Pluripotential Monge-Amp{è}re flows in big cohomology class
aa r X i v : . [ m a t h . DG ] F e b PLURIPOTENTIAL MONGE-AMP `ERE FLOWSIN BIG COHOMOLOGY CLASSES
QUANG-TUAN DANG
Abstract.
We study pluripotential complex Monge-Amp`ere flows in big cohomology classeson compact K¨ahler manifolds. We use the Perron method, considering pluripotential subso-lutions to the Cauchy problem. We prove that, under natural assumptions on the data, theupper envelope of all subsolutions is continuous in space and semi-concave in time, and providesa unique pluripotential solution with such regularity. We apply this theory to study pluripo-tential K¨ahler-Ricci flows on compact K¨ahler manifolds of general type as well as on K¨ahlervarieties with semi-log canonical singularities.
Contents
Introduction 11. Preliminaries 42. The envelope of subsolutions 113. Existence and uniqueness 244. Applications 32References 45
Introduction
The primary goal of this paper is to study pluripotential complex Monge-Amp`ere flowsmotivated by the Minimal Model Program (MMP) in algebraic geometry, whose aim is the(birational) classification of projective manifolds. In a recent celebrated work, Birkar-Cascini-Hacon-Mckernan [BCHM10] showed the existence of minimal models for a large class of varietiescalled varieties of general type. J. Song and G. Tian [ST12, ST17] have proposed an analyticanalogue making use of (twisted) K¨ahler-Ricci flows on compact K¨ahler manifolds. The latterrequires to develop a good theory of weak K¨ahler-Ricci flows. The first steps of a parabolicpluripotential theory have been developed by Guedj-Lu-Zeriahi [GLZ18, GLZ20], allowing usto define a unique weak K¨ahler-Ricci flow on compact K¨ahler varieties with Kawamata logterminal singularities. In this paper we aim at developing this theory further, extending it tothe most general singularities encountered in the MMP, and studying the geometric convergenceof the Monge-Amp`ere flows.Let X be a compact K¨ahler manifold of dimension n and fix a K¨ahler form ˆ ω . The (normal-ized) K¨ahler-Ricci flow on X starting at ˆ ω is the solution to the following evolution equation ∂θ t ∂t = − Ric( θ t ) − λθ t , θ t =0 = ˆ ω, (0.1) Date : February 17, 2021.2010
Mathematics Subject Classification.
Key words and phrases.
Parabolic Monge-Amp`ere equation, big cohomology class, K¨ahler-Ricci flow.The author is partially supported by the ANR projects GRACK and PARAPLUI. where the sign of λ ∈ R depends on that of c ( K X ). Solving the normalized K¨ahler-Ricciflow (0.1) turns out to be equivalent to solving the scalar complex Monge-Amp`ere flow ( ( ω t + dd c ϕ t ) n = e ∂ t ϕ t + λϕ t + h ( t,x ) dVω t + dd c ϕ t > , where ω t ∈ { θ t } ∈ H , ( X, R ) is fixed, and h is a smooth density.Since the MMP requires one to work on singular varieties, it is necessary to develop a finetheory dealing with weak solutions. One has indeed to deal with similar complex Monge-Amp`ereflows with various degeneracies: the reference forms ω t are no longer K¨ahler and the densities h no longer smooth, with integrability properties that depend on the type of singularities.A parabolic viscosity approach has been developed recently in [EGZ16], which requires thedensities to be continuous hence has a limited scope of applications. The first elements of aparabolic pluripotential theory has been laid down in [GLZ18, GLZ20] which are the parabolicanalogues of the pioneering work of Bedford and Taylor in the local case [BT76, BT82]. Weextend here this theory so as to be able to deal with big cohomology classes. Assumptions and Notations.
Before going further and stating the main results of the paper,let us fix some notations. Let X be a compact K¨ahler manifold of dimension n . We let X T := (0 , T ) × X denote the real (2 n + 1)-dimensional manifold with T ∈ (0 , + ∞ ]. We focusmostly on finite time intervals i.e. T < + ∞ . The parabolic boundary of X T is denoted by ∂X T := { } × X. We fix θ a smooth closed (1 , θ , Ω := Amp( θ ) , which is a non empty Zariski open subset of X .We assume that ( ω t ) t ∈ [0 ,T ) is a smooth family of closed (1 , X such that g ( t ) θ ≤ ω t , ∀ t ∈ [0 , T ) , where g ( t ) is an increasing smooth positive function on [0 , T ].Throughout the article we assume that there exists a K¨ahler form Θ such that − Θ ≤ ω t , ˙ ω t , ¨ ω t ≤ Θ . (0.2)We fix dV a smooth volume form on X . We shall always assume that • ≤ f ∈ L p ( X, dV ) for some p >
1, and f is strictly positive almost everywhere; • F : [0 , T ] × X × R → R is continuous on [0 , T ] × X × R , • the function r F ( ., ., r ) is increasing in r , • the function F is uniformly Lipschitz in ( t, x ) ∈ [0 , T ] × R , i.e. there exists a constant κ F > t, t ′ ∈ [0 , T ], x ∈ X , r, r ′ ∈ R , | F ( t, x, r ) − F ( t ′ , x, r ′ ) | ≤ κ F ( | t − t ′ | + | r − r ′ | ) . • the function ( t, r ) F ( t, ., r ) is convex.The purpose of this paper is to extend the results of [GLZ20] to big cohomology classes.More precisely, we consider the complex Monge-Amp`ere flows: dt ∧ ( ω t + dd c ϕ t ) n = e ˙ ϕ t + F ( t,.,ϕ t ) f dV ∧ dt (CMAF)in X T . Note that the equation (CMAF) should be understood in the weak sense of measuresin (0 , T ) × Ω (cf. [GLZ20] and Section 1.3).The existence of the weak K¨ahler-Ricci flow is often proved by using approximation argumentsand a priori estimates (cf. [ST17, GLZ20]). Big cohomology classes can not be approximated by
LURIPOTENTIAL MONGE-AMP`ERE FLOWS IN BIG COHOMOLOGY CLASSES 3
K¨ahler ones so this approach breaks down in our case. We shall instead use the Perron method,inspired by [GLZ18], considering the upper envelope U of all pluripotential subsolutions to theCauchy problem. We prove that this upper envelope is locally uniformly semi-concave in time: Theorem A.
Let ϕ be a ω -psh function with minimal singularities. Then the upper envelope U of all subsolutions to (CMAF) with initial data ϕ is a pluripotential solution to (CMAF) which is locally uniformly Lipschitz and locally uniformly semi-concave in t ∈ (0 , T ) . We prove Theorem A by following the arguments of [GLZ18] in the local context: • we first show that the upper envelope of all subsolutions is locally uniformly Lipschitzin t (Theorem 2.7) and that it is itself a pluripotential subsolution; • we then show that the envelope is locally uniformly semi-concave (Theorem 2.13); • we finally apply a balayage process and use the analogue result in the local context[GLZ18] to conclude the proof.We prove in Theorem 3.6 that the envelope U in Theorem A has minimal singularities andis continuous in (0 , T ) × Ω under an extra assumption:(0.3) ˙ ω t ≤ Aω t , t ∈ [0 , T ) , for some positive constant A . We also show that U is the unique pluripotential solution withsuch regularity by establishing the following comparison principle: Theorem B.
Assume that (0.3) holds. Let ϕ (resp. ψ ) be a pluripotential subsolution (resp.supersolution) to (CMAF) with initial data ϕ (resp. ψ ). We assume that ψ is locally uniformlysemi-concave in t ∈ (0 , T ) and ψ is continuous in (0 , T ) × Ω . We assume moreover that foreach t , ψ t has minimal singularities. Then ϕ ≤ ψ on [0 , T ) × X if ϕ ≤ ψ . The assumption that ψ t has minimal singularities means that for each t ∈ (0 , T ), there existsa constant C t such that | ψ t − V ω t | is bounded by C t , where V ω t is the largest negative ω t -pshfunction. The proof of Theorem Theorem B is provided in Section 3.2, generalizing some ideasfrom [GLZ20].Starting from a K¨ahler form ω , it follows from [Cao85, Tsu88, TZ06] that the (smooth)normalized K¨ahler-Ricci flow exists in [0 , T ) where T := sup { t > e − t { ω } + (1 − e − t ) c ( K X ) is K¨ahler } . The singularity time T is finite unless c ( K X ) is nef.It is an interesting question to know how to define the flow for t > T . This was formulatedin [FIK03, Section 10, Question 8] and a precise conjecture was made in [BT12]. Note that,for t > T the forms ω t remain big but are no longer nef and one can not hope to make senseof the flow equation in the classical sense. In [Tˆo19], it was proved that the flow can becontinued through the singularity time T in the viscosity sense and it eventually converges tothe unique singular K¨ahler-Einstein metric on X . Using the tools developed above, we establishthe pluripotential analogue of the main result of [Tˆo19]: Theorem C.
Let X be a compact n-dimensional K¨ahler manifold of general type. Then thenormalized pluripotential K¨ahler-Ricci flow, emanating from a K¨ahler metric ω ∂θ t ∂t = − Ric ( θ t ) − θ t , exists for all time t > . It coincides with the smooth flow on [0 , T ) and deforms ω towardsthe unique singular K¨ahler-Einstein metric ω KE , as t → + ∞ . We actually establish a more general result allowing to run the flow from an arbitrary closedpositive current with bounded potential (see Theorem 4.1).
QUANG-TUAN DANG
In the last part of the paper we study pluripotential K¨ahler-Ricci flows on K¨ahler varieties Y with semi-log canonical singularities (the most general class of singularities appearing in thelog MMP) and ample canonical line bundle.It has been proved by R. Berman and H. Guenancia [BG14] that Y admits a unique K¨ahler-Einstein current ω KE in the class c ( K Y ) which is smooth in the regular locus Y reg . We applyour theory to run the pluripotential K¨ahler-Ricci flow on Y and recover the canonical metric ω KE as the long time limit of the flow. More precisely, we have the following: Theorem D.
Let X be a projective complex algebraic variety with semi-log canonical singular-ities such that K X is ample. Then the normalized pluripotential K¨ahler-Ricci flow, emanatingfrom a K¨ahler metric ω ∂θ t ∂t = − Ric ( θ t ) − θ t , exists for all time t > . It deforms ω towards the unique singular K¨ahler-Einstein metric ω KE as t → + ∞ . Again we actually show that the flow can be run from an arbitrary positive closed currentwith bounded potentials (see Theorem 4.8).For varieties of log general type with log terminal singularities the pluripotential K¨ahler-Ricci flow (with non continuous data) was constructed in [GLZ20, § Q -factorial projective varieties with log canonical singularities:establishing higher order a priori estimates, they obtain a notion of weak K¨ahler-Ricci flowwhich is smooth in the regular locus of the variety. Organization of the paper.
In Section 1 we provide some backgrounds on pluripotentialtheory in big cohomology classes. In Section 2 we study the regularity properties of the envelopeof pluripotential subsolutions. In Section 3 we shall prove Theorem A and Theorem B. Westudy in Section 4 the normalized K¨ahler-Ricci flow on compact K¨ahler manifolds of generaltype (resp. stable varieties) and prove Theorem C (resp. Theorem D).
Acknowledgements.
The author would like to thank his advisors Vincent Guedj and Hoang-Chinh Lu for constant help and encouragement. The author also thanks Tˆat-Dat Tˆo for usefulconversations on his results in [Tˆo19]. The author is truly grateful to H. Guenancia and A.Zeriahi for several interesting dicussions.1.
Preliminaries
In this section we recall necessary definitions and background. Let X be a compact K¨ahlermanifold of complex dimension n , and Θ be a K¨ahler metric on X . We let H , ( X, R ) denotethe Bott-Chern cohomology of d -closed real (1 , ∂ ¯ ∂ -exact ones.1.1. Monge-Amp`ere operators in big cohomology classes.
Big cohomology classes.
Let θ be a smooth real closed (1 , − form on X . An uppersemi-continuous function ϕ : X → [ −∞ , + ∞ ) is called θ - plurisubharmonic ( θ -psh for short)if in any local holomorphic coordinates ϕ can be written as the sum of a psh and a smoothfunction, and θ + dd c ϕ ≥ , in the weak sense of currents, where d = ∂ + ¯ ∂ and d c = i π ( ¯ ∂ − ∂ ). LURIPOTENTIAL MONGE-AMP`ERE FLOWS IN BIG COHOMOLOGY CLASSES 5
We let PSH(
X, θ ) denote the set of all θ -psh functions on X which are not identically −∞ .This set is endowed with the weak topology which coincides with the L -topology. By Hartogs’lemma ϕ sup X ϕ is continuous in the L -topology.By the dd c -lemma any closed positive (1 , T cohomologous to θ can be written as T = θ + dd c ϕ for some θ -psh function ϕ which is moreover unique up to an additive constant.If T and T ′ are two closed positive (1 , X which are cohomologous, then T issaid to be less singular than T ′ if their global potentials satisfy ϕ ′ ≤ ϕ + O (1) (then we also saythat ϕ is less singular than ϕ ′ ). A positive current T is now said to have minimal singularities if it is less singular than any other positive current in its cohomology class. Definition 1.1. A θ -psh function ϕ is said to have minimal singularities if it is less singularthan any other θ -psh function on X . Such θ -psh functions always exists, one can consider, following Demailly, the upper envelope V θ := sup { ϕ : ϕ ∈ PSH(
X, θ ) , and ϕ ≤ } . Observe that V ∗ θ is a θ -psh function satisfying V ∗ θ ≤ V θ , hence V θ = V ∗ θ is a θ -psh function withminimal singularities.The cohomology class α = { θ } ∈ H , ( X, R ) is said to be big if there exists a closed (1 , T + = θ + dd c ϕ + , cohomologous to θ such that T + is strictly positive i.e T + ≥ ε Θ for some constant ε >
0. Wealso call T + a K¨ahler current.A function u has analytic singularities if it can locally be written as u = c N X j =1 | f j | + h, where the f ′ j are local holomorphic functions, the function h is smooth and c is a positiveconstant.From now on we assume that the class α = { θ } is big. By Demailly’s regularization theorem[Dem92], any θ -psh function u can be approximated from above by a sequence of ( θ + ε j ω ) − pshfunctions ( u j ) with analytic singularities. Applying this to the potential ϕ + of a K¨ahler current T + = θ + dd c ϕ + , one can moreover assume that the function ϕ + has analytic singularities. Sucha current T + is then smooth on a Zariski open subset, this motivates the following: Definition 1.2.
The ample locus Amp( α ) of α is the set of x ∈ X such that there exists aK¨ahler current with analytic singularities which is smooth around x . It follows from the Noetherian property of closed analytic subsets that Amp( α ) is a Zariskiopen set. Note that any θ -psh function ϕ with minimal singularities is locally bounded onthe ample locus Amp( α ) since it has to satisfy ϕ + ≤ ϕ + O (1). Moreover, ϕ + does not haveminimal singularities unless α is a K¨ahler class (cf. [Bou04, Proposition 2.5]).By the above analysis, there exists a θ -psh function χ on X with analytic singularities suchthat, for some δ > θ + dd c χ ≥ δ Θ . (1.1)Subtracting a large constant, we can always assume that χ ≤
0, thus χ ≤ V θ . Moreover, χ issmooth in the ample locus Amp( α ), and χ ( x ) → −∞ as x → ∂ Ω (cf. [Bou04, Theorem 3.17]).
QUANG-TUAN DANG
Full Monge-Amp`ere mass.
In [BEGZ10], the authors defined the non-pluripolar product T
7→ h T n i of any closed positive (1 , T ∈ α , which is shown to be well-defined as apositive measure on X putting no mass on pluripolar sets. In particular given a θ -psh function ϕ , one can define its non-pluripolar Monge-Amp`ere byMA θ ( ϕ ) = MA( ϕ ) := h ( θ + dd c ϕ ) n i . By definition the total mass of MA( ϕ ) is less than or equal to the volume Vol( α ) of the class α : Z X MA( ϕ ) ≤ Vol( α ) := Z X MA( V θ ) . A particular class of θ -psh functions that appears naturally is the one for which the lastinequality is an equality. We will say that such functions (or the associated currents) have full Monge–Amp`ere mass . For example, θ -psh functions with minimal singularities have fullMonge–Amp`ere mass (cf. [BEGZ10, Theorem 1.16]), but the converse is not true.When f ∈ L p ( X, dV ) for some p > ρ can be found by solving a complexMonge-Amp`ere equation as we now explain. We let c be the normalizing constant such that2 n e c f dV has total mass equal to Vol( α ). Theorem 1.3 ([BEGZ10, Theorem 4.1]) . There exists a unique θ -psh function ρ with fullMonge-Amp`ere mass such that ( θ + dd c ρ ) n = 2 n e c f dV, (1.2) and normalized by sup X ρ = 0 . Moreover, there exists a constant M > only depending on θ , dV , and p > such that ρ ≥ V θ − M k f k /np . Parabolic potentials.
In this section we define the parabolic pluripotential objects inbig cohomology classes necessary for our study. These are mainly taken from [GLZ18, GLZ20]but we need to be more precise when dealing with unbounded functions. Let ω = ( ω t ) t ∈ [0 ,T ) be asmooth family of closed smooth real (1 , Definition 1.4.
We let P ( X T , ω ) denote the set of functions ϕ : X T → [ −∞ , + ∞ ) such that • for each t ∈ (0 , T ) fixed, the slice x → ϕ ( t, x ) is ω t -psh on X , • for any compact subinterval J ⊂ (0 , T ) there exists a positive constant κ = κ J ( ϕ ) suchthat ∂ t ϕ ≤ κ − κ ( ρ + χ ) , ∀ t ∈ J, (1.3) in the sense of distribution in J × Ω , where ρ, χ are defined in (1.2) , (1.1) . We would like to have an interpretation of the last condition. For any compact subset K ⋐ Ω,there exists a constant C = C ( K ) > ∂ τ ( t, x ) ≤ C, ∀ ( t, x ) ∈ K. Hence for every x ∈ K , the function t ϕ ( t, x ) − Ct is decreasing in J , so the partial derivative ∂ t ϕ exists for almost everywhere t ∈ J (see e.g. [KK, Theorem 2.1.8]). Lemma 1.5.
Assume that ϕ is a ω -psh function and ϕ ∈ P ( X T , ω ) . If ϕ t → ϕ in L ( X ) as t → , then the extension ϕ : [0 , T ) × X → [ −∞ , + ∞ ) is upper semi-continuous in [0 , T ) × Ω .Proof. Recall that Ω T = (0 , T ) × Ω. The upper semicontinuity of ϕ inside Ω T follows from thesemicontinuity in space and Lipschitz regularity in time. It remains to prove that the extension LURIPOTENTIAL MONGE-AMP`ERE FLOWS IN BIG COHOMOLOGY CLASSES 7 ϕ is upper semi-continuous at (0 , x ) for any x ∈ Ω. Let ( t j , x j ) ∈ Ω T be a sequence whichconverges to (0 , x ). We will show thatlim sup j → + ∞ ϕ ( t j , x j ) ≤ ϕ ( x ) . Since ϕ is bounded from above we can assume the functions ϕ t are negative. Since the problemis local, we may assume that Ω = B is an open neighborhood of x . Let h t be a smooth localpotential for ω t in B i.e. dd c h t = ω t . Up to replacing ϕ t by ϕ t + h t , we may assume that thefunctions ϕ t are psh and negative on B . Fix r so small that B ( x, r ) ⋐ B . For any δ ∈ [0 , r ),there exists j such that x j ∈ B ( x , δ ) for all j ≥ j hence B ( x , r ) ⊂ B ( x j , r + δ ). We have ϕ ( t j , x j ) ≤ B ( x j , r + δ )) Z B ( x j ,r + δ ) ϕ ( t j , x ) dV ≤ B ( x j , r + δ )) Z B ( x ,r ) ϕ ( t j , x ) dV. Since lim sup j ϕ t j ( x ) ≤ ϕ ( x ) for all x ∈ X , Fatou’s lemma implies thatlim sup j → + ∞ ϕ ( t j , x j ) ≤ Vol( B ( x , r ))Vol( B ( x , r + δ )) 1Vol( B ( x , r )) Z B ( x ,r ) ϕ ( x ) dV ( x ) . Now we first let δ → r → (cid:3) Definition 1.6.
We say that ϕ ∈ P ( X T , ω ) has minimal singularities if ϕ t − V ω t is boundedfor each t ∈ (0 , T ) fixed. If ϕ ∈ P ( X T , ω ) ∩ L ∞ loc (Ω T ) the product dt ∧ ( ω t + dd c ϕ t ) n is well defined as a positive measure in Ω T as follows from the works of Bedford-Taylor [BT76,BT82]. This Monge-Amp`ere measure extendes trivially over X T since X \ ( { t }× Ω) is pluripolar(Ω is a Zariski open subset in X ). Since ω t ≤ Θ for all t (as explained in the introduction), thepositive Borel measures ( ω t + dd c ϕ t ) n have uniformly bounded masses on X: Z X ( ω t + dd c ϕ t ) n ≤ Z X (Θ + dd c ϕ t ) n ≤ Z X Θ n . (1.4)These can be considered as a family of currents of degree 2 n in the real (2 n + 1)-dimensionalmanifold X T = (0 , T ) × X . We now show that this family depends continuously on t : Lemma 1.7.
Let ϕ ∈ P ( X T ) ∩ L ∞ loc (Ω T ) and γ be a continuous test function in Ω T . Then thefunction t R Ω γ ( t, . )( ω t + dd c ϕ t ) n is continuous in (0 , T ) . Moreover sup ≤ t Fix compact sets J ⋐ (0 , T ), K ⋐ Ω such that J × K contains the support of γ .The continuity on (0 , T ) \ J is clear. Fix now t ∈ J . By definition of P ( X T ) there exists aconstant C > J, K such that, for each x ∈ K fixed, ϕ ( t, x ) − Ct is decreasingin t . Thus the continuity of t ω t and Bedford-Taylor’s convergence theorem (see e.g. [GZ,Theorem 3.18]) ensure that ( ω t + dd c ϕ t ) n → ( ω t + dd c ϕ t ) n weakly as t → t . On the otherhand γ ( t, . ) uniformly converges on Ω to γ ( t , . ), the first statement follows. The second onefollows from the inequality (1.4) above. (cid:3) QUANG-TUAN DANG Definition 1.8. Let ϕ ∈ P ( X T , ω ) ∩ L ∞ loc (Ω T ) . The map γ Z X T γdt ∧ ( ω t + dd c ϕ t ) n := Z T dt (cid:18)Z Ω γ ( t, . )( ω t + dd c ϕ t ) n (cid:19) defines a positive (2 n + 1) -current on Ω T , hence on X T , denoted by dt ∧ ( ω t + dd c ϕ t ) n , whichcan be identified with a positive Radon measure on X T . The following is a parabolic analogue of the convergence result of Bedford-Taylor [BT76,BT82]. Lemma 1.9. Assume that ( ϕ j ) is a monotone sequence of functions in P ( X T , ω ) which con-verges almost everywhere to a function ϕ ∈ P ( X T , ω ) ∩ L ∞ loc (Ω T ) on X T . Then dt ∧ ( ω t + dd c ϕ jt ) n → dt ∧ ( ω t + dd c ϕ t ) n in the sense of measures on Ω T .Proof. The proof is similar to that of [GLZ20, Proposition 1.12] but, because of its crucial rolein the sequel, we give the details here.Let γ ( t, x ) be a continuous test function in Ω T . By definition we have, for any j , Z Ω T γ ( t, . ) dt ∧ ( ω t + dd c ϕ jt ) n = Z T dt (cid:18)Z Ω γ ( t, . )( ω t + dd c ϕ jt ) n (cid:19) . We now apply Bedford-Taylor’s convergence theorem (see e.g. [GZ, Theorem 3.23]) to inferthat, for any t ∈ (0 , T ), Z Ω γ ( t, . )( ω t + dd c ϕ jt ) n → Z Ω γ ( t, . )( ω t + dd c ϕ t ) n . On the other hand, for all t ∈ (0 , T ), (cid:12)(cid:12)(cid:12)(cid:12)Z Ω γ ( t, . )( ω t + dd c ϕ jt ) n (cid:12)(cid:12)(cid:12)(cid:12) ≤ max Ω | γ ( t, . ) | Vol( ω t ) ≤ C ( γ ) Z X Θ n . The result follows from Lebesgue dominated convergence theorem. (cid:3) We say that ϕ : X T → R is locally uniformly semi-concave (resp. semi-convex ) in Ω T =(0 , T ) × Ω if for any compact subset J × K ⊂ (0 , T ) × Ω there exists κ = κ ( ϕ, J, K ) > κ < 0) such that for all x ∈ K , the function t → ϕ ( t, x ) − κt is concave (resp. convex) in t ∈ J . For any x ∈ Ω fixed, the left and right derivatives, ∂ + t ϕ ( t, x ) = lim s → + ϕ ( t + s, x ) − ϕ ( t, x ) s , and ∂ − t ϕ ( t, x ) = lim s → − ϕ ( t + s, x ) − ϕ ( t, x ) s exist for all t ∈ (0 , T ), and they coincide when ∂ t ϕ ( t, x ) exists.Let ℓ denote the Lebesgue measure on R and µ denote a positive Borel measure on X . Wehave the following result whose proof is identical to that of [GLZ18, Lemma 1.12]. Proposition 1.10. Let ϕ : Ω T → R be a continuous function which is locally uniformly semi-concave in (0 , T ) . Then ( t, x ) → ∂ − t ϕ ( t, x ) is upper semi-continuous while ( t, x ) → ∂ + t ϕ ( t, x ) is lower semi-continuous in Ω T . In particular, ∂ + t ϕ and ∂ − t ϕ coincide and are continuous in Ω T \ E , where E is a Borel set with ℓ ⊗ µ measure zero. LURIPOTENTIAL MONGE-AMP`ERE FLOWS IN BIG COHOMOLOGY CLASSES 9 By replacing ϕ with − ϕ one can obtain a similar conclusion for uniformly semi-convex func-tions.The following convergence results play a key role in the sequel. We omit their proofs andrefer the reader to [GLZ18, Section 2]. Proposition 1.11. Let D be a bounded open subset in R m , J ⊂ R be a bounded open interval,and ≤ f ∈ L p ( D ) with p > . Let ( v j ) be a sequence of Borel functions in J × D such that ( e v j f ) is uniformly bounded in L ( J × D, dt ∧ dV ) . Assume that for any x ∈ D , v j ( ., x ) convergesto a bounded Borel function v ( ., x ) in the sense of distributions on J and for all η ∈ C ∞ ( J × D )sup j ∈ N ,x ∈ D (cid:12)(cid:12)(cid:12)(cid:12)Z J η ( t, x ) v j ( t, x ) dt (cid:12)(cid:12)(cid:12)(cid:12) < + ∞ . (1.5) Then for any positive smooth test function η ∈ C ∞ ( J × D ) , lim inf j → + ∞ Z J × D η ( t, x ) e v j ( t,x ) f ( x ) dt ∧ dV ≥ Z J × D η ( t, x ) e v ( t,x ) f ( x ) dt ∧ dV. Proof. See [GLZ18, Proposition 2.6] (cid:3) Proposition 1.12. Let ( f j ) be a sequence of positive functions which converges to f in L ( X T , ℓ ⊗ µ ) . Let ( ϕ j ) be a sequence of functions in P ( X T , ω ) ∩ L ∞ loc (Ω T ) which • converges ℓ ⊗ µ -almost everywhere in X T to a function ϕ ∈ P ( X T , ω ) ∩ L ∞ loc (Ω T ) ; • is locally uniformly semi-concave in (0 , T ) × Ω .Then the limit lim j → + ∞ ˙ ϕ j ( t, x ) exists and is equal to ˙ ϕ ( t, x ) for ℓ ⊗ µ -almost every ( t, x ) ∈ Ω T ,and h ( ˙ ϕ j ) f j ℓ ⊗ µ → h ( ˙ ϕ ) f ℓ ⊗ µ, in the weak sense of measures on Ω T , for all h ∈ C ( R , R ) .Proof. See [GLZ18, Proposition 2.9] (cid:3) Pluripotential subsolutions/supersolutions. We assume that T < + ∞ . As explainedin the introduction, we assume here that g ( t ) θ ≤ ω t ≤ Θ, where θ is a big (1 , g is asmooth increasing positive function in t ∈ [0 , T ], and Θ is a K¨ahler form.Let us emphasize here that by comparison with [GLZ18, GLZ20], for an element u ∈ P ( X T , ω )the weak derivative ∂ τ ϕ ( t, . ) is merely locally bounded from above in Ω but u is not locallyuniformly Lipschitz in (0 , T ). This is natural as we are dealing with quasi-psh functions whichare bounded from above but not from below.Before defining pluripotential subsolutions (supersolutions), we need to make sense of thequantity ˙ ϕ t = ∂ τ ϕ ( t, . ), in order to define the right-hand side of (CMAF). By the definition of P ( X T , ω ), for any compact subset K ⊂ Ω, J ⋐ (0 , T ), there exists a constant C = C K,J > Ct − ϕ ( t, x ) is increasing in t ∈ J for every x ∈ K . Thus, for every x ∈ K , ∂ τ ϕ t ( x )is well defined for almost every t ∈ J (see e.g. [KK, Lemma 1.2.8]). This implies that theright-hand side of (CMAF) is well-defined almost everywhere in Ω T (using Fubini’s theorem).This analysis motivates the following: Definition 1.13. A parabolic potential ϕ ∈ P ( X T , ω ) is a subsolution to (CMAF) on X T if • for each t ∈ (0 , T ) fixed, the ω t -psh function ϕ ( t, . ) is locally bounded in Ω • the inequality ( ω t + dd c ϕ t ) n ∧ dt ≥ e ˙ ϕ t + F ( t,.,ϕ t ) f dV ∧ dt holds in the sense of measures in (0 , T ) × Ω . Definition 1.14. A parabolic potential ϕ ∈ P ( X T , ω ) is a supersolution to (CMAF) on X T if • for each t ∈ (0 , T ) fixed, the ω t -psh function ϕ ( t, . ) is locally bounded in Ω , • the inequality ( ω t + dd c ϕ t ) n ∧ dt ≤ e ˙ ϕ t + F ( t,.,ϕ t ) f dV ∧ dt holds in the sense of measures in (0 , T ) × Ω . Remark 1.15. In these definitions the left-hand side is well-defined by using Bedford-Taylor’stheory (see Definition 1.8). Lemma 1.16. Let ϕ ∈ P ( X T , ω ) be a parabolic potential such that the restriction of ϕ to { t } × Ω is an ω t -psh function which is locally bounded on Ω . Then1) ϕ is a pluripotential subsolution to (CMAF) if and only if for a.e. t ∈ (0 , T ) , ( ω t + dd c ϕ t ) n ≥ e ∂ τ ϕ ( t,. )+ F ( t,.,.ϕ t ) f dV, (1.6) in the sense of measures in Ω .2) ϕ is a pluripotential supersolution to (CMAF) if and only if for a.e. t ∈ (0 , T ) , ( ω t + dd c ϕ t ) n ≤ e ∂ τ ϕ ( t,. )+ F ( t,.,.ϕ t ) f dV, (1.7) in the sense of measures in Ω .Proof. We shall prove the result for subsolutions. The proof for supersolutions is similar.We first assume that (1.6) holds for almost every t . Let η be a positive continuous testfunction in (0 , T ) × Ω. We thus obtain Z Ω η ( t, · )( ω t + dd c ϕ t ) n ≥ Z Ω η ( t, · ) e ∂ t ϕ t + F ( t, · ,ϕ t ) f dV. Integrating with respect to t , we get Z T Z Ω η ( ω t + dd c ϕ t ) n ∧ dt ≥ Z T Z Ω ηe ∂ t ϕ t + F ( t, · ,ϕ t ) f dV dt, hence ϕ is a pluripotential subsolution to (CMAF).Conversely, if ϕ is a pluripotential subsolution to (CMAF), we consider positive test functionsthat can be decomposed as η ( t, x ) = λ ( t ) ξ j ( x ) , where ( ξ j ) is a sequence of positive test functions on Ω which generates a dense subspace of thespace C c (Ω) in C -topology. It follows from Fubini’s theorem that Z T (cid:18)Z Ω ξ j ( x )( ω t + dd c ϕ t ) n (cid:19) λ ( t ) dt ≥ Z T (cid:18)Z Ω ξ j ( x ) e ∂ t ϕ t + F ( t,.,ϕ t ) f dV (cid:19) λ ( t ) dt. Hence for any j there exists a subset E j which has full measure in (0 , T ) so that for all t ∈ (0 , T ) Z Ω ξ j ( x )( ω t + dd c ϕ t ) n ≥ Z Ω ξ j ( x ) e ∂ t ϕ t + F ( t,.,ϕ t ) f ( x ) dV ( x ) . (1.8)If we set E := ∩ j E j , then E has full measure in (0 , T ). Moreover, the inequality (1.8) holdsfor all t ∈ E and for all j . Let ξ be an arbitrary positive continuous function in Ω. We canapproximate this function by convex combinations of the ξ j , we infer that for all t ∈ E , Z Ω ξ ( x )( ω t + dd c ϕ t ) n ≥ Z Ω ξ ( x ) e ∂ t ϕ t + F ( t,.,ϕ t ) f ( x ) dV ( x ) , from which (1.6) follows. (cid:3) LURIPOTENTIAL MONGE-AMP`ERE FLOWS IN BIG COHOMOLOGY CLASSES 11 Lemma 1.17. Let ϕ, ψ ∈ P ( X T , ω ) be two pluripotential subsolutions to (CMAF) . Then { ϕ ≥ ψ } ∂ t max( ϕ, ψ ) = 1 { ϕ ≥ ψ } ∂ t ϕ, almost everywhere in Ω T and ( ω + dd c max( ϕ, ψ )) n ∧ dt ≥ { ϕ ≥ ψ } ( ω + dd c ϕ ) n ∧ dt in the sense of measures in Ω T .Proof. Fix K ⋐ Ω and J ⋐ (0 , T ). Then there exists a constant C = C K,J > x ∈ K fixed, Ct − ϕ ( t, x ) and Ct − ψ ( t, x ) are increasing in t ∈ J . They thus havederivatives in t almost everywhere on J (see e.g. [KK, Theorem 1.2.8]). Hence the first equalityfollows from [GT, Lemma 7.6].The second inequality is a simple consequence of the elliptic maximum principle in the localcontext (see e.g. [GZ, Theorem 3.27]). (cid:3) Lemma 1.18. For every λ ∈ [(1 + δ g (0)) − , , the function λg (0)( ρ + χ ) / is ω -psh. Inparticular, there exists a uniform constant C > such that λg (0) ρ + χ − C ≤ ϕ . Recall here that χ is a fixed θ -psh function with analytic singularities such that θ + dd c χ ≥ δ Θ, for some δ > Proof. By hypothesis (0.2), we first observe that ω + dd c λg (0) ρ + χ λ ω + g (0) dd c ρ ) + λ ω + g (0) dd c χ ) + (1 − λ ) ω ≥ λ g (0)( θ + dd c ρ ) + λ g (0)( θ + dd c χ ) + (1 − λ ) ω ≥ λg (0) δ Θ − (1 − λ )Θ ≥ λ . Thus the function λg (0)( ρ + χ ) / ω -psh. Since ϕ is ω -psh with minimal singularities, there exists a constant C > λg (0)( ρ ( x ) + χ ( x )) / − C ≤ ϕ ( x ) for all x ∈ X . (cid:3) The envelope of subsolutions Definition.Definition 2.1. A Cauchy datum for (CMAF) is a ω -psh function ϕ : X → R with minimalsingularities. We say ϕ ∈ P ( X T , ω ) is a subsolution to the Cauchy problem: ( ω t + dd c u t ) n = e ∂ t u t + F ( t,.,u t ) f dV, u (cid:12)(cid:12) { }× X = ϕ if ϕ is a pluripotential subsolution to (CMAF) such that lim sup t → ϕ ( t, x ) ≤ ϕ ( x ) .We let S ϕ ,f,F ( X T ) denote the set of subsolutions to the Cauchy problem above. Lemma 2.2. The set S ϕ ,f,F ( X T ) is non-empty, uniformly bounded from above on X T , andstable under finite maxima.Proof. Fix 1 ≥ λ ≥ (1 + g (0) δ ) − . Consider, for any ( t, x ) ∈ X T ,(2.1) u ( t, x ) := λg ( t ) ρ ( x ) + χ ( x )2 − C ( t + 1) , where ρ and χ are defined in (1.2) and (1.1), the uniform constant C > ω t we have λ ω t + g ( t ) dd c χ ) + (1 − λ ) ω t ≥ λ g ( t )( θ + dd c χ ) + (1 − λ ) ω t ≥ λg ( t ) δ Θ − (1 − λ )Θ= [ λ (1 + δ g ( t )) − ≥ . since g ( t ) is increasing, so λ ≥ (1 + g ( t ) δ ) − for all t . Therefore,(2.2) ( ω t + dd c u t ) n = (cid:18) λ ω t + g ( t ) dd c ρ ) + λ ω t + g ( t ) dd c χ ) + (1 − λ ) ω t (cid:19) n ≥ (cid:18) λ g ( t )( θ + dd c ρ ) (cid:19) n = ( λg ( t )) n e c f dV. We set C = C + M F + | n log( g ( T )) | + | c | , (2.3)it thus follows from (2.2) thatexp( ∂ t u t + F ( t, ., u t ( . ))) f dV = exp( λg ′ ( t )( ρ + χ ) / − C + F ( t, ., u t ( . ))) f dV ≤ exp( n log( λg ( t )) + c ) f dV ≤ ( ω t + dd c u t ) n using in the first inequality that g is increasing in t ∈ [0 , T ], sup X ρ = sup X χ = 0. It followsmoreover from the choice of C and Lemma 1.18 that u (0 , . ) ≤ ϕ on X , hence u ∈ S ϕ ,f,F ( X T ).Let now ϕ ∈ S ϕ ,f,F ( X T ) such that ϕ ≥ u . Consider the set G := { x ∈ X : u ( T, x ) > − M } , where M > µ ( G ) > µ ( X )2 , with µ := f dV . We observe that for every t ∈ (0 , T ), ϕ t ( x ) ≥ u ( t, x ) ≥ u ( T, x ) > − M, ∀ x ∈ G. Set − m F = inf [0 ,T ) × X F ( t, x, − M ) > −∞ . Since F ( ., ., r ) is non-decreasing in r we obtain Z G e ˙ ϕ t − m F dµ ≤ Z G e ˙ ϕ t + F ( t,.,ϕ t ) dµ ≤ Z G ( ω t + dd c ϕ t ) n ≤ Z G (Θ + dd c ϕ t ) n ≤ Z X Θ n . On the other hand, it follows from Jensen’s inequality thatexp (cid:18)Z G ˙ ϕ t dµµ ( G ) (cid:19) ≤ Z G e ˙ ϕ t dµµ ( G ) . Combining these two estimates we get Z G ˙ ϕ t f dV ≤ µ ( G ) log (cid:18) e m F R X Θ n µ ( G ) (cid:19) ≤ µ ( X ) log (cid:18) e m F R X Θ n µ ( X ) (cid:19) =: C. We then infer that the function t R G ϕ t dµ − Ct is non-increasing in (0 , T ), hence Z G ϕ t dµ ≤ Z G ϕ dµ + Ct ≤ Z G ϕ dµ + CT. (2.4)On the other hand, it follows from [GZ, Proposition 8.5] that, for some uniform constant C (depending on µ ), Z X ( ψ − sup X ψ ) dµ ≥ − C ′ , for all ψ ∈ PSH( X, Θ) . LURIPOTENTIAL MONGE-AMP`ERE FLOWS IN BIG COHOMOLOGY CLASSES 13 Thus for each t ∈ (0 , T ), − C ′ ≤ Z X ( ϕ t − sup X ϕ t ) dµ ≤ Z G ( ϕ t − sup X ϕ t ) dµ ≤ C ′′ − µ ( G ) sup X ϕ t . It follows that sup X ϕ t is uniformly bounded from above.The stability under finite maxima follows immediately from Lemma 1.17. (cid:3) From now on, we let M > ϕ to (CMAF) such that ϕ ≥ u on X T , and set M F := sup X T F ( ., ., M ) . (2.5)Lemma 2.2 allows us to define the upper envelope of subsolutions: Definition 2.3. We let U = U ϕ ,f,F,X T := sup { ϕ ∈ S ϕ ,f,F ( X T ) : u ≤ ϕ ≤ M } denote the upper envelope of subsolutions. Lemma 2.4. There exists ϕ ∈ S ϕ ,f,F ( X T ) such that for any x ∈ X , lim t → ϕ ( t, x ) = ϕ ( x ) . Proof. We set δ = δ g (0). For any ( t, x ) ∈ [0 , δ ) × X , consider ϕ ( t, x ) = (1 − α t ) ϕ ( x ) + α t g (0) ρ ( x ) + χ ( x )2 + nt (log( δ − t ) − − Ct, where α t = ( δ g (0)) − t , the functions ρ, χ are defined in (1.2), (1.1), and C := M F + δ − sup X (cid:18) g (0) ρ + χ − ϕ (cid:19) − min( c , . Lemma 1.18 with λ = 1 ensures that C < + ∞ . We compute ω t + dd c ϕ t = (1 − α t )( ω + dd c ϕ ) + α t ω + g (0) dd c ρ )+ α t ω + g (0) dd c χ ) + ω t − ω . Since ω t + t Θ is increasing, we have ω t − ω ≥ − t Θ, which gives α t ω + g (0) dd c χ ) + ω t − ω ≥ α t g (0)2 ( θ + dd c χ ) + ω t − ω ≥ α t g (0) δ Θ − t Θ = 0 . Computing the time derivative we obtain ∂ τ ϕ ( t, x ) = ( δ g (0)) − (cid:18) g (0) ρ ( x ) + χ ( x )2 − ϕ ( x ) (cid:19) + n log( δ − t ) − C. Hence, for all t ∈ (0 , δ ) we have( ω t + dd c ϕ t ) n ≥ (cid:18) α t g (0)2 ( θ + dd c ρ ) (cid:19) n = ( δ − ( t − t )) n e c f dV ≥ e ∂ t ϕ t + F ( t,.,ϕ t ) f dV, by the choice of the constant C . It thus follows that( ω t + dd c ϕ ( t, . )) n ≥ e ∂ t ϕ ( t,. )+ F ( t,.,ϕ ( t,. )) f dV. We divide [0 , T ] into N small intervals of the same length [ T k , T k +1 ], k = 0 , ..., N − | T k +1 − T k | ≤ δ := δ g (0), T = 0 and T N = T . For t ∈ [ T k , T k +1 ] we define ϕ ( k ) ( t, . ) : = (1 − α ( k ) t ) ϕ T k + α ( k ) t g ( T k ) ρ + χ − C ( k ) ( t − T k )+ n ( t − T k )(log( δ − ( t − T k )) − , where α ( k ) t = δ − ( t − T k ), and C ( k ) = M F + δ − sup X (cid:18) g ( T k ) ρ + χ − ϕ T k (cid:19) − min( c , . The subsolution constructed in the proof of Lemma 2.2 (see (2.1)) ensures that C ( k ) < + ∞ isa uniform positive constant. The same arguments as above ensure that ϕ ( k ) is a subsolution to(CMAF) in [ T k , T k +1 ] × X . Gluing these functions, we get our desired pluripotential subsolutiondefined on [0 , T ) × X . It is also clear from the definition that ϕ ( t, . ) converges to ϕ in L ( X, dV )as t → + . (cid:3) Lipschitz regularity in time. In this section, we study the regularity in time t of thePerron upper envelope by adapting some arguments in [GLZ18, Section 4]. Proposition 2.5. For all < S < T we have U ϕ ,f,F,X S = U ϕ ,f,F,X T in X S .Proof. For notational convenience we set U T = U ϕ ,f,F,X T and U S = U ϕ ,f,F,X S . We can assumethat | T − S | ≤ δ g (0)2 , since if we can show that U T = U S for such S we can restart the processto prove that U S = U S ′ for S − δ g (0)2 < S ′ < S .It suffices to show that U S ≤ U T because the reverse inequality is clear. Fix ϕ ∈ S ϕ ,f,F ( X S ).Fix 0 < t < S such that T − t < δ g (0). Set, for t ∈ ( t , T ), ψ ( t, . ) = (1 − α t ) ϕ t + α t g ( t ) ρ + χ − C ( t − t ) + n ( t − t )(log[ δ − ( t − t )] − , where α t = ( δ g ( t )) − ( t − t ) < 1, the functions ρ, χ are defined in (1.2), (1.1), and C := M F + ( δ g ( t )) − sup X (cid:18) g ( t ) ρ + χ − ϕ t (cid:19) + | c | . From (2.1), with λ = 1 and t = t , we see that C < + ∞ is a uniform constant. From (2.1)again we see that ∂ t ψ ( t, x ) satisfies (1.5). We compute ω t + dd c ψ t = (1 − α t )( ω t + dd c ϕ t ) + α t ω t + g ( t ) dd c ρ )+ α t ω t + g ( t ) dd c χ ) + ω t − ω t . Since ω t + t Θ is increasing, we have ω t − ω t ≥ − ( t − t )Θ for all t ≥ t . It thus follows that α t ω t + g ( t ) dd c χ ) + ω t − ω t ≥ α t g ( t )2 ( θ + dd c χ ) + ω t − ω t ≥ α t g ( t ) δ Θ − ( t − t )Θ = 0 . Hence, for all t ∈ [ t , T ),( ω t + dd c ψ t ) n ≥ (cid:18) α t g ( t )2 ( θ + dd c ρ ) (cid:19) n = ( δ − ( t − t )) n e c f dV ≥ e ∂ t ψ t + F ( t,.,ψ t ) f dV, LURIPOTENTIAL MONGE-AMP`ERE FLOWS IN BIG COHOMOLOGY CLASSES 15 by the choice of the constant C . Therefore the function( t, x ) u ( t, x ) := ( ϕ ( t, x ) , if t ∈ [0 , t ] ψ ( t, x ) , if t ∈ [ t , T )is a pluripotential subsolution to (CMAF) in [0 , T ) × X by using Lemma 1.16. We thus have u ∈ S ϕ ,f,F ( X T ) since u (0 , . ) = ϕ . This yields u ≤ U T in [0 , T ) × X . In particular ϕ ≤ U T in[0 , t ] × X , and it follows that U S ≤ U T on [0 , t ] × X by taking supremum over all subsolutions.We now let t → S to get U s ≤ U T in X S . (cid:3) Next we introduce the mixed type inequality: Lemma 2.6. Let θ , θ be two closed smooth (1 , -forms on X such that their cohomologyclasses are big. Let ϕ ( ϕ resp.) be a bounded θ -psh ( θ -psh resp.) function such that ( θ + dd c ϕ ) n ≥ e f µ and ( θ + dd c ϕ ) n ≥ e f µ where f , f are bounded measurable functions and µ is a positive Radon measure with L densitywith respect to Lebesgue measure. Then, for any λ ∈ (0 , , ( λ ( θ + dd c ϕ ) + (1 − λ )( θ + dd c ϕ )) n ≥ e λf +(1 − λ ) f µ. Proof. The proof is the same as that of [GLZ18, Lemma 2.10] using the convexity of theexponential together with the mixed Monge-Amp`ere inequalities; see e.g. [Din09]. (cid:3) Theorem 2.7. There exists a uniform constant L U > such that for all ( t, x ) ∈ X T , t | ∂ t U ( t, x ) | ≤ L U − L U ( ρ ( x ) + χ ( x )) . (2.6) Proof. Let ϕ ∈ S ϕ ,f,F ( X T ) such that ϕ ≥ u on X T , where u is defined in (2.1). Fix 0 < T ′ < T and ε > ε ) T ′ < T . Set, for all ( t, x ) ∈ X T ′ , s ∈ (1 − ε , ε ), u s ( t, x ) := α s s ϕ ( st, x ) + (1 − α s ) g ( t ) ρ ( x ) + χ ( x )2 − C | s − | ( t + 1) , where ρ, χ are defined in (1.2), (1.1), α s = 1 − A | s − | , and C = C ( A + 2) + κ F T + AM F + ( A + 2) C ( T + 1) + ( A + 2) M . The constant C is defined in (2.3), and the constant A will be chosen later that depends onlyon T . We will show that u s ∈ S ϕ ,f,F ( X T ). We compute ω t + dd c u s ( t, . ) = α s s ( ω st + dd c ϕ st )+ α s ω t − α s s ω st + (1 − α s ) (cid:18) ω t + g ( t ) dd c ρ + χ (cid:19) . Since ϕ is a subsolution to (CMAF), we have for almost every t ∈ (0 , T ′ ),( s − ( ω st + dd c ϕ st )) n ≥ e − n log s + ∂ τ ϕ ( st,. )+ F ( t,.,ϕ ( st,. )) f dV. Recalling the definition of ρ and ω t ≥ g ( t ) θ we also have (cid:18) 12 ( ω t + g ( t ) dd c ρ ) (cid:19) n ≥ (cid:18) g ( t )2 ( θ + dd c ρ ) (cid:19) n = e n log g ( t )+ c f dV. (2.7) On the other hand, since ˙ ω t ≥ − Θ, we have α s ω t − α s s ω st = α s s ( ω t − ω st ) + α s (1 − s − ) ω t ≥ − α s t (cid:12)(cid:12) s − − (cid:12)(cid:12) Θ − α s (cid:12)(cid:12) s − − (cid:12)(cid:12) Θ ≥ − ( t + 1) s − | s − | Θ ≥ − (2 T + 2) | s − | Θ , where the last line follows from s ≥ / 2. Recall that θ + dd c χ ≥ δ Θ for some δ > 0. If wechoose A ≥ T + 1)( δ g (0)) − , then α s ω t − α s s ω st + (1 − α s ) ω t + g ( t ) dd c χ ≥ − (2 T + 2) | s − | Θ + A | s − | g ( t ) δ Θ ≥ , since g is increasing in t . Combining these estimates with the mixed Monge-Amp`ere inequality(Lemma 2.6) we obtain( ω t + dd c u s ( t, . )) n ≥ ( α s ( s − ω st + dd c ϕ st ) + (1 − α s )(2 − ( ω t + dd c ρ ))) n ≥ exp( α s ∂ τ ϕ ( st, . ) + α s F ( st, ., ϕ ( st, . )) − α s n log s + (1 − α s )( n log g ( t ) + c )) f dV ≥ exp( ∂ τ u s ( t, . ) + F ( t, ., u s ( t, . )) f dV. where the last line follows from the choice of C as we now explain. Observe that α s ∂ τ ϕ ( st, . ) = ∂ τ u s ( t, . ) + C | s − | − (1 − α s ) g ′ ( t ) ρ + χ ≥ ∂ τ u s ( t, . ) + C | s − | . (2.8)Since g is non-decreasing, we also have g ( ts ) − g ( t ) ≤ κ g t | s − | ≤ κ g T ε g (0) − g ( t ) . It thus follows that g ( ts ) ≤ γg ( t ) where γ = 1 + ε κ g T g (0) − . Choosing ε small enough atthe beginning we can ensure that γ (1 + δ g (0)) − < A so that AA +2 ≥ γ (1 + δ g (0)) − , hence (cid:16) − α s s (cid:17) ϕ st = (cid:16) − α s s (cid:17) ( ϕ st − M ) + (cid:16) − α s s (cid:17) M ≥ ( A + 2) | s − | ( ϕ ts − M ) ≥ ( A + 2) | s − | (cid:18) A ( A + 2) γ g ( ts ) ρ + χ − C ( T + 1) − M (cid:19) ≥ A | s − | g ( t ) ρ + χ − (( A + 2) C ( T + 1) + ( A + 2) M ) | s − | , where the first inequality follows from the elementary one 1 − α s s ≤ ( A + 2) | s − | , while thesecond inequality follows the assumption ϕ ≥ u . This yields ϕ st ≥ α s s ϕ st + (1 − α s ) g ( t ) ρ + χ − C | s − | ≥ u s ( t, . ) . (2.9)Using (2.9) and the assumption that F is non-decreasing in r and uniformly Lipschitz in t , weget α s F ( st, ., ϕ ( st, . )) = F ( st, ., ϕ ( st, . )) − (1 − α s ) F ( st, ., ϕ ( st, . )) ≥ F ( t, ., ϕ ts ( . )) − κ F t | s − | − | s − | AM F ≥ F ( t, ., u s ( t, . )) − ( κ F T + AM F ) | s − | . (2.10) LURIPOTENTIAL MONGE-AMP`ERE FLOWS IN BIG COHOMOLOGY CLASSES 17 Combining (2.8), (2.10), and the definition of C , we obtain α s ∂ τ ϕ ( st, . ) + α s F ( st, ., ϕ ( st, . )) − α s n log s + (1 − α s )( n log g ( t ) + c ) ≥ ∂ τ u s ( t, . ) + C | s − | + F ( t, ., u s ( t, . )) − | s − | ( κ F T + AM F + A ( n log g ( t ) + c ) + 2 n ) ≥ u s ( t, . ) + F ( t, ., u s ( t, . )) . Hence u s is a subsolution to (CMAF) by Lemma 1.16. We now take care of the initial values.For any x ∈ X we have u s (0 , x ) = ϕ ( x ) − C | s − | + (1 − α s ) g (0) ρ ( x ) + χ ( x )2 − (cid:16) − α s s (cid:17) ϕ ( x ) ≤ ϕ ( x ) − C | s − | + A | s − | g (0) ρ ( x ) + χ ( x )2 − ( A + 2) | s − | ϕ ( x ) ≤ ϕ ( x ) − C | s − | + ( A + 2) | s − | (cid:18) AA + 2 g (0) ρ ( x ) + χ ( x )2 − ϕ ( x ) (cid:19) ≤ ϕ ( x )where the last line follows again from the choice of C . Thus, for any x ∈ X we also getlim sup t → u s ( t, x ) ≤ ϕ ( x ). Therefore u s ∈ S ϕ ,f,F ( X T ), so u s ≤ U in X T ′ . We thus obtain α s s ϕ ( st, x ) + (1 − α s ) g ( t ) ρ ( x ) + χ ( x )2 − C | s − | ( t + 1) ≤ U ( t, x ) . We now take the supremum over all subsolutions ϕ ∈ S ϕ ,f,F ( X T ) to get α s s U ( st, x ) + A | s − | g ( t )( ρ ( x ) + χ ( x )) − C | s − | ( t + 1) ≤ U ( t, x ) , ∀ ( t, x ) ∈ X T ′ . Letting s → 1, we infer, for all ( t, x ) ∈ X T ′ that t∂ t U ( t, x ) ≤ C ( T + 1) + AM − Ag ( T )( ρ ( x ) + χ ( x )) . We can now define L U := Ag ( T ) + C ( T + 1) + AM . Finally, letting T ′ → T and applyingProposition 2.5 to complete the proof. (cid:3) Convergence at initial time. We define the upper semicontinuous regularization U ∗ of U by the formula U ∗ ( t, x ) = lim sup X T ∋ ( s,y ) → ( t,x ) U ( s, y ) , ( t, x ) ∈ X T . We then prove that the upper envelope has the right initial values: Theorem 2.8. The upper semi-continuous regularisation of the upper envelope U := U ϕ ,f,F,X T satisfies, for all x ∈ X , lim X T ∋ ( t,y ) → (0 ,x ) U ∗ ( t, y ) = ϕ ( x ) . Proof. Thanks to Lemma 2.4, it suffices to show that for all x ∈ X ,lim sup X T ∋ ( t,y ) → (0 ,x ) U ∗ ( t, y ) ≤ ϕ ( x ) . Theorem 2.7 ensures that the upper envelope U is locally uniformly Lipschitz in (0 , T ). Arguingexactly as in the proof of [GLZ18, Lemma 1.4] we can show that U ∗ ( t, . ) = ( U t ) ∗ for all t ∈ (0 , T ).It remains then to prove that, for all x ∈ X ,lim sup t → U ∗ t ( x ) ≤ ϕ ( x ) . Fix M > G := { u T > − M } , where u is defined as in (2.1). We claim that there existsa positive constant C (depending also on M ) such that, for all t ∈ (0 , T ), Z G U ∗ t f dV ≤ Z G ϕ f dV + Ct. (2.11)Fix t ∈ (0 , T ). By Choquet’s lemma, there exists a sequence { ϕ j } in S ϕ ,f,F ( X T ) such that U ∗ t = (cid:18) lim j → + ∞ ϕ jt (cid:19) ∗ in X. Since the set S ϕ ,f,F ( X T ) is stable under finite maximum, we can moreover assume that thesequence { ϕ j } is increasing with ϕ j ≤ M on X . It follows from (2.4) that Z G ϕ jt f dV ≤ Z G ϕ f dV + Ct, ∀ t ∈ (0 , T ) , for a constant C = C ( M ) > ϕ j ). For t = t , letting j → + ∞ ,we obtain Z G U ∗ t f dV ≤ Z G ϕ f dV + Ct , thanks to a classical theorem of Lelong (see e.g. [GZ, Proposition 1.40]). Note that the sequence { ϕ jt } depends on t , but the constant C does not. Therefore the claim (2.11) follows.Let now u ∈ PSH( X, ω ) be any cluster point of U ∗ t as t → 0. We can assume that U ∗ t converges to u in L q ( X, dV ) for any q > 1. Then U ∗ t f converges to u f in L ( X ). Thus, theclaim above ensures that Z G u f dV ≤ Z G ϕ f dV. We infer that u ≤ ϕ almost everywhere on G with respect to f dV , hence everywhere on G by the assumption on f . Letting M → + ∞ , we can thus conclude that lim sup t → U ∗ t = ϕ onΩ, hence on the whole X . (cid:3) The envelope is a subsolution. We now consider the set of subsolutions which arelocally uniformly Lipschitz. Definition 2.9. Let κ be a fixed positive constant. We let S κϕ ,f,F ( X T ) denote the set of allfunctions ϕ ∈ S ϕ ,f,F ( X T ) such that, for all t ∈ (0 , T ) , x ∈ Ω , t∂ t ϕ ( t, x ) ≤ κ − κ ( ρ ( x ) + χ ( x )) . Set U κ := U κϕ ,f,F,X T := sup { ϕ : ϕ ∈ S κϕ ,f,F ( X T ) } . Proposition 2.10. For all < S < T we have U κϕ ,f,F,X S = U κϕ ,f,F,X T in X S .Proof. The proof is the same as that of Proposition 2.5. (cid:3) Theorem 2.11. We have, for all κ > and t ∈ (0 , T ) , t | ∂ t U κ | ≤ L U − L U ( ρ + χ ) , where L U is the constant defined in Theorem 2.7. LURIPOTENTIAL MONGE-AMP`ERE FLOWS IN BIG COHOMOLOGY CLASSES 19 Proof. The proof is the same as that of Theorem 2.7. In fact, if ϕ ∈ S κϕ ,f,F ( X T ) then thefunction u s in the proof of Theorem 2.7 satisfies t∂ t u s ≤ α s ( κ − κ ( ρ + χ )) ≤ κ − κ ( ρ + χ )because 0 < α s ≤ 1. It follows that u s ∈ S κϕ ,f,F ( X T ) and we argue as in the proof of Theorem 2.7to conclude. (cid:3) Theorem 2.12. The upper envelope U is a pluripotential subsolution to (CMAF) in X T .Proof. We will first show that U κ = ( U κ ) ∗ is a subsolution to (CMAF). Indeed, Choquet’slemma implies that there exists a sequence { ϕ j } in S κϕ ,f,F ( X T ) such that( U κ ) ∗ = (cid:18) sup j ∈ N ϕ j (cid:19) ∗ in X T . Since S κ is stable under finite maximum, we can assume that the sequence { ϕ j } is non-decreasing. We now claim that dt ∧ ( ω t + dd c ϕ jt ) n → dt ∧ ( ω t + dd c ( U κ ) ∗ t ) n in the sense of measures in (0 , T ) × Ω.Let K be a relatively compact open subset of Ω and J be a compact interval of (0 , T ). Thenthere exists a constant C = C ( J, K ) > j ∈ N , ϕ j ( t, x ) − Ct is decreasing in t ∈ J , for any x ∈ K . Moreover, the sequence of functions ϕ j increases towards u , so for any x ∈ K , u ( t, x ) − Ct is decreasing in t . Thus for each x ∈ K , there exists a countable subset E x ⊂ J such that u ( ., x ) is continuous on J \ E x . Now set E := { ( t, x ) ∈ J × K : t ∈ E x } . Note that E has zero (2 n + 1)-dimensional Lebesgue measure by using the Fubini theorem. Let N be the set of t ∈ J such that E t = { x ∈ K : ( t, x ) ∈ E } has positive Lebesgue measure. Wemust have that N has zero Lebesgue measure. Thus for any t ∈ J ′ := J \ N , the set E t has zeroLebesgue measure, and lim s → t u ( s, x ) = u ( t, x ) for all x ∈ K \ E t . Fixing ( t, x ) ∈ J ′ × K , wewant to show that lim sup ( s,y ) → ( t,x ) u ( s, y ) ≤ ( U κt ) ∗ ( x ) , (2.12)where the upper semicontinuous regularization in the RHS is in the x -variable only. Since theproblem is local we may assume that the functions u s are psh and negative in a neighborhood B ( x, r ) ⊂ K . Fix δ ∈ (0 , r ). For y so close to x that B ( x, r ) ⊂ B ( y, r + δ ) we have ϕ j ( s, y ) ≤ B ( y, r + δ )) Z B ( y,r + δ ) ϕ j ( s, z ) dV ( z ) ≤ B ( y, r + δ )) Z B ( y,r + δ ) u ( s, z ) dV ( z ) . Letting j → + ∞ we get u ( s, y ) ≤ B ( y, r + δ )) Z B ( y,r + δ ) u ( s, z ) dV ( z ) ≤ Vol( B ( x, r ))Vol( B ( y, r + δ )) 1Vol( B ( x, r )) Z B ( x,r ) u ( s, z ) dV ( z ) . Since lim s → t u ( s, z ) = u ( t, z ) for almost every z ∈ B ( x, r ) ⊂ K , Fatou’s lemma yieldslim sup ( s,y ) → ( t,x ) u ( s, y ) ≤ Vol( B ( x, r ))Vol( B ( x, r + δ )) 1Vol( B ( x, r )) Z B ( x,r ) u ( t, z ) dV ( z ) . Now, we first let δ → r → U κ . The reverse inequality is clear, hence we get the equality. Therefore, for each t ∈ J ′ wehave that ϕ jt increase almost everywhere towards ( U κt ) ∗ = ( U κ ) ∗ t on K , so the Bedford-Taylorconvergence theorem yields ( ω t + dd c ϕ jt ) n → ( ω t + dd c ( U κ ) ∗ t ) n in the weak sense of measures in K . Thus the claim follows directly from the Fubini theorem.On the other hand, for each x ∈ K fixed, the sequence { t ∂ t ϕ j ( t, x ) + F ( t, x, ϕ j ( t, x )) } converges to ∂ t ( U κ ) ∗ ( t, x ) + F ( t, x, ( U κ ) ∗ ( t, x )) in the sense of distributions in J , with the laterbeing bounded in J × K . Applying Proposition 1.11 we obtainlim j → + ∞ e ∂ t ϕ j + F ( t,.,ϕ j ) f dt ∧ dV ≥ e ∂ t ( U κ ) ∗ + F ( t,., ( U κ ) ∗ ) f dt ∧ dV in the weak sense of measures in J × K . It thus follows that ( U κ ) ∗ is a subsolution to (CMAF)in Ω T , and hence ( U κ ) ∗ ∈ S κϕ ,f,F ( X T ). We thus deduce that U κ = ( U κ ) ∗ on X T .We have shown that, for some κ > U κ = U κ for all κ > κ (by Theorem 2.11). It thusremains to prove that U = U κ in X T . We first assume that ϕ = P ω h := sup { ψ ∈ PSH( X, ω ) : ψ ≤ h } for some continuous function h . Fix 0 < S < T , s > ϕ ∈ S ϕ ,f,F ( X T ).For ( t, x ) ∈ [0 , S ] × X , we define u s ( t, x ) := α s ϕ ( t + s, x ) + (1 − α s ) g ( t + s ) ρ ( x ) + χ ( x )2 − Cs ( t + 1) − η ( s ) , where α s = 1 − ( δ g (0)) − s , η ( s ) := sup X ( α s ϕ s − h ) and C = ( δ g (0)) − C ( T + 1) + 2 δ − M F + n | log( g ( T )) | + | c | , with C > ω t + dd c u s ( t, . ) = α s ( ω t + s + dd c ϕ t + s ) + 1 − α s ω t + s + g ( t + s ) dd c ρ )+ 1 − α s ω t + s + g ( t + s ) dd c χ ) + ω t − ω t + s . It follows from the assumption (0.2) that ω t − ω t + s ≥ − s Θ . Since θ + dd c χ ≥ δ Θ we thus obtain1 − α s ω t + s + dd c χ ) + ω t − ω t + s ≥ ( δ g (0)) − sg ( t + s )( θ + dd c χ ) − s Θ ≥ . We thus get( ω t + dd c u s ( t, . )) n ≥ ( α s ( ω t + s + dd c ϕ t + s ) + (1 − α s ) g ( t + s )( θ + dd c ρ ) / n ≥ e α s ( ∂ t ϕ t + s + F ( t + s,.,ϕ t + s ( . ))+(1 − α s )( n log g ( t + s )+ c ) f dV LURIPOTENTIAL MONGE-AMP`ERE FLOWS IN BIG COHOMOLOGY CLASSES 21 where we apply Lemma 2.6 in the last line. Since ( t, r ) F ( t, ., r ) is uniformly Lipschitz itfollows that α s F ( t + s, ., ϕ t + s ( . )) = F ( t + s, ., ϕ t + s ( . )) − (1 − α s ) F ( t + s, ., ϕ t + s ( . )) ≥ F ( t, ., ϕ t + s ) − κ F s − ( δ g (0)) − sM F . Since ϕ ≥ u on X T we have( δ g (0)) − sϕ t + s ≥ ( δ g (0)) − s (cid:18) g ( t + s ) ρ + χ − C ( T + 1) (cid:19) , which yields (1 − α s ) ϕ t + s ≥ (1 − α s ) g ( t + s ) ρ + χ − ( δ g (0)) − C ( T + 1) s. Consequently, it follows from the choice of C that ϕ t + s ≥ u s ( t, . ) for all t ∈ [0 , S ]. Thus, α s F ( t + s, ., ϕ t + s ( . )) ≥ F ( t, ., u s ( t, . )) − s ( κ F + ( δ g (0)) − M F ) , since the function r F ( ., ., r ) is increasing. Observe now that α s ∂ t ϕ t + s = α s ∂ t u s + Cs − α s g ′ ( t + s ) ρ + χ . It thus follows from the choice of C and the estimates above that( ω t + dd c u st ) n ≥ e ∂ t u st + F ( t,.,u s ( t,. )) f dV, which means that u s is a subsolution to (CMAF). By definition of u s we have u s (0 , . ) ≤ h on X since sup X ρ = sup X χ = 0. Since u s (0 , . ) is ω -psh, we infer u s (0 , . ) ≤ ϕ = P ω h on X . It follows that u s ∈ S ϕ ,f,F ( X S ), and hence u s ∈ S κϕ ,f,F ( X S ) for some κ > u s ≤ U κ = U κ in X S by Proposition 2.10. On the other hand it follows fromHartogs’ Lemma that lim s → η ( s ) ≤ 0. Letting s → ϕ ≤ U κ in X S . Finally, letting S → T to obtain ϕ ≤ U κ , so U ≤ U κ on X T (see Proposition 2.5). Therefore U = U κ is themaximal subsolution to (CMAF) with initial data ϕ .We now remove the extra assumption on ϕ . Let ϕ j = P ω ( h j ) be a decreasing sequence of ω -psh functions in X converging to ϕ , where the h j are continuous on X . We thus obtainthat the upper envelope U j := U ϕ j ,f,F,X T is also a subsolution to (CMAF) by the previousarguments. We also provide a uniform Lipschitz constant for U j . Since ϕ j decreases to ϕ , weobtain U ≤ U j decreases to some V ∈ P ( X T ) which is a subsolution to (CMAF). On the otherhand we have V (cid:12)(cid:12) { }× X ≤ ϕ . Hence V = U . (cid:3) The envelope is locally uniformly semi-concave in time.Theorem 2.13. There exists a uniform constant C U > such that t ∂ t U ( t, x ) ≤ C U − C U ( ρ ( x ) + χ ( x )) , (2.13) in the sense of distributions in X T .Proof. Fix 0 < T ′ < T and ε > ε ) T ′ < T , s ∈ [1 − ε , ε ].Set, for any ( t, x ) ∈ X T ′ , u s ( t, x ) := α s s − U ( st, x ) + sU ( s − t, x )2 + (1 − α s ) g ( t ) ρ ( x ) + χ ( x )2 − C | s − | ( t + 1) , where α s = 1 − A ( s − with the constant A to be chosen later, and C := ( A + 1) C + κ F T + A ( n | log( g ( T )) | + | c | ) . We are going to prove that u s is a subsolution to (CMAF). We compute ω t + dd c u s ( t, . ) = α s (cid:18) s ( ω st + dd c U st ) + s ( ω s − t + dd c U s − t ) (cid:19) + (1 − α s ) ω t + g ( t ) dd c ρ α s ω t − α s s − ω st + sω s − t ) + (1 − α s ) ω t + g ( t ) dd c χ . Consider, for s ∈ [1 − ε , ε ], h ( s ) := ω t − 12 ( ω st + ω s − t ) . We have h (1) = h ′ (1) = 0, and | h ′′ ( s ) | ≤ ( T + 2) Θ on [1 − ε , ε ]. Hence α s ω t − α s s − ω st + sω s − t ) ≥ − ( T + 2) Θ . Recall that χ is a θ -psh function on X such that θ + dd c χ ≥ δ Θ. If we take A > A ≥ ( T + 2) ( δ g (0)) − then α s ω t − α s s − ω st + sω s − t ) + (1 − α s ) ω t + dd c χ ≥ , hence, ω t + dd c u s ( t, . ) ≥ α s (cid:18) s ( ω st + dd c U st ) + s ( ω s − t + dd c U s − t ) (cid:19) + (1 − α s ) ω t + g ( t ) dd c ρ . It follows from Theorem 2.12 that U is a subsolution to (CMAF). We then have for almostevery t ∈ (0 , T ′ ) ( s − ( ω st + dd c U st )) n ≥ e ∂ τ U st + F ( st,.,U st ) − n log s f dV, and ( s ( ω s − t + dd c U st )) n ≥ e ∂ τ U s − t + F ( s − t,.,U s − t )+ n log s f dV. Combining these together with Lemma 2.6, we obtain( ω t + dd c u s ( t, . )) n ≥ e α s ( a ( s )+ a ( s − ))+(1 − α s )( n log g ( t )+ c ) f dV, where a ( s ) = α s ∂ τ U st + F ( st, ., U st )) . Since F is a convex function in r we get12 F ( st, ., U st ) + 12 F ( s − t, ., U s − t ) ≥ F (cid:18) ( s + s − ) t , ., U ( st, . ) + U ( s − t, . )2 (cid:19) . (2.14)Now we use the same arguments as in Theorem 2.7 to show that for each t ∈ [0 , T ) U ( st, . ) + U ( s − t, . )2 ≥ u s ( t, . )(2.15)It is equivalent to show that12 (cid:2) (1 − s − α s ) U st + (1 − sα s ) U s − t (cid:3) ≥ (1 − α s ) g ( t ) ρ + χ − C ( t + 1)( s − . (2.16)The left-hand side can be rewritten as12 (cid:2) (1 − s − (1 − A ( s − )) U st + (1 − s (1 − A ( s − )) U s − t (cid:3) = 12 (cid:2) ( s − U st − U s − t ) + ( A − s − ( s − U st + As ( s − U s − t (cid:3) . LURIPOTENTIAL MONGE-AMP`ERE FLOWS IN BIG COHOMOLOGY CLASSES 23 By Theorem 2.7, we have( s − U st − U s − t ) ≥ s − ( L U ( ρ + χ ) − L U ) . The same arguments as in the proof of Theorem 2.7 give( A − s − U st ≥ AU st ≥ ( A − L U /g (0)) g ( t ) ρ + χ − C ( T + 1) ,AsU s − t ≥ ( A + 1) U s − t ≥ ( A − L U /g (0)) g ( t ) ρ + χ − C ( T + 1) . where A is large enough so that ( A − L U /g (0)) / ( A + 1) ≥ (1 + δ g (0)) − γ ( γ = 1 ε κ g T g (0) − ).Combining these estimates, it follows from the choice of C that (2.16) holds. Since F is non-decreasing in r and uniformly Lipschitz in t , it follows from (2.14) and (2.15) that12 F ( st, x, U st ) + 12 F ( s − t, ., U s − t ) ≥ F (cid:18) t, x, U ( st, x ) + U ( s − t, x )2 (cid:19) − κ F t (cid:18) s + s − − (cid:19) ≥ F ( t, x, u s ( t, x )) − κ F T ( s − , so α s (cid:0) F ( st, x, U st ) + F ( s − t, ., U s − t ) (cid:1) ≥ F ( t, x, u s ( t, x )) − κ F T ( s − − AM F ( s − . Therefore, a ( s ) + a ( s − ) + (1 − α s )( n log g ( t ) + c ) ≥ ∂ τ u s ( t, . ) + F ( t, ., u s ( t, . )) . On the other hand, the choice of C ensures, for any ( t, x ) ∈ X T ′ that u s (0 , x ) ≤ ϕ ( x ) − C ( s − + (1 − α s ) g (0) ρ ( x ) + χ ( x )2 − (cid:18) − s + s − α s (cid:19) ϕ ( x ) ≤ ϕ ( x ) − C ( s − + A ( s − g (0) ρ ( x ) + χ ( x )2 − ( A + 1)( s − ϕ ( x ) ≤ ϕ ( x ) − C ( s − + ( A + 1)( s − (cid:18) AA + 1 g (0) ρ ( x ) + χ ( x )2 − ϕ ( x ) (cid:19) ≤ ϕ ( x ) − C ( s − + ( A + 1) C ( s − ≤ ϕ ( x ) . Therefore, we conclude that u s ∈ S ϕ ,f,F ( X T ′ ), and we obtain for any ( t, x ) ∈ X T ′ that α s s − U ( st, x ) + sU ( s − t, x )2 − U ( t, x ) + A/ s − ( ρ ( x ) + χ ( x )) ≤ C ( T + 1)( s − , hence s − U ( st, x ) + sU ( s − t, x )2 − U ( t, x ) + A ( s − ( ρ ( x ) + χ ( x )) ≤ ( C ( T + 1) + 2 AM )( s − . From this, we obtain for all ( t, x ) ∈ X T ′ , U ( st, x ) + U ( s − t, x )2 − U ( t, x ) + A ( s − ( ρ ( x ) + χ ( x )) ≤ ( C ( T + 1) + (2 A + 1) M + 2 L U − L U ( ρ ( x ) + χ ( x )))( s − . Letting s → t, x ) ∈ X T ′ t ∂ t U ( t, x ) ≤ ( C ( T + 1) + (2 A + 1) M + 5 L U ) − ( A + L U )( ρ ( x ) + χ ( x )) . We finally let T ′ → T and apply Proposition 2.5 to complete the proof. (cid:3) Existence and uniqueness Existence of solutions. We shall prove in this section that U ϕ ,f,F,X T is the uniquepluripotential solution to the Cauchy problem (see Definition 2.1). Theorem 3.1. The upper envelope U := U ϕ ,f,F,X T is a pluripotential solution to the Cauchyproblem for the parabolic complex Monge-Amp`ere equation (CMAF) in X T . Moreover, U islocally uniformly semi-concave in (0 , T ) × Ω .Proof. We have shown in Theorem 2.13 that U is locally uniformly semi-concave in t ∈ (0 , T ),and U ∈ S ϕ ,f,F ( X T ) and it satisfies the initial condition. It remains to show that U solves theparabolic equation (CMAF). We apply a local balayage process to modify the function U on agiven ’small ball’ B ⋐ Ω by constructing a new ω t -psh function U B so that it satisfies the localMonge-Amp`ere flow ( ω t + dd c U B ) n = e ∂ t U B ( t,. )+ F ( t,.,U B ( t,. )) f dV on B T = (0 , T ) × B and U B ≥ U on X and U B = U on X \ B .Indeed, we choose complex coordinates z = ( z , ..., z n ) identifying B with the complex unitball B ⊂ C n . We can write ω t = dd c g t in a local holomorphic coordinate chart B ⊂ X , for somesmooth potential g t . Set ˜ f = f ◦ z − ∈ L p ( B ) and d ˜ V is the restriction of the volume dV to B .We consider the following complex Monge-Amp`ere flow dt ∧ ( dd c u t ) n = e ∂ t u ( t,. )+ ˜ F ( t,.,u ( t,. )) ˜ f d ˜ V ∧ dt (3.1)in B T := (0 , T ) × B with the Cauchy-Dirichlet boundary data h being the restriction of U defined on the parabolic boundary of B T denoted by ∂ P B T := ([0 , T ) × ∂ B ) ∪ ( { } × B ). Here˜ F ( t, x, r ) = F ( t, x, r − g t ( x )) − ∂ t g t satisfies the same assumptions as F . We have shown that h is locally uniformly Lipschitz (Theorem 2.7) and locally uniformly semi-concave (Theorem2.13) i.e. for all 0 < T ′ < T , and for all ( t, z ) ∈ (0 , T ′ ) × ∂ B , there exist constants L = L ( T ′ ),and C = C ( T ′ ) such that t | ∂ t h ( t, z ) | ≤ L, t | ∂ t h ( t, z ) | ≤ C. (3.2)Using mollifiers we can find a sequence h j of continuous function on [0 , T ) × ∂ B such that h j decreases pointwise to h . The function h j is the Cauchy-Dirichlet boundary data satisfying thesame assumption (3.2) as h .Then it follows from [GLZ18, Theorem 6.4] that there exists a sequence of functions u j solvingthe Monge-Amp`ere flows (3.1) with boundary data h j . Moreover, u j is locally uniformly semi-concave in t ∈ [0 , T ). Since h j decreases to U ◦ z − on ∂ P B T , so U ◦ z − ≤ u j and the sequence u j decreases to some function v . The function v solves (3.1) by using Proposition 1.12, andlim sup t → v ( t, z ) ≤ U ◦ z − . But the comparison principle (see [GLZ18, Theorem 6.5]) ensuresthat U ◦ z − ≤ v in B T . Hence lim t → v ( t, z ) = U ◦ z − . We then set U B ( t, x ) = ( v ( t, z ) in B T U ( t, x ) in X T \ B T The function U B is upper semi-continuous and U B ≥ U on B T . Moreover, U B is a pluripotentialsubsolution to (CMAF), so U B = U in B T , hence in Ω T .Moreover, by Theorem 2.7 and Theorem 2.13, U is locally uniformly uniformly Lipschitz andsemi-concave in t . (cid:3) The comparison principle. We first establish a version of the comparison principlewhich requires relatively strong regularity assumptions: Proposition 3.2. Let ϕ (resp. ψ ) be a subsolution (resp. supersolution) to (CMAF) withinitial value ϕ (resp. ψ ). We assume that LURIPOTENTIAL MONGE-AMP`ERE FLOWS IN BIG COHOMOLOGY CLASSES 25 a) ϕ is C in t and continuous on (0 , T ) × Ω , b) ψ is locally uniformly semi-concave in t c) ϕ t → ϕ and ψ t → ψ in L ( X ) , as t → , d) for any t ∈ [0 , T ) , ψ t has minimal singularities, e) the function ( t, x ) ψ ( t, x ) is continuous on [0 , T ) × Ω .Then ϕ ≤ ψ ⇒ ϕ ≤ ψ in X T . Proof. Fix 0 < T ′ < T , in particular T ′ < + ∞ . We shall prove that ϕ ≤ ψ on [0 , T ′ ] × X .The result thus follows by letting T ′ → T . We fix λ, ε > t, x ) ∈ [0 , T ′ ] × X , ϕ λ ( t, x ) := (1 − λ ) ϕ ( t, x ) + λg ( t ) ρ ( x ) + χ ( x )2 , where ρ, χ are θ -psh functions defined in (1.2), (1.1). One can moreover impose χ < { θ } , with analytic singularities, and such that χ ( x ) → −∞ as x → ∂ Ω. We will show that ϕ λ ≤ ψ and we then let λ → w ( t, x ) := ϕ λ ( t, x ) + λg (0) δ χ ( x ) − ψ ( t, x ) − εt. Observe that by Lemma 1.5, this function is upper semi-continuous on [0 , T ′ ] × Ω. By theassumption d) we have ϕ λ ( t. ) ≤ ψ ( t, . ) + O (1) for each t . Since χ is continuous in Ω and tendsto −∞ on ∂ Ω, we have that w tends to −∞ on ∂ Ω. Hence w attains its maximum at somepoint ( t , x ) ∈ [0 , T ′ ] × Ω.We want to show that w ( t , x ) ≤ 0. Assume by contradiction that it is not the case i.e w ( t , x ) > 0, with t > 0. The set K := { x ∈ Ω : w ( t , x ) = w ( t , x ) } is a compact subset of Ω ′ since w ( t , x ) tends to −∞ as x → ∂ Ω ′ . The classical maximumprinciple ensures for all x ∈ K that(1 − λ ) ∂ t ϕ ( t , x ) ≥ ∂ − t ψ ( t , x ) + 3 ε, since g ′ ( t ) ≥ t . The partial derivative ∂ t ϕ ( t, x ) is continuous in Ω by assumption. Sincethe function t ψ ( t, x ) is locally uniformly semi-concave, for any t ∈ (0 , T ), the left derivative ∂ − t ψ ( t, · ) is upper semi-continuous in Ω (see Proposition 1.10). We can thus find η > D := { x ∈ Ω : w ( t , x ) > w ( t , x ) − η } is an open set containing K . We have for all x ∈ D (1 − λ ) ∂ t ϕ ( t , x ) > ∂ − t ψ ( t , x ) + 2 ε. (3.3)Set u := ϕ λ ( t , . ) + λg (0) δ χ and v = ψ ( t , . ). We observe that λ ω t + dd c g ( t ) χ λg (0) dd c χ ≥ λg (0) δ Θ + λg (0) δ dd c χ ≥ . Since ϕ is a pluripotential subsolution to (CMAF), we infer by using Lemma 2.6 that( ω t + dd c u ) n ≥ ((1 − λ )( ω t + dd c ϕ t ) + λ ( g ( t ) θ + g ( t ) dd c ρ ) / n ≥ e (1 − λ )( ∂ t ϕ ( t ,. )+ F ( t ,.,ϕ ( t ,. )))+ λ ( n log g ( t )+ c ) f dV ≥ e (1 − λ ) ∂ t ϕ ( t ,. )+ F ( t ,.,ϕ ( t ,. )) − λ ( M F + | n log g ( t )+ c | ) f dV in the weak sense of measures in D . Choosing λ sufficiently small so that λ < min [0 ,T ] { ( M F + | n log g ( t ) + c | ) − ε } , it follows from (3.3) and the increasing property of F that( ω t + dd c u ) n ≥ e ∂ − t ψ ( t ,. )+ F ( t ,.,u ( . ))+ ε f dV in the weak sense of measures in D . On the other hand, ψ is a pluripotential supersolutionto (CMAF), thus ( ω t + dd c v ) n ≤ e ∂ − t ψ ( t ,. )+ F ( t ,.,ψ ( t ,. )) f dV in the weak sense of measures in D . The last two inequalities yield( ω t + dd c u ) n ≥ e F ( t ,.,u ( . )) − F ( t ,.,v ( . ))+ ε ( ω t + dd c v ) n . Shrinking D if necessary, we can assume that u ( x ) > v ( x ) for any x ∈ D . We thus get( ω t + dd c u ) n ≥ e ε ( ω t + dd c v ) n in the sense of positive measures in D .Setting ˜ u := u + min ∂D ( v − u ). We observe that v ≥ ˜ u on ∂D , hence the elliptic comparisonprinciple (see Proposition 3.3) yields Z { v
Let D be an open subset of X . Assume that u, v are θ -psh functions withminimal singularities in X , which satisfy lim sup D ∋ x → ∂D ( u − v )( x ) ≥ . Then Z { u Proof. We know that the boundary condition means that for any ε > { u < v − ε } is compact in D . For each C > u C := max( u, V θ − C ). Then the function max( u C , v − ε )is θ -psh bounded in D and it coincides with u C in a neighborhood of ∂ Ω. Let { u C < v − ε } ⋐ D ′ ⋐ D . This implies that Z D ′ ( θ + dd c u C ) n = Z D ′ ( θ + dd c max( u C , v − ε )) n . (3.5)Indeed, set w := max( u C , v − ε ), using local regularization of plurisubharmonic functions,we observe that ( θ + dd c w ) n − ( θ + dd c u ) n = dd c S , in the sense of currents on D , where S := ( w − u C )(( θ + dd c w ) n − + · · · + ( θ + dd c u C ) n − ) is a well-defined current with compactsupport in D . Pick any test function γ which is identically 1 in a neighborhood of the supportof S . Then Z D ′ dd c S = Z D ′ γdd c S = Z D ′ S ∧ dd c γ = 0 , where we have known that dd c γ = 0 on the support of S . This implies the identity (3.5).On the other hand, we apply [GZ, Theorem 3.27] to get Z { u C Proposition 3.4. Let ϕ (resp. ψ ) be a subsolution (resp. supersolution) to (CMAF) withinitial value ϕ (resp. ψ ). We assume that a) ϕ is C in t and continuous on (0 , T ) × Ω , b) ψ is locally uniformly semi-concave in t , c) ϕ t → ϕ and ψ t → ψ , as t → , d) for any t ∈ (0 , T ) , ψ t has minimal singularities, e’) the function ( t, x ) ψ ( t, x ) is continuous on (0 , T ) × Ω .Then ϕ ≤ ψ ⇒ ϕ ≤ ψ in X T . Proof. We proceed as in the proof of Theorem 2.7. We fix s > v s ( t, x ) = ψ ( t + s, x ) + Cs ( t + 1) − Cs log δ − s, and u s ( t, x ) := α s ϕ ( t, x ) + (1 − α s ) g ( t ) ρ ( x ) + χ ( x )2 − Cs ( t + 1) . Here α s = 1 − As ∈ (0 , A > ρ, χ are definedin (1.2), (1.1) and C is a positive constant which will be chosen later. We want to showthat for C > u s is a subsolution while v s is a supersolution to (CMAF) and u s (0 , . ) ≤ v s (0 , . ). We can then apply Proposition 3.2 and let s → ω t + s + dd c u s = α s ( ω t + dd c ϕ t ) + 1 − α s ω t + g ( t ) dd c ρ )+ 1 − α s ω t + g ( t ) dd c χ ) + ω t + s − ω t . By assumptions, we observe that for s small enough ω t + s − ω t ≥ − s Θ . Since θ + dd c χ ≥ δ Θwe thus obtain 1 − α s ω t + g ( t ) dd c χ ) + ω t + s − ω t ≥ Asg ( t ) δ Θ − s Θ ≥ . Hence, ( ω t + s + dd c u s ) n ≥ ( α s ( ω t + dd c ϕ t ) + (1 − α s ) g ( t )( θ + dd c ρ ) / n ≥ e α s ( ∂ t ϕ t + F ( t,.,ϕ t ))+(1 − α s )( n log g ( t )+ c ) f dV applying Lemma 2.6 in the last line. Since α s = 1 + O ( s ) and F is bounded above andlocally uniformly Lipschitz, by choosing C > δ , κ F , M F , c (seeTheorem 2.7), we then have ( ω t + s + dd c u s ) n ≥ e ∂ t u s + F ( t,.,u s ) f dV. On the other hand, since ψ is a supersolution, we have( ω t + s + dd c v s ) n ≤ e ∂ t v s − Cs + F ( t + s,.,ψ ( t + s,. )) f dV ≤ e ∂ t v s − Cs + F ( t,.,v s ( t,. ))+ κ F s f dV ≤ e ∂ t v s + F ( t,.,v s ( t,. )) f dV, where the second line follows from the Lipschitz condition and the increasing monotonocity of F , the last line follows from the choice of C . Up to increasing C > x ∈ X , u s (0 , x ) ≤ (1 − As ) ϕ (0 , x ) + Ag (0) s ( ρ + χ ) / ≤ ψ s ( x ) + Cs − Cs log Ag (0) s = v s (0 , x ) . It then follows from Proposition 3.2 for all ( t, x ) ∈ X T that u s ( t, x ) ≤ v s ( t, x ) . Letting s → (cid:3) LURIPOTENTIAL MONGE-AMP`ERE FLOWS IN BIG COHOMOLOGY CLASSES 29 Lemma 3.5. With the same assumptions of ψ as in Proposition 3.4. Then there exist uniformconstants A > , C > , and t > small enough such that for all ( t, x ) ∈ (0 , t ) × X , ψ ( t, x ) ≥ (1 − At ) ψ ( x ) + C ( t log( Ag (0) t ) − t ) + Ag (0) t ( ρ ( x ) + χ ( x )) / . Proof. The proof is similar to that of [GLZ20, Lemma 3.14]. We recall that the function F satisfies the Lipschitz condition i.e. there exists a constant κ F > t, t ′ ∈ [0 , T / x ∈ X , r ∈ R , | F ( t, x, r ) − F ( t ′ , x, r ) | ≤ κ F | t − t ′ | . Set t = min(1 , A, T / 4) with A > s > t, x ) ∈ (0 , t ) × X , u s ( t, x ) := (1 − At ) ψ s ( x ) + Atg ( s )( ρ ( x ) + χ ( x )) / n ( t log Ag (0) t − t ) − Ct,v s ( t, x ) := ψ ( t + s, x ) + 2 κ F ts, where ρ, χ are θ -psh functions on X defined in (1.2),(1.1), and C is a positive constant to bechosen later. We see that u s is of class C in t , and for any t ∈ (0 , t ) fixed, u s ( t, . ) is continuousin Ω as ρ is continuous in Ω (see e.g. [GZ, Theorem 12.23]). Now let A ≥ ( δ g (0)) − . One has,for any t ∈ (0 , t ) ,ω t + s + dd c u s ( t, . ) = (1 − At )( ω s + dd c ψ s ) + At ( ω s + g ( s ) dd c ρ ) / At ( ω s + g ( s ) dd c χ ) / ω t + s − ω s . By hypothesis (0.2), we have ω t + s − ω s ≥ − t Θ , and since θ + dd c χ ≥ δ Θ, we thus get At ( ω s + g ( s ) dd c χ ) / ω t + s − ω s ≥ Atg ( s ) δ Θ − t Θ ≥ . We then obtain ( ω t + s + dd c u s ) n ≥ ( At ( ω s + g ( s ) dd c ρ ) / n ≥ ( Ag (0) t ) n e c f dV, since g ( s ) ≥ g (0). We now choose C = A sup X ( ρ + χ − ϕ ) / M F − min( c , ω t + s + dd c u s ) n ≥ e ∂ t u s ( t,. )+ F ( t,.,u s ( t,. )) f dV. It follows from the definition that u s ( t, . ) converges in L ( X, dV ) to u s (0 , . ) = ψ s . On the otherhand, since ψ is a supersolution to (CMAF), we have( ω t + s + dd c v s ) n ≤ e ∂ t ψ t + s + F ( t + s,.,ψ ( t + s,. )) f dV = e ∂ t v s − κs + F ( t + s,.,ψ ( t + s,. )) f dV. Since the function F is Lipschitz in t and is increasing in r , for all t, s ∈ (0 , t ) , x ∈ X , F ( t + s, x, ψ ( t + s, x )) ≤ F ( t, x, ψ ( t + s, x )) + κ F s ≤ F ( t, x, v s ( t, x )) + κ F s. We thus obtain, ( ω t + s + dd c v s ) n ≤ e ∂ t v s ( t,. )+ F ( t,.,v s ( t,. )) f dV. Since for each s that ψ s is continuous on Ω hence v s is continuous on [0 , t ) × Ω and it is clearthat v s ( t, . ) converges to v s (0 , . ) = ψ s in L ( X, dV ) as t → 0. We moreover see that for each t , v s ( t, . ) has minimal singularities because ψ t has. We can now apply Proposition 3.2 and get u s ≤ v s on (0 , t ) × X . Letting s → t, x ) ∈ (0 , t ) × X ,(1 − At ) ψ ( x ) + Ag (0) t ( ρ ( x ) + χ ( x )) / C ( t log( Ag (0) t ) − t ) ≤ ψ ( t, x ) , as desired. (cid:3) Space regularity. In this section, we shall use the extra assumption˙ ω t ≤ Aω t , ∀ t ∈ [0 , T ) , (3.6)for some constant A > Theorem 3.6. Under the extra assumption (3.6) , the envelope U has minimal singularitiesand U t is moreover continuous in Ω , for each t ∈ (0 , T ) .Proof. Observe that η t := e − At ω t is decreasing in t . By [BEGZ10, Theorem 6.1], for each t ∈ [0 , T ) there exists a unique η t -pshfunction φ t with full Monge-Amp`ere mass such that( η t + dd c φ t ) n = e φ t + c f dV, for c > X φ = 0. For 0 < s ≤ t we have( η s + dd c φ t ) n ≥ ( η t + dd c φ t ) = e φ t + c f dV. It follows that φ t is a subsolution to ( η s + dd c φ s ) n = e φ s + c f dV , so a classical comparisonprinciple (see e.g. [BEGZ10, Proposition 6.3]) ensures that φ t ≤ φ s . Therefore t φ t isdecreasing in [0 , T ) × X . Since t η t is decreasing in t , we may assume that η t ≥ θ ′ for somebig (1 , θ ′ . By [BEGZ10, Theorem 4.1], there exists a unique θ ′ -psh function ρ ′ withminimal singularities such thatsup X ρ ′ = 0 , ( θ ′ + dd c ρ ′ ) n = 2 n e c f dV, for some normalization constant c . Since θ ′ is big, we can find a θ ′ -psh function χ ′ normalizedby sup X χ ′ = 0 such that θ ′ + dd c χ ′ ≥ ε Θ for some ε > 0. Up to multiplying a positiveconstant we may assume that − Θ ≤ ˙ η t ≤ Θ, where Θ is a K¨ahler form. Fix 0 < ε < T ′ < T ,let s ∈ ( − ε, ε ). For any ( t, x ) ∈ [ ε, T ] × X , setΦ( t, x ) := (1 − α s ) φ t + s ( x ) + α s ρ ′ ( x ) + χ ′ ( x )2 − C | s | , where α s = B | s | with B > Bε ≥ 2, and C > C > B ( c + c ). Then α s ( θ ′ + dd c χ ′ ) ≥ B | s | εθ ≥ | s | Θ . Since ˙ η t ≥ − Θ we have η t − η t + s ≥ −| s | Θ. We thus obtain η t + dd c Φ t = (1 − α s )( η t + s + dd c φ t + s ) + α s η t + s + dd c ρ ′ α s η t − s + dd c χ ′ η t − η t + s ≥ (1 − α s )( η t + s + dd c φ t + s ) + α s θ ′ + dd c ρ ′ . Combining this together with Lemma 2.6 we obtain( η t + dd c Φ t ) n ≥ e (1 − α s )( φ t + s + c )+ α s c f dV ≥ e Φ t + c f dV where the last line follows from the fact that sup X ρ ′ = sup X χ ′ = 0 and the choice of theconstant C . Thus Φ t is a subsolution to ( η t + dd c φ t ) n = e φ t + c f dV , hence Φ t ≤ φ t in X by thecomparison principle. Letting s → ∂ t φ t ( x ) ≤ C − B ( ρ ′ ( x ) + χ ′ ( x )) , for all ( t, x ) ∈ [ ε, T ′ ] × X . As explained in Section 1.3, ˙ φ t = ∂ t φ t is well-defined almosteverywhere. Set u ( t, x ) := e At φ t ( x ) − C ( t + 1) with C > c so LURIPOTENTIAL MONGE-AMP`ERE FLOWS IN BIG COHOMOLOGY CLASSES 31 that sup X φ = 0. We infer ˙ u ( t, . ) = e At ˙ φ t + Ae At φ t − C ≤ φ t − C since t φ t is deacreasing,so ( ω t + dd c u t ) n = e nAt + φ t + c f dV ≥ e ˙ u t + F ( t,.,u ( t,. )) f dV, for C > C > M F . Hence u t is a subsolution to (CMAF). Moreover, we canchoose C > u (0 , . ) = φ − C ≤ ϕ since ϕ is a η -psh function with minimalsingularities. Therefore u ∈ S ϕ ,f,F ( X T ). By [BEGZ10, Theorem 6.1] we have φ t ≥ V η t − C ( t )for some time-dependent constant C ( t ) we infer u t ≥ V ω t − C ′ ( t ), hence U t ≥ V ω t − C ′ ( t ) for all t ∈ [0 , T ).It remains to show that for each t ∈ (0 , T ), U t is continuous in Ω. We have that for each t > ∂ t U + F ( t, ., U t ) ≤ κ − κ ( ρ + χ ) + M F on X , for some κ > 0. The continuity of U t in Ωthus follows from [Dan21, Theorem 3.2]. (cid:3) We now show that the solution constructed in Theorem 3.1 is unique: Theorem 3.7. Let Φ be a pluripotential solution to the Cauchy problem for (CMAF) withinitial data ϕ . Assume that (3.6) holds and • Φ is locally uniformly semi-concave in (0 , T ) ; • for each t , Φ t has minimal singularities; • Φ t is continuous in Ω .Then Φ = U .Proof. Since Φ is locally uniformly Lipschitz in t we infer that Φ is continuous on (0 , T ) × Ω.We would like to apply Proposition 3.4 but U is not C in t . We are going to regularize it bytaking convolution in t as in [GLZ20, Proposition 3.16]. Fix 0 < T ′ < T , s > t, x ) ∈ X T ′ , V s ( t, x ) := α s s U ( st, x ) + (1 − α s ) g ( t ) ρ ( x ) + χ ( x )2 − C | s − | ( t + 1) , where ρ, χ are θ -psh functions defined in (1.2), (1.1). The proof of Theorem 2.7 implies that V s ∈ S ϕ ,f,F ( X T ′ ) where C > η be a smooth function with compactsupport in [ − , 1] such that R η ( t ) dt = 1. Set, for ε > η ε ( t ) = ε − η ( t/ε ), and we definefor any ( t, x ) ∈ X T ′ u ε ( t, x ) := Z R V s ( t, x ) η ε ( s − ds − Bε ( t + 1) . This function is a pluripotential subsolution to (CMAF) which is C in t (see [GLZ20, Proposi-tion 3.16]). For s close to 1, V s is a pluripotential subsolution to (CMAF), hence Lemma 1.16yields ( ω t + dd c V s ( t, . )) n ≥ exp( ∂ t V s + F ( t, ., V s ( t, . ))) f dV. We know that the function A (det A ) /n is concave on the convex cone of non negativehermitian matrices. Jensen’s inequality ensures that, for any t ∈ (0 , T ),(det( ω t + dd c u ε )) /n = (cid:18) det (cid:18)Z R ( ω t + dd c V s ) η ε ( s − ds (cid:19)(cid:19) /n ≥ (cid:18)Z R exp (cid:18) n ( ∂ t V s ( t, . ) + F ( t, ., V s ( t, . ))) (cid:19) η ε ( s − ds (cid:19) f /n ≥ exp (cid:18)Z R n ( ∂ t V s + F ( t, ., V s ( t, . ))) η ε ( s − ds (cid:19) f /n ≥ exp (cid:18) n (cid:18) ∂ t u ε ( t, . ) + Bε + F (cid:18) t, ., Z R V s ( t, . ) η ε ( s − ds (cid:19)(cid:19)(cid:19) f /n ≥ exp (cid:18) n ( ∂ t u ε ( t, . ) + F ( t, ., u ε )) (cid:19) f /n . The second line follows from the Main Theorem in [GLZ19], the third and fourth ones followfrom the convexity of the exponential and F , and the last one follows from the monotonicityof F . Using Lemma 1.16 again, we infer that u ε is a subsolution to (CMAF).On the other hand, it follows from the proof of Theorem 2.7 that V s (0 , x ) ≤ ϕ on X , for B > u ε (0 , . ) ≤ ϕ on X . We can thus apply Proposition 3.4 to obtain u ε ≤ Φ on [0 , T ′ ] × Ω, hence on [0 , T ′ ] × X . Letting ε → T ′ → T we get U ≤ Φ on[0 , T ) × X . Hence the equality holds. (cid:3) Applications We apply the tools we have developped in two related geometrical settings. We first defineand study the long time behavior of the normalized K¨ahler-Ricci flow (NKRF) on a manifoldof general type. We then analyze the normalized K¨ahler-Ricci flow on a variety with semi-logcanonical singularities and ample canonical bundle.4.1. Manifolds of general type. We study in this section the (normalized) K¨ahler-Ricci flowon a manifold of general type. We try to run such flow in a weak sense beyond the maximalexistence time by using the results obtained in the previous section. We then study the long-time behavior of this flow.Let ( X, ω ) be a compact K¨ahler manifold of general type i.e. the canonical divisor K X is big,and ω be a K¨ahler (1 , ∂ϑ t ∂t = − Ric( ϑ t ) − ϑ t , ϑ | t =0 = ω . (4.1)Let T be the maximal existence time of smooth flows which is defined by T := sup { t > e − t { ω } + (1 − e − t ) c ( K X ) is K¨ahler } . Note that T = + ∞ if and only if the canonical divisor K X is nef. In this case, the normalizedK¨ahler-Ricci flow exists in the classical (smooth) sense and converges to a singular K¨ahler-Einstein metric (see [Tsu88, TZ06]).When K X is not nef, the flow has a finite time singularity at T ( < + ∞ ). The limit class at T of the flow α T := lim t → T { ϑ t } = e − T { ω } + (1 − e − T ) c ( K X )is big and nef. In [CT15, Theorem 1.5], Collins and Tosatti showed that the flow θ t exists onthe maximal time interval [0 , T ) and develops singularities precisely on the Zariski closed set LURIPOTENTIAL MONGE-AMP`ERE FLOWS IN BIG COHOMOLOGY CLASSES 33 X \ Amp( α T ) as t → T − . For t > T , the cohomology class { ϑ t } is still big, but no longer nef,we can not continue the flow in the classical sense.In [FIK03, Section 10], Feldman, Ilmanen and Knopf have asked the question: can onedefine and construct weak solutions of the K¨ahler-Ricci flow beyond the singular time? In[BT12, Theorem 4], Boucksom and Tsuji have constructed the normalized K¨ahler-Ricci flow onsmooth projective varieties with pseudoeffective canonical class for all times. They used thediscretization of the K¨ahler-Ricci flow and some algebro-geometric tools. In the end, they haveconjectured the same result for the case of K¨ahler manifolds (cf. [BT12, Conjecture 1]). In[Tˆo19], To used the viscosity theory to show that the weak K¨ahler-Ricci flow exists for all timein the viscosity sense and converges to the unique singular K¨ahler-Einstein metric in the class c ( K X ) constructed in [EGZ09, BEGZ10].In this section, we shall apply the parabolic pluripotential theory established in the previoussections to answer the question of Feldman-Ilmanen-Knopf and the conjecture of Boucksom-Tsuji. Moreover, we also study the long-term behavior of the K¨ahler-Ricci flow.Now let θ be a smooth closed (1 , c ( K X ). Set ω t := e − t ω + (1 − e − t ) θ .Since dV X is a smooth volume form on X , then − Ric( dV ) ∈ − c ( K X ), and so there is a smoothfunction f such that θ = − Ric( dV X ) + dd c f . We then define µ = e f dV which is a smooth positive volume with θ = Ric( µ ). Thus the normalized K¨ahler-Ricci flow (4.1)can be written as the complex Monge-Amp`ere flow( ω t + dd c ϕ t ) n = e ˙ ϕ t + ϕ t dµ (4.2)In [BEGZ10], it is shown that there exists a unique θ -psh function ϕ KE with minimal singu-larities such that ( θ + dd c ϕ KE ) n = e ϕ KE µ. (4.3)The singular K¨ahler- Einstein metric ω KE := θ + dd c ϕ KE ∈ c ( K X ) is smooth in Ω = Amp( K X ),where it satisfies Ric( ω KE ) = − ω KE . We shall show that the (normalized) pluripotential K¨ahler-Ricci flow continuously deforms anyinitial K¨ahler form ω towards ω KE , as t → + ∞ . The result even holds for an initial datum S which is a positive current with bounded potentials : Theorem 4.1. Let S ∈ { ω } be an arbitrary positive current with bounded potential. Thenthe normalized K¨ahler-Ricci flow (4.1) starting at S exists and converges exponentially fasttowards ω KE , as t → + ∞ .Proof. The problem is equivalent to solving and studying the parabolic complex Monge-Amp`ereequation (4.2) with initial data ϕ , where S = ω + dd c ϕ and ω t = e − t ω + (1 − e − t ) θ . Recallthat θ is a big form representing c ( K X ). Since ω is a K¨ahler form, there exists a small constant c > ω ≥ cθ . Hence ω t = e − t ω + (1 − e − t ) θ ≥ g ( t ) θ , where g ( t ) = ce − t + 1 − e − t isa smooth positive function with g ′ ( t ) = e − t (1 − c ) > c > ϕ by constructing a subsolution to the Cauchy Problem. Set, for any ( t, x ) ∈ (0 , T ) × X , u ( t, x ) := e − t ϕ + (1 − e − t ) ϕ KE + h ( t ) , where h is a C function in R and h (0) = 0 to be chosen later so that u is a subsolution tothe Cauchy Problem. We observe that u (0 , x ) = ϕ ( x ), and for all t > ω t + dd c u t = e − t ( ω + dd c ϕ ) + (1 − e − t )( θ + dd c ϕ KE ) ≥ u t is θ t -psh and( ω t + dd c u t ) n ≥ (1 − e − t ) n ( θ + dd c ϕ KE ) n = e n log(1 − e − t ) e ϕ KE f dV. On the other hand ∂ τ u t + u = ϕ KE + h ′ ( t ) + h ( t ) hence u is a subsolution if n log(1 − e − t ) ≥ h ( t ) + h ′ ( t ) . We thus choose h to be the unique solution of the ODE: h ( t ) + h ′ ( t ) = n log(1 − e − t ) , h (0) = 0 . We compute ( e t h ( t )) ′ = e t ( h ( t ) + h ′ ( t )) = ne t log(1 − e − t ), hence h ( t ) = ne − t (cid:20)Z e t log(1 − e − t ) dt (cid:21) = ne − t (cid:2) ( e t − 1) log( e t − − te t + c (cid:3) . Since h (0) = 0, we have h ( t ) = ne − t (cid:2) ( e t − 1) log( e t − − te t (cid:3) = O ( te − t ) as t → ∞ . It follows from the comparison principle that u ≤ ϕ t hence ϕ KE + e − t ( ϕ − ϕ KE ) + h ( t ) ≤ ϕ t (4.4)on (0 , ∞ ) × X . For the upper bound, we argue as in [Tˆo19, Theorem 4.4]. Since the cohomologyclass c ( K X ) is big, we can find a θ -psh function χ with analytic singularities such that θ + dd c χ ≥ εω for some small constant ε > 0. We can assume that ε ≤ 1. Substracting a large constant, wecan always assume that χ ≤ χ ≤ V θ . We then have ω t + dd c ϕ t = e − t ( ω − ε − dd c χ ) + (1 − e − t ) θ + dd c ( ϕ t + ε − e − t χ ) ≤ [ e − t ε − + (1 − e − t )] θ + dd c ( ϕ t + ε − e − t χ ) . (4.5)Set u ( t, x ) = ϕ t ( x ) + e − t ( ε − χ ( x ) − C ) and g ( t ) = 1 + ( ε − − e − t . It follows from (4.5) that( g ( t ) θ + dd c u t ) n ≥ ( ω t + dd c ϕ t ) n ≥ e ˙ ϕ t + ϕ t µ = e ˙ u t + u t µ, Let φ be a ε − θ -psh function with minimal singularities. We can find a constant C > φ − ε − χ ≥ ϕ − C . Therefore u is a subsolution of the following Cauchy problem(4.6) ( ( g ( t ) θ + dd c φ t ) n = e ∂ t φ t + φ t µφ (0 , . ) = φ , where φ denotes the pluripotential solution to (4.6). Thus Proposition 3.4 yields u ≤ φ on[0 , ∞ ) × Amp( K X ), i.e. ϕ t + e − t ( ε − χ − C ) ≤ φ t On the other hand, the function φ t converges to ϕ KE as t → + ∞ by Lemma 4.2 below.Combining this with (4.4), we infer that ϕ t converges to ϕ KE in Amp( K X ) as t → + ∞ . (cid:3) Lemma 4.2. The solution φ t of (4.6) converges locally exponentially fast towards ϕ KE in theample locus Amp( K X ) as t → + ∞ . LURIPOTENTIAL MONGE-AMP`ERE FLOWS IN BIG COHOMOLOGY CLASSES 35 Proof. Set ˜ φ t := g ( t ) − ( φ t − a ( t )) , where g ( t ) = 1 + ( ε − − e − t , and h is the unique solution of the ODE: a ( t ) + a ′ ( t ) = n log g ( t ),with a (0) = 0. An easy computation shows that a ( t ) = O ( te − t ). Now the flow (4.6) becomes ( ( θ + dd c ˜ φ t ) n = e g ( t ) ∂ t ˜ φ t + ˜ φ t µ ˜ φ (0 , . ) = εφ . (4.7)We renormalize in time ψ ( t, . ) = ˜ φ ( s ( t ) , . ), where s ( t ) is the unique solution of the ODE s ′ ( t ) = g ( s ( t )) and s (0) = 0. Then the flow (4.6) can be written as ( ( θ + dd c ψ t ) n = e ∂ t ψ t + ψ t µψ (0 , . ) = εφ . (4.8)We set for ( t, x ) ∈ (0 , + ∞ ) × X , u ( t, x ) := e − t ψ + (1 − e − t ) ϕ KE + h ( t ) , where h is the uniquesolution to the ODE h ′ ( t ) + h ( t ) = n log(1 − e − t ), with h (0) = 0 . As in the proof of Theorem4.1 we can check that u is a subsolution to (4.8).On the other hand, since ϕ KE is a θ -psh function with minimal singularities we can choosea constant C > ϕ KE + C ≥ ψ . Set for ( t, x ) ∈ (0 , + ∞ ) × X , v ( t, x ) := ϕ KE ( x ) + Ce − t . One can check that v is subsolution to (4.8). Therefore, e − t ψ + (1 − e − t ) ϕ KE + O ( te − t ) ≤ ψ t ≤ ϕ KE + Ce − t , which implies ψ t → ϕ KE in Amp( K X ) as t → + ∞ . So does the flow φ t since g ( t ) → a ( t ) → t → + ∞ . (cid:3) Remark 4.3. The uniqueness of the flow (4.2) follows directly from Theorem 3.7.4.2. Stable varieties. Log canonical pairs. Let X be a compact K¨ahler normal complex variety such that K X is Q -Cartier. We say that X has log canonical singularities (lc) if for some (or equivalently any) logresolution π : Y → X , we have K Y = π ∗ K X + X a i E i where E i are exceptional divisors, and the coefficients a i satisfy the inequality a i ≥ − Semi-log canonical singularities. We give here a short overview of the notion of semi-log canon-ical singularities and stable varieties. We refer to the survey [Kov13, § X will be a reduced and equidimensional scheme of finite type over C unlessstated otherwise, and we set n := dim C X . In order to study the normalized K¨ahler-Ricci flow,one needs a canonical sheaf (or a canonical divisor). Let us stress that the dualizing sheaf, evenif it exists, is not necessarily a line bundle (or a divisor).We say that the scheme (variety) X is Cohen-Macaulay if for every x ∈ X the local ring O X,x is Cohen-Macaulay, that is its depth is equal to its Krull dimension. If X is Cohen-Macaulay,then it admits a dualizing sheaf ω X .We say that X is Gorenstein if X is Cohen-Macaulay ( X admits a dualizing sheaf ω X ) and ω X is a line bundle. A scheme (variety) X is called G if it is Gorenstein in codimension 1,which means that there exists an open subset U ⊂ X such that codim X ( X \ U ) ≥ U isGorenstein. We say that X satisfies the S condition of Serre if for all x ∈ X , we have depth( O X,x ) ≥ min(dim O X,x , ı : Z ֒ → X of codimension at least two, the natural map O X → ı ∗ O X \ Z is an isomorphism.We now want to have an interpretation of ω X in terms of Weil divisor. If X satisfies theconditions G and S , and U is a Gorenstein open subset whose complement has codimension atleast 2, one may define the ”canonical=dualizing” sheaf ω U as the determinant of the cotangentbundle, i.e., the sheaf of top differential forms, ω U = det Ω U . One can then define the canonicalsheaf ω X by ω X = ∗ ω U where : U ֒ → X is the open embedding.As U is non-singular, ω U is a line bundle, hence corresponds to a Cartier divisor. Let K U := P a i K i be a Weil divisor associated to this Cartier divisor such that for all i , K i doesnot contain any component of X sing of codimension 1. Let ¯ K i denote the closure of K i and K X := X a i ¯ K i . Since codim X ( U ) ≥ 2, this is the unique Weil divisor for which K X | U = K U . We see that thedivisorial sheaf O X ( K X ) := { f ∈ K ( X ) : K X + div ( f ) ≥ } is reflexive, and coincides with ω U = ω X | U , hence the S condition implies that ω X ≃ O X ( K X ) . Remark 4.4. The condition G guarantees the existence of the canonical sheaf ω X , and thecondition S ensures its uniqueness. When X is projective, we know that it admits a dualizingsheaf, as it is reflexive, it coincides with ω X by the S condition.We let ω [ m ] X denote the m -th reflexive power of the canonical sheaf ω X (defined by ω [ m ] X :=( ω ⊗ mX ) ∗∗ ). As above, we obtain ω [ m ] X ≃ O X ( mK X ). Thus the Weil divisor K X is Q -Cartier ifand only if ω X is a Q -line bundle, i.e., for some m > ω [ m ] X is a line bundle. From now on wework with the canonical divisor K X instead of its associated canonical sheaf ω X .We say that a closed point x ∈ X is double crossing if it is locally analytically isomorphic tothe singularity { ∈ ( z z = 0) ⊂ C n +1 } . A scheme X is called demi-normal if it satisfies the S condition and has only double crossingsingularities in codimension 1. We now give the definition of semi-log canonical models: Definition 4.5. We say that X has semi-log canonical (slc) singularities if K X is Q -Cartierand there exist two Zariski open sets U, V such that • X = U ∪ V , • U is a normal variety with log canonical singularities, • V has only double crossing points. We mention that semi-log canonical models may not be normal varieties. Let µ : X n → X be a normalization of X . We emphasize again that X is not irreducible in general, soits normalization is defined to be the disjoint union of the normalization of its irreduciblecomponents. The conductor ideal sheaf I C X := Ann O X ( µ ∗ O X n / O X )is defined to be the largest ideal sheaf on X that is also an ideal sheaf on X n . If we considerthe affine case where A n is the integral closure of some integral ring A , then one can see thatthe annihilator Ann A ( A n /A ) := { a ∈ A : aA n ⊂ A } is the largest ideal in A that is also anideal in B . LURIPOTENTIAL MONGE-AMP`ERE FLOWS IN BIG COHOMOLOGY CLASSES 37 For the case of schemes (varieties), we let I C Xn denote the corresponding conductor idealsheaf on X n , and we define the conductor subscheme as C X := Spec X ( O X / I C X ) on X and C X n := Spec X n ( O X n / I C Xn ) on X n . If X is seminormal (i.e. every finite morphism X ′ → X ,with X ′ is reduced, that is a bijection on points is an isomorphism) and S , then one canshow that these subschemes have pure codimension 1 hence they define Weil divisors which aremoreover reduced (cf. [KSS10, 4.5]).If X is demi-normal and K X is Q -Cartier, then we have the following relation µ ∗ K X = K X n + C C n . (4.9)Under the previous seminormality and S assumptions, the G condition is equivalent to thedemi-normality.In other words, we may alternatively define slc models as follows: Proposition 4.6. We say that X has semi-log canonical singularities if and only if • X is G and S , • K X is Q -Cartier (of index m ), • The pair ( X n , C X n ) is log-canonical. Note that there are many schemes satisfying the S condition and the seminormality but notdemi-normality. For instance, a reduced scheme consisting of the three axes in A does nothave double crossings in codimension 1, but is both S and seminormal.We can finally give the definition of stable variety: Definition 4.7. A projective variety X is called stable if • X has semi-log canonical singularities, • and K X is an ample Q -Cartier divisor. From Definition 4.5, we can see that X is a stable variety if K X is ample and X = U ∪ V ,where U, V are Zariski open sets, U is a normal variety with log canonical singularities, and V has only double crossing singularities.4.3. Convergence of NKRF on stable varieties. We now study the normalized K¨ahler-Ricci flow on a complex projective variety with semi-log canonical singularities (stable variety).This is the evolution equation: ∂θ t ∂t = − Ric( θ t ) − θ t , θ | t =0 = ω . (4.10)After passing to a suitable resolution of singularities, we may as well assume that X issmooth if we study the setting of log pairs ( X, D ), where D = P Ni =1 a i D i is the Q -divisor on X with simple normal crossing (snc), where the role of the canonical line bundle is playedby the log canonical line bundle K X + D (which occurs as the pull-pack to the resolution ofthe original canonical line bundle). In this setting the original variety has semi-log canonicalsingularities precisely when the log pair ( X, D ) is log canonical (lc) in the usual sense of MinimalModel Program (MMP), i.e. the coefficients of D are at most equal to one (but negativecoefficients allowed). Let us mention that even if the original canonical line bundle is ample,the corresponding log canonical line bundle is merely semi-ample and big on the resolution,since it is trivial along the exceptional divisors of the corresponding resolution.Let X be a compact K¨ahler manifold and ( X, D ) be a log canonical pair such that K X + D is semi-positive and big (i.e., ( K X + D ) n > unique closed positive current ω KE = θ + dd c ψ KE in c ( K X + D ) which is smooth on a Zariskiopen set U of X and satisfies Ric( ω KE ) = − ω KE + [ D ]in the weak sense on X . The current ω KE is called the singular K¨ahler-Einstein metric .The main result of this section is the following: Theorem 4.8. Let S = ω + dd c ϕ be a closed positive current with bounded potential. Thenthe normalized K¨ahler-Ricci flow (4.10) on X admits a unique pluripotential solution definedon R + × X . Moreover the flow (4.10) converges towards ω KE = θ + dd c ψ KE as t → + ∞ . Observe that Theorem 4.8 implies Theorem D from the introduction: indeed, if Y is aprojective variety with semi-log canonical singularities (stable variety) and π : ( X, D ) → Y isa log resolution of the normalization (endowed with its conductor), then the exceptional locusof π is contained in the complement of the ample locus of K X + D . Remark 4.9. A similar result has been obtained in [CGLS19, Theorem 1.3] with a very differentapproach. These authors generalize the a priori estimates of Song-Tian [ST17] to the case of Q -factorial projectives varieties with log canonical singularities. They also show that if X isstable, then the normalized K¨ahler-Ricci flow (4.10) has a unique maximal weak solution on[0 , + ∞ ) which is smooth in (0 , + ∞ ) × X reg and converges to the singular K¨ahler-Einstein metric ω KE both in the sense of currents and in the C ∞ loc ( X reg )-topology as t tends to infinity.Our approach allows one to treat more general equations, avoiding any projectivity assump-tion on the variety nor any integrality on the initial cohomology class, and applies to big classesfor which no smooth deformation is available. Proof of Theorem 4.8. By definition, D = P Ni =1 a i D i is a simple normal crossings R -divisorwith a i ∈ ( −∞ , 1] and defining section s i . The normalized K¨ahler-Ricci flow (4.10) can bewritten as the complex Monge-Amp`ere flow( ω t + dd c ϕ t ) n = e ˙ ϕ t + ϕ t dµ, (4.11)where ω t := e − t ω + (1 − e − t ) θ , with θ being a smooth semi-positive and big (1 , c ( K X + D ) and dµ is a volume form on X which can be locally written dµ = dV X Q Ni =1 | s i | a i = f dV X , where dV X is a smooth volume form on X , f := Q i | s i | − a i . Here, s i are non-zero sectionsof O X ( D i ), | · | i are smooth hermitian merics on O X ( D i ). We let D lc := ∪ a k =1 D k denote the”non-klt” locus. Step 1: constructing a subsolution. We let Ω denote the ample locus of the class { θ } .Since the latter is big, there exists a θ -psh function χ such that˜ θ := θ + dd c χ ≥ δω on Ω for some δ > χ → −∞ near ∂ Ω . Up to multiplying by a positive constant we can assume that | s i | ≤ /e so that − log( | s i | ) ≥ D i . Note also that dd c ( − log( | s i | )) extends as a smooth real (1 , X whosecohomology class is 2 πc ( D ). We compute − dd c (log( λ − log( | s i | ))) = − dd c ( λ − log( | s i | )) λ − log( | s i | ) + ds i ∧ d c s i | s i | ( λ − log( | s i | )) . The second term is a semipositive (1 , − log( | s i | ) goes to ∞ near D j , we inferthat ˜ θ − dd c log( λ − log | s i | ) is positive on Ω \ D j when λ is big enough. Replacing ˜ θ by N ˜ θ and LURIPOTENTIAL MONGE-AMP`ERE FLOWS IN BIG COHOMOLOGY CLASSES 39 increasing λ if necessary one has N ˜ θ − dd c log( λ − log | s i | ) > 0, hence P ( N ˜ θ − dd c log( λ − | s i | ))defines a K¨ahler form on Ω \ D . Then for suitable positive constants λ, A the function v := − N X i =1 log( λ − log | s i | ) + χ − A (4.12)is a subsolution of the elliptic Monge-Amp`ere equation:( θ + dd c ψ KE ) n = e ψ KE dV Q i | s i | a i . (4.13)By the arguments above, we get the lower bound (see also [BG14, 5.5.2]): ψ KE ≥ χ − X a k =1 log( − log | s k | ) − A for some uniform constant A > 0. Here the hermitian metrics | · | k are chosen conveniently.Since θ is semi-positive we see that for all t , ω t ≥ cθ for some c > c = 1. We can check that(4.14) u ( t, x ) := ψ KE ( x ) − C e − t is a subsolution to (4.11), for C > u (0 , . ) ≤ ϕ . Step 2: the approximating flows. We now establish the existence of the flow (4.11) byan approximation argument using ideas from [DiNL15, Theorem 4.5]. Fix T < + ∞ . Thedifficulty is that the density f = Π i | s i | − a i is not in L p , p > 1, (not even in L ) since someof the coefficients a j are equal to 1. For each j ∈ N , Theorems A and B provide a unique ϕ t,j ∈ P ( X T , ω ) such that(4.15) ( ω t + dd c ϕ t,j ) n = e ˙ ϕ t,j + ϕ t,j min( f, j ) dµ X , ϕ ,j = ϕ . Since ω t is the pull-back of a smooth family of K¨ahler forms, we have − Aω t ≤ ˙ ω t ≤ Aω t , for some uniform constant A . We can proceed as in the proof of Theorem 2.7 and Theorem 2.13to establish the following uniform bounds: for each T ∈ (0 , + ∞ ) and any compact K ⊂ Ω \ D lc ,there is a constant C ( T, K ) such that t | ∂ t ϕ t,j ( x ) | ≤ C ( T, K ) , and t ∂ t ϕ t,j ( x ) ≤ C ( T, K ) , ∀ ( t, x ) ∈ (0 , T ) × K. Indeed, on (0 , T ) the forms ω t satisfy ω t ≥ g ( t ) θ , where g ( t ) = c > F in our case is defined by r F ( t, x, r ) := r which satisfies the assumptions in theintroduction. More precisely, we have the following: Proposition 4.10. Let J = [ a, b ] be a compact interval of (0 , T ) . There exist constants uni-forms C , C , C > such that for all j ∈ N , t ∈ J , (1) C ≥ ϕ t,j ( x ) ≥ ψ KE ( x ) − C e − t , (2) | ∂ t ϕ t,j | ≤ C + P a k =1 log( − log | s k | ) − χ , (3) ∂ t ϕ t,j ≤ C + P a k =1 log( − log | s k | ) − χ . Proof. We first prove (1). For the lower bound, we can check that the function u in (4.14) isalso a subsolution to (4.15). For the upper bound, we pick C > ω nt ≤ e C f dV for all t ∈ [0 , T ]. The domination principle (see e.g. [BEGZ10, Corollary 2.5]) yields ϕ t,j ≤ C holds everywhere. We next prove (2). Fix ε > ε ) b < T . For all t ∈ J and s ∈ (1 − ε , ε )there exists a constant A > ω t ≥ (1 − A | s − | ) ω ts . (4.16)For s small enough we set λ s := | − s | s , α s := s (1 − λ s )(1 − A | s − | ) ∈ (0 , , (4.17)hence γ s := λ s / (1 − α s ) ≥ ε > 0. Shrinking ε we may assume that γ s ω t ≥ ε θ . Let v be asolution to the following equation ( ε θ + dd c v ) n = e v µ. (4.18)The same argument above (in the Step 1) yields v ≥ ε χ − ε X a k =1 log( − log | s k | ) − A, for some uniform constant A > 0. For any ( t, x ) ∈ J × X we consider u s ( t, x ) := α s s ϕ j ( ts, x ) + (1 − α s ) v ( x ) − C | s − | e − t , for C > ω t + dd c u s ( t, . )) n = h (1 − λ s ) ω t + α s s dd c ϕ ts + (1 − λ s ) ω t + (1 − α s ) dd c v i n ≥ [ α s ( ω ts + dd c ϕ ts,j ) + (1 − α s )( γ s ω t + dd c v )] n ≥ e α s ( ∂ t ϕ j ( ts,. )+ ϕ j ( ts,. ))+(1 − α s ) v min( f, j ) dV = e ∂ t u s ( t,. )+ u s ( t,. ) min( f, j ) dV where we use (4.16) in the second line and Lemma 2.6 in the third one. Therefore u s is asubsolution to (4.15). Since ϕ is bounded we can choose C > large enough so that u s (0 , . ) ≤ ϕ on X . Hence the comparison principle (Proposition 3.4) ensures that u s ≤ ϕ j in J × X , i.e., α s s ϕ j ( ts, x ) + (1 − α s ) v − C | s − | e − t ≤ ϕ j ( t, x ) , ∀ ( t, x ) ∈ J × X. Letting s → t, x ) ∈ J × X , | ∂ t ϕ j ( t, x ) | ≤ C − C v ( x ) , with C > t, x ) ∈ J × X , v s ( t, x ) := α s s − ϕ j ( ts, x ) + sϕ j ( ts − , x )2 + (1 − α s ) v ( x ) − C | s − | e − t , where C > v s (0 , . ) ≤ ϕ . We can check as above that v s is a subsolutionto (4.15). By the same arguments we can obtain the estimate (3) . (cid:3) We now complete the proof of Step 2. For each t fixed, ϕ t,j is decreasing as j → ∞ by thecomparison principle (Proposition 3.4). It follows from Proposition 4.10 that ϕ t,j ( x ) ≥ ψ KE ( x ) − C , ∀ ( t, x ) ∈ [0 , T ) × X, for a large constant C > 0. It has been shown in [BG14, Theorem 4.2] that ψ KE ∈ E ( X, ω t ) foreach t since 0 ≤ θ ≤ ω t , hence ϕ t,j ∈ E ( X, ω t ). We will prove that lim j ϕ t,j = ϕ t is a solutionto the flow (4.11).Fix a compact subinterval J ⋐ (0 , T ), a compact subset K ⊂ (Ω \ D lc ). Proposition 4.10implies that there exists a constant C = C J > t → ϕ j ( t, x ) − Ct LURIPOTENTIAL MONGE-AMP`ERE FLOWS IN BIG COHOMOLOGY CLASSES 41 is concave in J , for all x ∈ K . Moreover, the function x ϕ j ( t, x ) is ω t -psh and uniformlybounded on K for all j . We obtain the same properties for the limiting function ϕ ( t, x ) by letting j → + ∞ . It follows from Proposition 1.10 that ˙ ϕ j , ˙ ϕ are well-defined and lim ˙ ϕ j ( t, . ) = ˙ ϕ ( t, . ).Consider G := { x ∈ X : f ( x ) > M } ∪ { x ∈ X : − v ( x ) > M } , where v is a solution to (4.18). Since − v is locally bounded outside a divisor, for any ε > M > G has small Monge-Amp`ere capacity Cap Θ ( G ) < ε for some K¨ahler form Θ. Hence for all t ∈ J, j ∈ N , ˙ ϕ t,j is uniformly bounded from above on X \ G . Therefore Lebesgue dominated convergence theorem ensures thatlim j → + ∞ Z J Z X \ G e ˙ ϕ t,j + ϕ t,j min( f, j ) dV dt = Z J Z X \ G e ˙ ϕ t + ϕ t f dV dt. Using the notations from [DiNL15, Section 2], it follows from [DiNL15, Theorem 2.9] that, forall t ∈ (0 , T ), j ∈ N , Z G ( ω t + dd c ϕ t,j ) n ≤ Cap ψ KE − C , ( G ) ≤ h ( ε ) , for some continuous function h : [0 , + ∞ ) → [0 , + ∞ ) with h (0) = 0.Hence Z J Z X e ˙ ϕ t + ϕ t f dV dt ≥ Z J Z X \ G e ˙ ϕ t + ϕ t f dV dt = lim j → + ∞ Z J Z X \ G ( ω t + dd c ϕ t,j ) n = lim j →∞ Z T Z X ( ω t + dd c ϕ t,j ) n − lim j → + ∞ Z J Z G ( ω t + dd c ϕ t,j ) n ≥ Z J Z X ω nt − T h ( ε ) . Letting ε → R J R X e ˙ ϕ t + ϕ t f dV dt ≥ R J R X ω nt . On the other hand, since ( ω t + dd c ϕ t,j ) n converges to ( ω t + dd c ϕ t ) n , Fatou’s lemma yields dt ∧ ( ω t + dd c ϕ t ) n ≥ e ˙ ϕ t + ϕ t f dV dt in the sense of measures in (0 , T ) × X , whence equality. It follows that ϕ is a solution of theflow (4.11) on [0 , T ) × X . Proposition 4.11. For each t , the solution ϕ t of the Monge-Amp`ere equation (4.11) is con-tinuous on Ω \ D lc .Proof. It follows from Proposition 4.10 that e ˙ ϕ t + ϕ t f ≤ exp C + X a k =1 log( − log | s k | ) − χ − X i log( | s i | ) ! . The proof thus follows from [Dan21, Theorem 3.2]. (cid:3) Step 3: convergence at time zero. Using similar arguments as in the proof of Theorem 2.8,we now check that the solution ϕ t of the Monge-Amp`ere equation (4.11) converges pointwisetowards ϕ as t → + .Arguing as at the beginning of the proof of Theorem 4.1, we can check that ϕ t ≥ u ( t, . ) := e − t ϕ + (1 − e − t ) ψ KE + h ( t ) , ∀ t ∈ (0 + ∞ ) , where ψ KE is the solution of (4.13) and h ( t ) = ne − t (cid:2) ( e t − 1) log( e t − − te t (cid:3) . It thus remains to show that for all x ∈ X , lim t → ϕ t ( x ) ≤ ϕ ( x ).Fix T < + ∞ and consider G := { x ∈ X : u ( T, x ) > − M } , where M > µ ( G ) > µ ( X ) / ψ K E is smooth outside adivisor). Observe that ϕ ( t, x ) ≥ u ( t, x ) ≥ u ( T, x ) > − M for all x ∈ G , t ∈ (0 , T ). Followingthe proof of Theorem 2.8, we obtain as in (2.11) that Z G ϕ t dµ ≤ Z G ϕ dµ + Ct, (4.19)with C > G .Let now u ∈ PSH( X, ω ) be any cluster point of ϕ t as t → 0. We can assume that ϕ t converges to u in L q ( X, dV ) for any q > 1. On the other hand, dµ = Q i | s i | − a i dV X hasdensity f := Q i | s i | − a i ∈ L ploc ( X \ D ). Hence, ϕ t f converges to u f in L ( K ) for any compactsubset K of X \ D . Thus, the claim above ensures that Z G u f dV ≤ Z G ϕ f dV. We infer that u ≤ ϕ almost everywhere on G with respect to f dV , hence everywhere on G .Letting M → + ∞ , we conclude that lim sup t → ϕ t = ϕ on X \ D , hence on the whole X . Step 4: uniqueness of the flow. By the previous steps, we have shown that there exists asolution ϕ to (4.11) with initial data ϕ . This function satisfies the following properties: • ϕ is locally uniformly semi-concave in t , • ( t, x ) → ϕ ( t, x ) is continuous on (0 , + ∞ ) × U , where U := Ω \ D lc , • ϕ t → ϕ pointwise as t → + .We are going to show that such a solution is unique. Let Φ be a solution to (4.11) with thesame properties as above. We shall prove that ϕ ≤ Φ on [0 , + ∞ ) × X , whence equality. Theproof follows step by step the uniqueness result obtained in Section 3.2. Step 4.1. Assume moreover that:(1) ϕ is C in t ,(2) Φ is continuous on [0 , + ∞ ) × U .Since θ is semi-positive we fix c > ω t ≥ cθ for all t . For simplicity we againassume that c = 1. Let χ be a θ/ U , suchthat χ = −∞ on ∂U , normalized by sup X χ = 0. We will use this function in order to applythe clasical maximum principle in U . The standard strategy is to replace ϕ by (1 − λ ) ϕ + λχ .Nevertheless, the time derivative ˙ ϕ t may blow up as t → ρ ∈ PSH( X, θ/ 2) be the unique solution to( θ/ dd c ρ ) n = e ρ dµ, (4.20)normalized by sup X ρ = 0, where dµ = Q i | f i | − a i dV . It follows from [Dan21, Corollary 3.5]that ρ is continuous in U .Fix 0 < T < + ∞ . For ε, λ > w ( t, x ) := (1 − λ ) ϕ ( t, x ) + λ ( ρ ( x ) + χ ( x )) − Φ( t, x ) − εt By Lemma 1.5, this function is upper semi-continuous on [0 , T ] × U . Since ρ + χ is a θ -pshfunction which is continuous in U and tends to −∞ on ∂U , the function w attains its maximumat some point ( t , x ) ∈ [0 , T ] × U . LURIPOTENTIAL MONGE-AMP`ERE FLOWS IN BIG COHOMOLOGY CLASSES 43 We want to show that w ( t , x ) ≤ 0. Assume by contradiction that it is not the case i.e., w ( t , x ) > t > 0. The set K := { x ∈ U : w ( t , x ) = w ( t , x ) } is a compact subset of U since w ( t , x ) tends to −∞ as x → ∂U . The classical maximumprinciple ensures that for all x ∈ K ,(1 − λ ) ∂ t ϕ ( t , x ) ≥ ∂ − t Φ( t , x ) + 3 ε, The partial derivative ∂ t ϕ ( t, x ) is continuous on U by assumption. Since the function t Φ( t, x )is locally uniformly semi-concave, for any t ∈ (0 , T ), the left derivative ∂ − t Φ( t, · ) is upper semi-continuous in Ω (see Proposition 1.10). We can thus find η > D := { x ∈ U : w ( t , x ) > w ( t , x ) − η } is an open set containing K . We have for all x ∈ D (1 − λ ) ∂ t ϕ ( t , x ) > ∂ − t Φ( t , x ) + 2 ε. (4.21)Set u := (1 − λ ) ϕ ( t , . ) + λ ( ρ + χ ) and v = Φ( t , . ). Since ϕ is a pluripotential solution to (4.11),using Lemma 2.6 we infer that( ω t + dd c u ) n ≥ [(1 − λ )( ω t + dd c ϕ t ) + λ ( θ/ dd c ρ )] n ≥ e (1 − λ )( ∂ t ϕ ( t ,. )+ ϕ ( t ,. ))+ λρ dµ ≥ e ∂ − t Φ( t ,. )+(1 − λ ) ϕ ( t ,. )+ λ ( ρ + χ ) dµ. On the other hand, Φ is a pluripotential solution to (4.11), hence ( ω t + dd c v ) n ≤ e ∂ − t Φ( t ,. )+Φ( t ,. ) dµ in the weak sense of measures in D . The last two inequalities yield( ω t + dd c u ) n ≥ e u − v + ε ( ω t + dd c v ) n . We then repeat the arguments in the proof of Proposition 3.2 to obtain a contradiction. There-fore, we must have t = 0, hence(1 − λ ) ϕ + λ ( ρ + χ ) − Φ − εt ≤ λ sup X (( ρ + χ ) − ϕ ) , in [0 , T ] × U . Letting λ → ϕ ≤ Φ + 3 εt in [0 , T ] × U , hence in [0 , T ] × X . We thusconclude the proof by letting ε → T → + ∞ . Step 4.2. We next remove the continuity assumption on Φ in Step 4.1.Fix s > u s ( t, x ) := e − s ϕ t ( x ) + (1 − e − s ) ψ KE ( x ) + h ( s )where h is defined as in Theorem 4.1. We observe that ω t + s = e − s ω t + (1 − e − s ) θ, ∀ t ∈ [0 , + ∞ ) , hence ( ω t + s + dd c u s ) n = (cid:2) e − s ( ω t + dd c ϕ t ) + (1 − e − s )( θ + dd c ψ KE ) (cid:3) n ≥ e e − s ( ∂ τ ϕ t + ϕ t )+(1 − e − s ) ψ KE dµ where the last inequality follows from Lemma 2.6. Since h ( s ) ≤ ω t + s + dd c u s ) n ≥ e ∂ τ u s + u s dµ. On the other hand, ( ω t + s + dd c v s ) n = e ∂ τ v s + v s dµ, where v s ( t, x ) := Φ( t + s, x ) for ( t, x ) ∈ (0 , + ∞ ) × X . By Lemma 4.12 we have u s (0 , x ) ≤ v s (0 , x )for all x ∈ X . Since v s is continuous on [0 , + ∞ ) × U , it follows from Step 4.1 that u s ( t, x ) ≤ v s ( t, x ) , ( t, x ) ∈ [0 , + ∞ ) × X. Letting s → ϕ ≤ Φ on [0 , + ∞ ) × X . Lemma 4.12. For all ( t, x ) ∈ (0 , T ) × X , (4.22) Φ t ( x ) ≥ e − t ϕ ( x ) + (1 − e − t ) ψ KE ( x ) + h ( t ) , where h is the unique solution to the ODE: h ′ ( t ) + h ( t ) = log(1 − e − t ) , h (0) = 0 .Proof. Fix ε > 0, and consider w ε ( t, x ) = e − t Φ ε + (1 − e − t ) ψ K E + h ( t )A direct computation shows that( ω t + ε + dd c w ε ) n = (cid:0) e − t ( ω ε + dd c Φ ε ) + (1 − e − t )( θ + dd c ψ KE ) (cid:1) n ≥ e log(1 − e − t )+ ψ KE µ. where we have used ω ε + dd c Φ ε ≥ 0. Since h ′ ( t ) + h ( t ) = n log(1 − e − t ) we have( ω t + ε + dd c w ε ) n ≥ e ∂ t w ε + w ε µ. It is also clear from the definition that w ε ( t, . ) converges in L ( X ) to w ε (0 , . ) = Φ ε as t → + .On the other hand, w ε is C in t and Φ t + ε is continuous on [0 , + ∞ ) × U . We can thus applyStep 4.1 and obtain w ε ( t, x ) ≤ Φ( t + ε, x ) for ( t, x ) ∈ (0 + ∞ ) × X . Letting ε → (cid:3) Step 4.3. We are now ready to treat the general case by removing the extra assumption on ϕ .For s > t, x ) ∈ (0 , T ) V s ( t, x ) := α s s ϕ ( ts, x ) + (1 − α s ) v ( x ) − C | s − | e − t , where α s is defined as in (4.17), and v is a solution to (4.18). For C > V s is a subsolution to (4.11) that satisfies V s (0 , . ) ≤ ϕ on X . Let { η ε } ε> be a family of smoothing kernels in R approximating the Dirac mass δ . For ε > ϕ ε ( t, x ) := Z R V s ( t, x ) η ε ( s ) ds We proceed as in the proof of Theorem 3.7 to show that ϕ ε − O ( ε ) is again a subsolution andapply the previous step to conclude. Step 5: the long-term behavior of the flow. It remains to establish the convergence at t = + ∞ . We have seen that u ( t, x ) := e − t ϕ + (1 − e − t ) ψ KE ( x ) + h ( t )is a subsolution to (4.11). The comparison principle (see Step 4) yields for any t > x ∈ X , ψ KE ( x ) − C ( t + 1) e − t ≤ u ( t, x ) ≤ ϕ ( t, x )for some uniform constant C > θ = θ + dd c χ is a K¨ahler current we can fix a constant A > ω ≤ (1 + A )˜ θ on Ω, thus ω t ≤ (1 + Ae − t )˜ θ for all t . Set v ( t, x ) := (1 + Ae − t ) ψ KE ( x ) + Be − t LURIPOTENTIAL MONGE-AMP`ERE FLOWS IN BIG COHOMOLOGY CLASSES 45 where B is chosen so that v ≥ ϕ . 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Laboratoire de Math´ematiques D’Orsay, Universit´e Paris-Saclay, CNRS, 91405 Orsay, France Email address : [email protected] Institut de Mathematiques de Toulouse, Universit´e de Toulouse; CNRS, 118 route de Nar-bonne, 31400 Toulouse, France Email address ::