F µ | g,h + n τ − (cid:18) |∇ f − | + R − | F µ | g,h (cid:19)(cid:21) udV g = ˆ Σ " τ (cid:12)(cid:12)(cid:12)(cid:12) Rc − F µ + ∇ f − − g τ (cid:12)(cid:12)(cid:12)(cid:12) + τ (cid:12)(cid:12) d ∗ g F µ + ∇ f − y F µ (cid:12)(cid:12) − | F µ | g,h udV g , as claimed. (cid:3) While the result of Proposition 4.7 is suggestive, the presence of the one negative term preventsimmediate application as a monotonicity formula with attendant estimates. This can be rectifiedin the presence of a bounded subsolution of the heat equation with inhomogeneous term − | F | g,h (cf. [9] Ch. 6 where one employs a ‘torsion-bounding subsolution’). In the setting of a trivial torusbundle over a Riemann surface, this is provided by |∇ f | g t ,h using Lemma 3.12. Proposition 4.8.
Let T k → M → Σ be the trivial T k bundle over a Riemann surface Σ . Let ( M n , g t , µ t , h ) be a solution to Ricci-Yang-Mills flow, with associated potential functions f t . Let f − denote an associated solution of the conjugate heat equation. Then, for T > ,setting τ = T − t , ddt (cid:20) W − ( g, µ, h, f − , τ ) − ˆ Σ |∇ f | g t ,h udV g (cid:21) ≥ ˆ Σ " τ (cid:12)(cid:12)(cid:12)(cid:12) Rc − F + ∇ f − − g τ (cid:12)(cid:12)(cid:12)(cid:12) + τ (cid:12)(cid:12) d ∗ g F + ∇ f − y F (cid:12)(cid:12) udV g ≥ . Proof.
Using that ∂∂t ( udV g ) = − ∆ udV g , it follows from Lemma 3.12 that ddt ˆ Σ |∇ f | g t ,h udV g = ˆ Σ (cid:18) ∂∂t − ∆ (cid:19) |∇ f | g t ,h udV g = ˆ Σ (cid:16) − (cid:12)(cid:12) F µ f (cid:12)(cid:12) − (cid:12)(cid:12) ( ∇ f ) , , (cid:12)(cid:12) − (cid:12)(cid:12) F µ f (cid:12)(cid:12) g,h |∇ f | g t ,h (cid:17) udV g . Combining this with Proposition 4.7 gives the result. (cid:3)
Using Proposition 4.8 it is possible to obtain a uniform κ -noncollapsing result for Ricci-Yang-Mills flow in this setting, as well as the existence of a blowup limit at any finite time singularitywhich is uniformly κ -noncollapsed (cf. [21] for the definitions) Theorem 4.9.
Let T k → M → Σ be the trivial T k bundle over a Riemann surface Σ . Let ( M n , g t , µ t , h ) be a solution to Ricci-Yang-Mills flow. Then g t is not locally collapsing at T < ∞ .Furthermore, assume ( x i , t i ) satisfies (1) Λ i := | Rm | ( x i , t i ) → ∞ , (2) sup M × [0 ,t i ] | Rm | ( x i , t i ) ≤ C Λ i ,then the sequence of pointed solutions (cid:8) (Λ i g i ( t i + Λ − i t ) , µ ( t i + Λ − i t ) , Λ i h ( t i + Λ − i t ) (cid:9) convergessubsequentially to a complete ancient solution to Ricci-Yang-Mills flow which is κ -noncollapsed onall scales for some κ > .Proof. The proof of no-local-collapsing follows the original argument of Perelman using the mono-tonicity formula of Proposition 4.8 (cf. [9] Ch. 6 for the modifications coming from the inclusion ofthe |∇ f | g t ,h term). Using this, the construction of the blowup limit follows from the compactnesstheory for Ricci-Yang-Mills flow/generalized Ricci flow ([39], [9] Ch. 5). (cid:3) Remark 4.10.
The blowup limits constructed in Theorem 4.9 will necessarily be solutions to Ricci-Yang-Mills flow on R k -bundles, since the fiber metric is fixed along the flow, and thus blowing upalong the rescaled sequence.4.3.2. Global existence and convergence.
Proposition 4.11.
Let T k → Σ × T k → Σ denote the trivial principal T k -bundle over a compactRiemann surface with χ (Σ) > . Given G = π ∗ g + tr h µ ⊗ µ an invariant metric on Σ × T k ,the solution to (2.1) with λ = 0 and initial condition ( g , µ , h ) exists on a finite time interval [0 , T ) , and satisfies ICCI-YANG-MILLS FLOW ON SURFACES AND PLURICLOSED FLOW ON ELLIPTIC FIBRATIONS 19 (1) lim t → T ( T − t ) | F µ t | g t ,h = 0 , (2) lim t → T ( T − t ) − g t = g Σ , g Σ denotes a metric of constant curvature .Proof. By taking a double cover if necessary we can reduce to the case Σ ∼ = S . We first claim thatthe existence time is finite. We describe the proof in the case k = 1 for simplicity, the general caseis directly analogous. As in the proof of Proposition 4.12, we can extend the principal connection µ using a flat connection to obtain a Hermitian connection µ on the trivial T bundle on S × T .This defines a choice of complex structure on T , which then yields a complex structure on theproduct S × T . Furthermore, by ([35] Proposition 5.3) the data ( g , µ , h ) defines a Hermitianmetric on S × T , which is furthermore pluriclosed by ([35] Lemma 5.6). By ([35] Proposition6.3, cf. also Remark 6.4) it follows that the pluriclosed flow with this initial data will reduce toRicci-Yang-Mills flow. The S slices are all holomorphic curves in this complex manifold, and itfollows from Stokes Theorem (cf. [36] Proposition 3.9) that for the solution ω t to pluriclosed flowone has ddt ˆ S ω t = − π. It follows that the area goes to zero in finite time for any initial data, and thus the flow must gosingular at a time
T < (8 π ) − Area ω ( S ).Fix a solution ( g t , µ t ) to Ricci-Yang-Mills flow as in the statement, with associated potentialfunctions f t . If the flow exists on a maximal time interval [0 , T ), T < ∞ , by an elementarypoint-picking argument we can choose a sequence ( x i , t i ) of points satisfying conditions (1) and(2) of Theorem 4.9, and construct a corresponding blowup limit ( g ∞ t , µ ∞ t , h ∞ ) as described inTheorem 4.9. We note that by construction, the sequence of functions f i ( x, t ) = f ( x, t i + Λ − i t ) arepotential functions for the rescaled flows, and the f i will converge to a limit function f ∞ satisfying F µ ∞ t = dd c f ∞ t . On the other hand, since |∇ f | g t ,h has a uniform upper bound by Proposition3.13, it follows by the scaling law that |∇ f ∞ | g ∞ ,h ≡
0. Thus f ∞ is constant, and so F µ ∞ ≡ g ∞ t is an ancient 2-dimensional κ -solution to Ricci flow, which by ([21] Corollary11.3) is isometric to the standard round shrinking sphere solution. Using this blowup sequence itis straightforward to show that the whole flow line converges to a round point and satisfies thescale-invariant decay of curvature claimed in the statement. (cid:3) The case χ (Σ) > , nontrivial bundle.Proposition 4.12. Let T k → P → Σ denote a principal T k -bundle over a compact Riemannsurface with χ (Σ) > , c ( P ) = 0 . Given G = π ∗ g + tr h µ ⊗ µ an invariant metric on P ,the solution to (2.1) with λ = 0 and initial condition ( g , µ , h ) exists on [0 , ∞ ) , and there is aone-parameter family of diffeomorphisms φ t such that (1) lim t →∞ φ ∗ t g t = γ g Σ , (2) lim t →∞ φ ∗ t F µ t = γ ω Σ ,where g Σ denotes a metric of constant curvature and ω Σ denotes the associated area form.Proof. First, by lifting to a double cover it suffices to consider the case Σ = S . We first employ agauge-fixing procedure (cf. [40]) to account for the noncompact group of conformal diffeomorphismsof S . For such a diffeomorphism φ we will set φ ∗ g = φ ∗ ( e u g S ) = e u ◦ φ φ ∗ g Σ = e v g Σ = ˆ g, where v = u ◦ φ + log det dφ. (4.5) The condition we require, aiming at application of the sharp Sobolev inequality of Aubin [2], isthat ˆ S xdV ˆ g = 0 , (4.6)where x denotes the position vector in R . By ([5] Lemma 2) we can find a conformal transformation φ so that (4.6) is satisfied at time t = 0. To solve for the relevant gauge transformations we firstset ξ t = (cid:0) φ − t (cid:1) ∗ ˙ φ , which is a vector field on S . Differentiating equation (4.5) gives˙ v = ˙ u ◦ φ + ξv + div g S ξ. Differentiating the defining condition (4.6) yields (cf. [40] Lemma 6.2)0 = ˆ S h x (cid:16) − ve − v + (cid:12)(cid:12) F µ f (cid:12)(cid:12) g,h (cid:17) − ξ i dV ˆ g . As explained in ([40] § ξ . Thus we obtain the required familyof diffeomorphisms φ t such that φ ∗ t g t = φ ∗ t ( e u g S ) = e v t g S = ˆ g t , where condition (4.6) is satisfied at all times. These diffeomorphisms define new curvature densitiesˆ F t = φ ∗ t F t , realized as the curvature of a family of connections ˆ µ t satisfying ∂∂t ˆ µ t = − d ∗ ˆ g F ˆ µ + i ξ t F ˆ µ . Next we observe the key qualitative influence of the hypothesis that c ( P ) = 0, namely the apriori volume lower bound provided by Lemma 3.14. It follows that ˆ g t also has a uniform lowervolume bound. It follows from [2] that for any ǫ > C = C ( ǫ ) such that c ≤ S e v dV S ≤ C exp (cid:20)(cid:0) + ǫ (cid:1) S | dv | g S dV S + R Σ S vdV S (cid:21) . Choosing for instance ǫ = , we can rearrange this to obtain the estimate616 S | dv | g S dV S + R Σ S vdV S ≥ − C. (4.7)We also note that the functional F is bounded above along the ungauged flow by Proposition 2.4.Since the Liouville energy is invariant under the action of conformal diffeomorphisms by ([5] Lemma1), it follows that the Liouville energy of v is uniformly bounded above. Using this together with(4.7) gives || dv || L ≤ C. Using this, it follows that e | v | is uniformly bounded in any L p space, also also that v is bounded in L . Thus by Lemma 4.2 we obtain a uniform estimate of the Sobolev constant of e v g . ApplyingProposition 2.5 we obtain the long time existence.To obtain the convergence we further analyze the monotonicity of F . Using the Sobolev constantestimate for ˆ g and the uniform L (ˆ g ) estimate for F µ f it follows that that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | F ˆ µ | g,h (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L (ˆ g ) ≤ C (cid:18)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∇ ˆ g F ˆ µ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L (ˆ g ) + || F ˆ µ || L (ˆ g ) (cid:19) ≤ C (cid:18)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∇ ˆ g F ˆ µ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L (ˆ g ) + 1 (cid:19) , (4.8)where the last line follows from the upper bound for F . In particular, since F is bounded belowalong the flow, there is a sequence t i → ∞ such that lim i →∞ ddt F ( v t , ˆ µ t , h ) = 0, thus, for sufficientlylarge i , at any time t i we have ǫ ≥ ˆ Σ (cid:16) R ˆ g − | F ˆ µ | g,h (cid:17) dV ˆ g + 2 ˆ Σ (cid:12)(cid:12)(cid:12) ∇ ˆ g F ˆ µ (cid:12)(cid:12)(cid:12) g,h dV ˆ g . (4.9) ICCI-YANG-MILLS FLOW ON SURFACES AND PLURICLOSED FLOW ON ELLIPTIC FIBRATIONS 21
This implies that (cid:12)(cid:12)(cid:12)(cid:12) ∇ ˆ g F ˆ µ (cid:12)(cid:12)(cid:12)(cid:12) L ( g,h ) is bounded, thus applying (4.8) we obtain a uniform L (ˆ g ) esti-mate for F ˆ µ , and it follows easily then from (4.9) that the Calabi energy of ˆ g is bounded. Since ´ M e | v | dV Σ < C , it follows that the curvature does not concentrate in L , and thus by ([40] Theo-rem 3.2, cf. also [6]), v is uniformly bounded in H . By following the arguments of Proposition 2.5,we obtain a uniform H estimate for v on [ t i , t i + 1]. Since there exists a relevant time t i in everytime interval of the form [ T, T + 1] for all sufficiently large T , it follows that there is a uniform H estimate for v . Using this and a bootstrapping argument as described in Proposition 4.1, itfollows that there are uniform C k,α estimates for v and ˆ F . Returning to the monotonicity formulafor F , it follows that ∇ ˆ g ˆ F →
0, and R ˆ g → const. Thus ˆ g is converging to a round metric, and ˆ F is converging to a multiple of the area form, finishing the proof. (cid:3) Proofs of Main Theorems.
Proof of Theorem 1.1.
The individual cases are proved in Propositions 4.1, 4.3, 4.11 and 4.12. (cid:3)
Proof of Corollary 1.4.
This follows from Proposition 2.6 and Theorem 1.1. (cid:3)
Proof of Corollary 1.6.
This follows from Proposition 2.7 and Theorem 1.1. In the case of the Hopfsurface, it is easy to check that the limiting structure in fact has vanishing Bismut-Ricci tensor.In dimension 4 it is known ([10], cf. [9] Theorem 8.26) that the metric is either Calabi-Yau orisometric to the Hopf metric. Since the Hopf surface does not admit K¨ahler metrics, it must bethe Hopf metric, as claimed. (cid:3)
References [1] Vestislav Apostolov and Jeffrey Streets,
The nondegenerate generalized K¨ahler Calabi-Yau problem ,arXiv:1703.08650.[2] Thierry Aubin,
Meilleures constantes dans le th´eor`eme d’inclusion de Sobolev et un th´eor`eme de Fredholm nonlin´eaire pour la transformation conforme de la courbure scalaire , J. Functional Analysis (1979), no. 2, 148–174.MR 534672[3] J. Bartz, M. Struwe, and R. Ye, A new approach to the Ricci flow on S , Ann. Scuola Norm. Sup. Pisa Cl. Sci.(4) (1994), no. 3, 475–482. MR 1310637[4] C. G. Callan, D. Friedan, E. J. Martinec, and M. J. Perry, Strings in background fields , Nuclear Phys. B (1985), no. 4, 593–609. MR 819433[5] Sun-Yung Alice Chang,
The Moser-Trudinger inequality and applications to some problems in conformal geome-try , Nonlinear partial differential equations in differential geometry (Park City, UT, 1992), IAS/Park City Math.Ser., vol. 2, Amer. Math. Soc., Providence, RI, 1996, pp. 65–125. MR 1369587[6] Xiuxiong Chen,
Weak limits of Riemannian metrics in surfaces with integral curvature bound , Calc. Var. PartialDifferential Equations (1998), no. 3, 189–226. MR 1614627[7] Bennett Chow, The Ricci flow on the -sphere , J. Differential Geom. (1991), no. 2, 325–334. MR 1094458[8] M. Garcia-Fernandez, Ricci flow, Killing spinors, and T-duality in generalized geometry , Adv. Math. (2019),1059–1108.[9] Mario Garcia-Fernandez and Jeffrey Streets,
Generalized Ricci Flow , University Lecture Series, AMS, 2020.[10] P. Gauduchon and S. Ivanov,
Einstein-Hermitian surfaces and Hermitian Einstein-Weyl structures in dimension
4, Math. Z. (1997), no. 2, 317–326. MR 1477631[11] Steven Gindi and Jeffrey Streets,
Structure of collapsing solutions of generalized Ricci flow , J. Geom. Anal.(2020).[12] Richard S. Hamilton,
The Ricci flow on surfaces , Mathematics and general relativity (Santa Cruz, CA, 1986),Contemp. Math., vol. 71, Amer. Math. Soc., Providence, RI, 1988, pp. 237–262. MR 954419[13] Tom Ilmanen and Dan Knopf,
A lower bound for the diameter of solutions to the Ricci flow with nonzero H ( M n ; R ), Math. Res. Lett. (2003), no. 2-3, 161–168. MR 1981893[14] Michael Jablonski and Andrea Young, Ricci Yang-Mills solitons on nilpotent Lie groups , J. Lie Theory (2013),no. 1, 177–202. MR 3060772[15] Man-Chun Lee and Jeffrey Streets, Complex manifolds with negative curvature operator , arXiv:1903.12645, toappear IMRN.[16] John Lott,
On the long-time behavior of type-III Ricci flow solutions , Math. Ann. (2007), no. 3, 627–666.MR 2336062 [17] ,
Dimensional reduction and the long-time behavior of Ricci flow , Comment. Math. Helv. (2010), no. 3,485–534. MR 2653690[18] J. Moser, A sharp form of an inequality by N. Trudinger , Indiana Univ. Math. J. (1970/71), 1077–1092.MR 301504[19] T. Oliynyk, V. Suneeta, and E. Woolgar, A gradient flow for worldsheet nonlinear sigma models , Nuclear Phys.B (2006), no. 3, 441–458. MR 2214659[20] Fabio Paradiso,
Generalized Ricci flow on nilpotent lie groups , arXiv:2002.01514.[21] Grisha Perelman,
The entropy formula for the Ricci flow and its geometric applications , arXiv:0211159.[22] Johan R˚ade,
On the Yang-Mills heat equation in two and three dimensions , J. Reine Angew. Math. (1992),123–163. MR 1179335[23] Alberto Raffero and Luigi Vezzoni,
On the dynamical behaviour of the generalized Ricci flow , arXiv:2012.06792.[24] Jian Song and Gang Tian,
The K¨ahler-Ricci flow through singularities , Invent. Math. (2017), no. 2, 519–595.MR 3595934[25] Jeffrey Streets,
Pluriclosed flow and the geometrization of complex surfaces , arXiv:1808.09490.[26] ,
Regularity and expanding entropy for connection Ricci flow , J. Geom. Phys. (2008), no. 7, 900–912.MR 2426247[27] , Singularities of renormalization group flows , J. Geom. Phys. (2009), no. 1, 8–16. MR 2479257[28] , Ricci Yang-Mills flow on surfaces , Adv. Math. (2010), no. 2, 454–475. MR 2565538[29] ,
Pluriclosed flow, Born-Infeld geometry, and rigidity results for generalized K¨ahler manifolds , Comm.Partial Differential Equations (2016), no. 2, 318–374. MR 3462132[30] , Pluriclosed flow on manifolds with globally generated bundles , Complex Manifolds (2016), no. 1, 222–230. MR 3550300[31] , Generalized geometry, T -duality, and renormalization group flow , J. Geom. Phys. (2017), 506–522.MR 3610057[32] , Generalized K¨ahler-Ricci flow and the classification of nondegenerate generalized K¨ahler surfaces , Adv.Math. (2017), 187–215. MR 3672905[33] ,
Global viscosity solutions of generalized K¨ahler-Ricci flow , Proc. Amer. Math. Soc. (2018), no. 2,747–757. MR 3731708[34] ,
Pluriclosed flow on generalized K¨ahler manifolds with split tangent bundle , J. Reine Angew. Math. (2018), 241–276. MR 3808262[35] ,
Classification of solitons for pluriclosed flow on complex surfaces , Math. Ann. (2019), no. 3-4,1555–1595. MR 4023384[36] Jeffrey Streets and Gang Tian,
A parabolic flow of pluriclosed metrics , Int. Math. Res. Not. IMRN (2010), no. 16,3101–3133. MR 2673720[37] ,
Generalized K¨ahler geometry and the pluriclosed flow , Nuclear Phys. B (2012), no. 2, 366–376.MR 2881439[38] ,
Regularity results for pluriclosed flow , Geom. Topol. (2013), no. 4, 2389–2429. MR 3110582[39] Jeffrey D. Streets, Ricci Yang-Mills flow , ProQuest LLC, Ann Arbor, MI, 2007, Thesis (Ph.D.)–Duke University.MR 2709943[40] Michael Struwe,
Curvature flows on surfaces , Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) (2002), no. 2, 247–274.MR 1991140[41] Neil S. Trudinger, On imbeddings into Orlicz spaces and some applications , J. Math. Mech. (1967), 473–483.MR 0216286[42] Andrea Young, Stability of Ricci Yang-Mills flow at Einstein Yang-Mills metrics , Comm. Anal. Geom. (2010),no. 1, 77–100. MR 2660458[43] Andrea Nicole Young, Modified Ricci flow on a principal bundle , ProQuest LLC, Ann Arbor, MI, 2008, Thesis(Ph.D.)–The University of Texas at Austin. MR 2712036
Rowland Hall, University of California, Irvine, Irvine, CA 92617
Email address ::