Gromov-Hausdorff limits of immortal Kähler-Ricci flows

GR OMO V-HA USDORFF LIMITS OF IMMOR T AL K ¨ AHLER-RICCI FLO WS MAN-CHUN LEE, V ALENTINO TOSA TTI, AND JUNSHENG ZHANG Abstract. W e show that the normalized K¨ ahler-Ricci flow on a com- pact K¨ ahler manifold with semiample canonical bundle con verges in the Gromov-Hausdorff top ology to the metric completion of the twisted K¨ ahler-Einstein metric on the canonical model, as conjectured by Song- Tian’s analytic mimimal mo del program. 1. Introduction Let ( X n , ω 0 ) b e a compact K¨ ahler manifold, and let ω ( t ) be the solution of the normalized K¨ ahler-Ricci flow (1.1)    ∂ ∂ t ω ( t ) = − Ric( ω ( t )) − ω ( t ) , ω (0) = ω 0 , starting at ω 0 . W e are interested in the case when the solution ω ( t ) is immortal , i.e. it exists for all t ⩾ 0. By a result of Tian-Zhang [41], this happ ens if and only if the canonical bundle K X is nef, i.e. c 1 ( K X ) is a limit of K¨ ahler classes. W e are in terested in the b eha vior of the metrics ω ( t ) as t → ∞ , and refer to the second-named author’s survey on this topic [44]. The Abundance Conjecture in birational geometry , and its natural exten- sion to K¨ ahler manifolds, predicts that if K X is nef then K X is semiample, whic h means that K X is base-p oin t free for some  ⩾ 1. Abundance is curren tly known for n ⩽ 3, and w e will assume from no w on that K X is indeed semiample. In this case, the Ko daira dimension m := κ ( X ) satisfies 0 ⩽ m ⩽ n , and the linear system | K X | for  sufficien tly divisible gives a fib er space (the Iitak a fibration of X ) (1.2) f : X → Y ⊂ P N with K X = f ∗ ( O P N (1) | Y ) , on to a normal pro jectiv e v ariet y Y of dimension dim Y = m . This implies that the generic fib ers of f are Calabi-Y au ( n − m )-folds. The v ariet y Y = Pro j ⊕ k ⩾ 0 H 0 ( X , kK X ) is the canonical mo del of X . W e define D ⊂ Y as the closed subv ariet y given by the union of the singularities of Y together with the discriminant lo cus of f , so that if we call X ◦ := X \ f − 1 ( D ) and Y ◦ := Y \ D , then f : X ◦ → Y ◦ is a prop er holomorphic submersion with connected ( n − m )-dimensional Calabi-Y au fib ers. 1 2 MAN-CHUN LEE, V ALENTINO TOSA TTI, AND JUNSHENG ZHANG In the case when m = 0, w e hav e that Y is a p oint, D = ∅ , X itself is Calabi-Y au, and the behavior of the flo w has been fully understoo d since the 1980s [3]: the flow (1.1) shrinks the metric smo othly to zero as t → ∞ (in particular, ( X, ω ( t )) conv erges to a p oin t in the Gromo v-Hausdorff top ol- ogy), and reparametrizing the flow to ha v e constant v olume, it con v erges smo othly to a Ricci-flat K¨ ahler metric on X . The case m = n is also sp ecial, b ecause in this case the flow is volume non-collapsed, in the sense that V ol( X , ω ( t )) ⩾ c > 0 for all t ⩾ 0. In this case K X is nef and big, and X is known as a minimal mo del of general type. In this case, Tsuji [43] and Tian-Zhang [41] prov ed that the flow conv erges smo othly on compact subsets of X ◦ to a negativ e K¨ ahler-Einstein metric ω can on Y ◦ (pulled back via f : X ◦ → Y ◦ , which is an isomorphism in this case). The remaining cases when 0 < m < n turned out to b e v astly more complicated, due to the fact that the flow is no w volume collapsing as t → ∞ . This setup was first considered by Song-Tian in [33, 34]. In this case, after m uc h w ork b y man y p eople (see [44] for references), it w as recen tly sho wn b y Hein and the first and second-named authors [24] that as t → ∞ , we hav e ω ( t ) → f ∗ ω can smo othly on X ◦ , where ω can is a twisted K¨ ahler-Einstein metric on Y ◦ , constructed b y Song-Tian [34]. In this pap er w e are interested in the global limiting b eha vior of ( X , ω ( t )) as t → ∞ , in the Gromov-Hausdorff top ology . The follo wing basic conjecture is due to Song-Tian [33, P .651], [36, Conjectures 6.2 and 6.3], and it fits into their picture of analytic minimal mo del program: Conjecture 1.1. L et ( X n , ω 0 ) b e a c omp act K¨ ahler manifold with K X semi- ample, and let ω ( t ) b e the solution of the K¨ ahler-Ric ci flow (1.1) . Then as t → ∞ , ( X, ω ( t )) c onver ges in the Gr omov-Hausdorff top olo gy to the metric c ompletion of ( Y ◦ , ω can ) . This is a c omp act metric sp ac e home omorphic to the c anonic al mo del Y . This conjecture has receiv ed muc h attention since it was first p osed in 2006. As men tioned ab o v e, we now kno w [24] that ω ( t ) conv erges lo cally smo othly to f ∗ ω can on X ◦ . Ho w ev er, obtaining reasonable estimates for the flo w near f − 1 ( D ) = X \ X ◦ has pro v ed to b e very challenging. The case m = n w as studied first, and the conjecture is known in low dimensions b y Guo-Song-W einko v e [23] and Tian-Zhang [42] or under the additional assumption of a uniform Ricci low er b ound by Guo [18]. When m = n , Conjecture 1.1 was settled by W ang [50]. The harder case when 0 < m < n has also b een muc h studied. In particular, Conjecture 1.1 was pro v ed by Song-Tian-Zhang [37] when m = 1 and the generic fib ers of f are tori, and by Li and the second-named author [29] in arbitrary dimensions assuming that Y is smo oth and the divisorial comp onen ts of D hav e simple normal crossings (whic h alw ays holds when m = 1). Let us remark that the uniform diameter b ound and existence of subse- quen tial Gromo v-Hausdorff limits of ( X , ω ( t )) was only achiev ed recently in GROMO V-HAUSDORFF LIMITS OF IMMOR T AL K ¨ AHLER-RICCI FLOWS 3 [26], and even more generally in [20, 21, 17, 49] assuming only that K X is nef. Lastly , Sz ´ ekelyhidi [40] has very recen tly sho wn that in the setting of Conjecture 1.1, the metric completion of ( Y ◦ , ω can ) is homeomorphic to Y , is a non-collapsed R CD( − 1 , 2 m )-space, and inside this metric space, Y \ Y ◦ has real Hausdorff codimension at least 2. W e will denote this metric completion b y ( Y , d can ). Our main result finally resolv es Conjecture 1.1 in general: Theorem 1.2. In the setting of Conje ctur e 1.1, we have that as t → ∞ the flow ( X , ω ( t )) c onver ges in the Gr omov-Hausdorff top olo gy to ( Y , d can ) . In p articular, Conje ctur e 1.1 holds. In particular, in the case when m = n , our arguments are different from those in [50], so we obtain a new pro of of that case. F or general m , our pro of follo ws a strategy introduced in [29], where a k ey input is P erel- man’s monotonicit y of the reduced volume, view ed as a parab olic analogue of the Bishop-Gromo v v olume comparison for Riemannian metrics with Ricci curv ature b ounded b elo w. This allows for some control ov er how muc h time minimizing L -geo desics sp end in a neighborho o d of the “bad region” X \ X ◦ = f − 1 ( D ). Ho w ev er, to push this strategy to the end, [29] had to use a result from [15] which shows that when Y is smo oth and the divisorial comp onen ts of D hav e simple normal crossings, ω can is quasi-isometric to a c onic al K¨ ahler metric near these comp onen ts, up to a small logarithmic error. Giv en that no such conical description is av ailable for general Y , in this pap er w e refine the analysis in [29] to establish the existence of L -geo desics that almost av oid a small neighborho od of f − 1 ( D ) (or more precisely , they visit this neighborho o d a controlled num ber of times for a controlled length of time). After carefully c ho osing appropriate neigh b orhoo ds of f − 1 ( D ) (here the RCD prop ert y of ( Y , d can ) is crucial), and combining this almost- a v oidance prop ert y with a H¨ older estimate for d can , we complete the pro of of Theorem 1.2. The pap er is organized as follo ws. In Section 2, we collect v arious results from the literature concerning estimates for ( X , ω ( t )) and ( Y ◦ , ω can ). In Section 3, we establish a H¨ older estimate for the p oten tial of ω can follo wing [19], and then derive a H¨ older estimate for d can using [28]. In Section 4, follo wing [29], we reduce the main theorem to an estimate asserting that the d can -distance can b e b ounded ab o v e by the L -distance. Section 5 con tains the core of the pap er. W e first prov e a general almost-a voidance principle, roughly stating that for suitably small neighborho o ds of f − 1 ( D ), a generic L -geo desic cannot intersect such neigh b orhoo ds to o many times. Then we carefully choose the small neigh b orhoo ds of f − 1 ( D ) to apply the almost- a v oidance principle to, and w e use the R CD property of ( Y , d can ) to establish an upp er volume b ound for these neighborho o ds. Finally com bining the almost-a v oidance principle with the H¨ older estimate obtained in Section 3, w e complete the pro of. 4 MAN-CHUN LEE, V ALENTINO TOSA TTI, AND JUNSHENG ZHANG Ac kno wledgemen ts. W e are grateful to G´ ab or Sz´ ek elyhidi for v ery useful discussions, and to Shouhei Honda, W ensh uai Jiang, Xiaoch un Rong, and Song Sun for email comm unications. The third-named author thanks Yifan Guo for discussions ab out P oincar´ e inequalities on minimal surfaces. The first-named author is supp orted b y Hong Kong RGC grants No. 14300623 and No. 14304225, and an Asian Y oung Scientist F ello wship. The second- named author w as partially supp orted by NSF grant DMS-2404599. 2. Preliminaries In this section we collect some known results from the literature whic h will b e used in our proof of Theorem 1.2. The setup is as given in the In tro duction. Notation. Throughout the pap er, Ψ ( ε 1 , . . . , ε k | a 1 , . . . , a ℓ ) will denote an R -v alued function of these parameters, such that whenever we fix the pa- rameters a 1 , . . . , a ℓ , w e ha ve (2.1) lim ε 1 ,...,ε k → 0 Ψ = 0 . 2.1. The t wisted K¨ ahler-Einstein metric ω can . By assumption, K X is semiample, so w e can choose  ⩾ 1 sufficiently divisible such that K X is base-p oin t free and the linear system | K X | defines the Iitak a fibration f : X → Y ⊂ P N of X . Pulling bac k the co ordinate functions on P N w e obtain a basis { s i } of H 0 ( X , K X ), and using these we can define a smo oth p ositiv e volume form M on X by (2.2) M = ( − 1) ℓn 2 2 X i s i ∧ s i ! 1 ℓ . In the follo wing, we will let ω Y := ω FS | Y b e the restriction of the F ubini- Study metric on P N , which satisfies [ f ∗ ω Y ] = c 1 ( K X ). It is show ed in [34] that there is a unique function ϕ ∈ C 0 ( Y ) ∩ C ∞ ( Y ◦ ) which is ω Y -psh and solv es the complex Monge-Amp ` ere equation (2.3) ( ω Y + i∂ ∂ ϕ ) m = e φ f ∗ ( M ) , p oin t wise on Y ◦ and also globally on Y in the sense of plurip oten tial theory (con tin uity of ϕ is prov ed in [7, 9, 12]). On Y ◦ w e ha ve that (2.4) ω can := ω Y + i∂ ∂ ϕ, is a K¨ ahler metric, which satisfies the twisted K¨ ahler-Einstein equation (2.5) Ric( ω can ) = − ω can + ω WP , where ω WP ⩾ 0 is a W eil-Petersson form on Y ◦ whic h measures the v ariation of the complex structure of the fib ers of f . Of course, in the case when m = n , the fib ers of f ov er Y ◦ are just p oin ts, and in this case ω WP ≡ 0, so that ω can is a K¨ ahler-Einstein metric in this case. GROMO V-HAUSDORFF LIMITS OF IMMOR T AL K ¨ AHLER-RICCI FLOWS 5 2.2. Kno wn results about ω ( t ) . Let ω ( t ) , t ⩾ 0, b e the solution of the K¨ ahler-Ricci flo w (1.1) on X . On the region X ◦ ⊂ X , the b eha vior of the flo w is completely understo o d: after muc h work b y a num b er of p eople, it w as finally pro ved in [24] that (2.6) ω ( t ) → f ∗ ω can , it the smo oth top ology (with resp ect to ω 0 ) on compact subsets of X ◦ and with lo cally uniformly b ounded Ricci curv ature. In this pap er, we will only use the lo cally uniform conv ergence which was prov ed earlier in [46], and more precisely w e will need the following result from [46, Theorem 1.2 and p.685]: given a compact subset K ⊂ X ◦ , on K we hav e (2.7) ∥ ω ( t ) − ( f ∗ ω can + e − t ω SRF ) ∥ C 0 ( K,ω ( t )) → 0 , where ω SRF is a “semi-Ricci flat” closed real (1 , 1)-form on X ◦ whic h restricts to a Ricci-flat K¨ ahler metric on ev ery fib er of f in X ◦ (in the case when m = n , these fib ers are p oin ts, and ω SRF ≡ 0). Observe that given K ⋐ X ◦ , w e can find T > 0 such that f ∗ ω can + e − t ω SRF is a K¨ ahler metric on K for all t ⩾ T , uniformly equiv alen t to f ∗ ω Y + e − t ω 0 . F rom (2.7) we immediately deduce that for ˆ ω t := f ∗ ω can + e − t ω SRF , (2.8) (1 − Ψ( t − 1 | K )) ˆ ω t ⩽ ω ( t ) ⩽ (1 + Ψ( t − 1 | K )) ˆ ω t . The ab o ve b ounds hold on compact subsets of X ◦ . W e also hav e the follo wing b ounds which hold on all of X × [0 , ∞ ): by [35] (and the earlier [52] when m = n ), we know that the scalar curv ature of ω ( t ) is uniformly b ounded, i.e. (2.9) sup X | R ( ω ( t )) | ⩽ C, for all t ⩾ 0, and also that the volume form of ω ( t ) satisfies (2.10) C − 1 e − ( m − n ) t ω n 0 ⩽ ω ( t ) n ⩽ C e − ( m − n ) t ω n 0 , on X × [0 , ∞ ) . By the “parabolic Sc h warz Lemma” estimate [35, Prop osition 2.2] (and also [47, (3.4)] for the case when Y is singular), we know that on X × [0 , ∞ ) , (2.11) ω ( t ) ⩾ C − 1 f ∗ ω Y . W e will also need the follo wing diameter and volume non-collapsing esti- mates, whic h w ere recently established recently in [20, 21, 22, 17, 49]: (2.12) diam( X, ω ( t )) ⩽ C , t ⩾ 0 , (2.13) V ol ω ( t )  B ω ( t ) ( x, r )  ⩾ C − 1 δ r 2 n + δ V ol( X , ω ( t )) , for any 0 < δ < 1 and x ∈ X , 0 < r < diam( X , ω ( t )) . Note that when m = n , one can even take δ = 0 b y [50], although we will not need this, all w e will use is that V ol ω ( t ) ( B ω ( t ) ( x,r ) ) V ol( X,ω ( t )) ⩾ F ( r ) > 0 for all x ∈ X and 0 < r < 1. 6 MAN-CHUN LEE, V ALENTINO TOSA TTI, AND JUNSHENG ZHANG 2.3. Kno wn results ab out ω can . As for the structure of ω can , the following theorem w as recen tly established in [40, Theorem 15] for general 0 < m ⩽ n . The case m = n , i.e., the volume non-collapsing case, was prov ed earlier in [32, Theorem 1.2]. Theorem 2.1. The metric c ompletion of ( Y ◦ , ω can ) is home omorphic to Y and is a non-c ol lapse d R CD( − 1 , 2 m ) -sp ac e, which we wil l denote by ( Y , d can ) . Mor e over, (2.14) dim H ( Y \ Y ◦ ) ⩽ 2 m − 2 . F rom the RCD prop erty it follo ws in particular that there exists κ > 0 suc h that for an y y ∈ Y and 0 < r < 1, w e ha v e (2.15) H 2 m ( B d can ( y , r )) ⩾ κr 2 m , and also that for any sequence of op en sets V ε of Y suc h that T ε V ε = D , w e ha ve (2.16) lim ε → 0 H 2 m ( V ε ) = 0 . Remark 2.2. When m = n , it is kno wn that ( Y , d can ) is a non-collapsed Ricci limit space by [32]. Moreov er, in general combining [37, Theorem 1.3] and [40, Theorem 4], w e know that ( Y , d can ) is a also Ricci limit space. Ho w ever, when m < n , we do not kno w whether ( Y , d can ) is a non-collapsed Ricci limit space in general, although this is true when Y is smo oth, see e.g. [29, Prop osition 2.1]. 3. H ¨ older bound on the potential and dist ance function of ω can In this section we prov e H¨ older b ounds for the K¨ ahler p oten tial of ω can and for the distance function d can . The pro of of the p oten tial H¨ older b ound follo ws closely the recen t work of Guo-Ko lo dziej-Song-Sturm [19], while the argumen t to deduce from this the H¨ older b ound for d can follo ws the metho d disco v ered b y Li [28]. In [19] the potential H¨ older bound was prov ed for families of p olarized K¨ ahler manifolds and for smo othable pro jective v ari- eties, under natural assumptions. Our observ ation is that using the R CD prop ert y of ( Y , d can ), we can run their argumen ts with only small c hanges ev en though in our case Y is singular and need not b e smo othable. F or the reader’s con v enience, we provide most of the details. 3.1. H¨ older b ound for the p oten tial of ω can . Recall that on Y ◦ w e ha v e the K¨ ahler metric ω can = ω Y + i∂ ∂ ϕ , which solv es the complex Monge- Amp ` ere equation in (2.3), and ϕ extends to a con tin uous function on Y with sup Y ϕ = 0. Since the righ t hand side of (2.3) is known to b e in L p ( Y , ω m Y ) for some p > 1 (see [34]), if Y w as smo oth then [25] would imply that ϕ is H¨ older contin uous on Y . It is widely b elieved that this statement still holds for solutions of such Monge-Amp ` ere equations on normal compact K¨ ahler analytic spaces. The main result of this subsection is that this do es indeed hold for our ω can : GROMO V-HAUSDORFF LIMITS OF IMMOR T AL K ¨ AHLER-RICCI FLOWS 7 Theorem 3.1. The function ϕ on Y is H¨ older c ontinuous (with r esp e ct to the distanc e function d Y ), i.e. we have (3.1) | ϕ ( x ) − ϕ ( y ) | ⩽ C d Y ( x, y ) α , for some c onstants C > 0 and 0 < α < 1 and for al l x, y ∈ Y . Here d Y denotes the asso ciated distance function induced from ω Y . F or ease of notation, we will denote by L = O P N (1) | Y , and up to replacing it b y a suitable p ositiv e m ultiple (without relab eling), we ma y assume that for an y k ,  ∈ N > 0 , (3.2) Sym k H 0 ( Y , L ) → H 0 ( Y , k L ) is surjectiv e. Recall that ω Y = ω FS | Y , and let also h Y = h FS | Y and define a singular Hermitian metric on L by h L := h Y e − φ , whose curv ature form is ω can . Then we obtain L 2 -Hermitian metrics Hilb k on global holomorphic sections H 0 ( Y , k L ) for an y k ∈ N > 0 b y (3.3) ⟨ s, σ ⟩ Hilb k := ˆ Y ⟨ s, σ ⟩ h k L ( k ω can ) m . The densit y of states function is giv en b y (3.4) ρ k = X α | s α | 2 h k L , where { s α } is any L 2 -orthonormal basis of H 0 ( X , k L ) with resp ect to Hilb k . With these preparations, w e hav e the following uniform b ounds for the densit y of states function: Theorem 3.2. Ther e exists an r ∈ N > 0 and A > 0 such that for any k ∈ N > 0 and x ∈ Y , we have (3.5) A − 1 ⩽ ρ rk ( x ) ⩽ A. Pr o of. The upp er b ound for the densit y of states is standard in the smo oth setting and follows from Moser iteration. In the present singular setting, it can b e pro v ed using the RCD prop ert y of ( Y , ω can ), see [39, Prop osition 19]. The low er b ound is essentially con tained in [30, Prop osition 3.1], com bined with an observ ation in [51, Theorem 1.5]. As discussed in [40, p.11], the argumen t in [30, Prop osition 5.1] implies that any metric cone V that arises as a p oin ted Gromov-Hausforff limit of ( Y , k i ω i , x i ) for some k i → ∞ and p oin ts x i ∈ Y , satisfies that V \ R ε ( V ) has zero 2-capacity for any ε > 0, where R ε is defined in analogy with (5.26) b elo w. Therefore [30, Prop osition 3.1] applies to the spaces { ( Y , k ω can ) | k ∈ N > 0 } , and it shows that there exist K ∈ N > 0 and b > 0 suc h that for an y k ∈ N > 0 and x ∈ Y , there exists an in teger  =  ( x, k ) ∈ [1 , K ] such that (3.6) ρ ℓ,kL ( x ) ⩾ b. Here we use ρ ℓ,kL to emphasize that the density of states function for the line bundle k L , which clearly satisfies (3.7) ρ ℓ,kL = ρ ℓk,L . 8 MAN-CHUN LEE, V ALENTINO TOSA TTI, AND JUNSHENG ZHANG If we let r = K !, then the lo w er b ound for ρ rk ,L follo ws from (3.6), (3.7) and the rough lo wer b ound in [10, Lemma 3.1] for the densit y of states function when raising p o w ers. □ Replacing L by r L , we ma y assume in the following that r = 1. Since Y is a pro jective v ariety , we know that Y reg is an essen tially Stein manifold, i.e. there exists an analytic hypersurface V ⊂ Y reg suc h that Y reg \ V is Stein. T o see this, w e can choose an effective ample divisor D suc h that Y sing ⊂ supp( D ) and let V = supp( D ) \ Y sing . Then (3.8) Y reg \ V = Y \ supp( D ) is Stein. By [48, Theorem 2.1, Theorem 2.2], w e kno w that the Skoda division theorem still holds on Y reg and since Y is normal, an y holomorphic section in H 0 ( Y reg , M ) where M is a line bundle on Y extends to a section in H 0 ( Y , M ). Then one can argue exactly as in [19, Theorem 5] (see also [27, Prop osition 7]), using the b ound on the density of states function to get the follo wing: Theorem 3.3. L et  0 ,  1 ⩾ 1 and k ⩾ ( m + 2 +  1 )  0 . L et s 0 , · · · , s N ℓ 0 b e an orthonormal b asis of H 0 ( Y ,  0 L ) with r esp e ct to Hilb ℓ 0 . Then for any U ∈ H 0 ( Y , k L ) , we c an write (3.9) U = N ℓ 0 X α 1 ,...,α ℓ 1 =0 U ( α 1 , . . . , α ℓ 1 ) s α 1 · · · s α ℓ 1 , with the estimate (3.10) ∥ U ( α 1 , · · · , α ℓ 1 ) ∥ 2 Hilb k − ℓ 1 ℓ 0 ⩽ ( m +  1 )! m !  1 ! ( k −  1  0 ) m k m A 2 m +2+ ℓ 1 ∥ U ∥ 2 Hilb k . Let { s 0 , · · · , s N 1 } b e an orthonormal basis of H 0 ( Y , L ) with resp ect to Hilb 1 , where N 1 = N . By increasing A if necessary , we hav e that (3.11) − A ⩽ ϕ ⩽ 0 , on Y . Then, as in [19], we can define the approximating p oten tials (3.12) ϕ k = 1 k log N k X i =0 | σ i | 2 h k Y ! = 1 k log N k X i =0 | σ i | 2 h k L ! + ϕ, where { σ i } is an orthonormal basis of H 0 ( Y , k L ) with respect to Hilb k . Thanks to the uniform b ound for the density of states function in T heorem 3.2, w e kno w that (3.13) ∥ ϕ k − ϕ ∥ L ∞ ( Y ) ⩽ log A k . In the following, all connections on Y reg are taken with respect to ω Y and h Y . GROMO V-HAUSDORFF LIMITS OF IMMOR T AL K ¨ AHLER-RICCI FLOWS 9 Lemma 3.4. Ther e exists a B > 0 such that for any k ∈ N > 0 and σ ∈ H 0 ( Y , k L ) , and x ∈ Y reg we have (3.14) ∥∇ σ ∥ L ∞ ( Y reg ,ω Y ,h k Y ) ⩽ B k ∥ σ ∥ Hilb k . Pr o of. After rescaling, we can assume ∥ σ ∥ Hilb k = 1. When k = 1 , after a unitary transformation we can assume that σ is the restriction of E 0 , the first co ordinate function on C N 1 +1 . Then we hav e that for any x ∈ Y reg , (3.15) |∇ σ | ω Y ,h Y ( x ) ⩽ ∥∇ E 0 ∥ L ∞ ( P N 1 ,ω FS ,h FS ) ⩽ C 1 . F or general k , since w e hav e the surjective map (3.2), w e can assume that σ is the restriction of a homogeneous p olynomial p = p ( E 0 , · · · , E N 1 ). Since any t w o norms on a finite dimensional vector space are equiv alent, the co efficien ts of p are b ounded b y a uniform constan t C k . Then a similar argumen t as in (3.15) giv es (3.16) ∥∇ σ ∥ L ∞ ( Y reg ,ω Y ,h k Y ) ⩽ C k = C k ∥ σ ∥ Hilb k . W e thus need to show that we can take C k = B k . Clearly we may assume without loss that k > m + 2. Recall that we hav e fixed { s 0 , · · · , s N 1 } , an orthonormal basis of H 0 ( Y , L ) with resp ect to Hilb 1 . By Theorem 3.2 and the normalization sup Y ϕ = 0, w e kno w that (3.17) ∥ s i ∥ L ∞ ( Y reg ,h Y ) ⩽ ∥ s i ∥ L ∞ ( Y reg ,h L ) ⩽ A, 0 ⩽ i ⩽ N 1 . W e then apply Theorem 3.3 with (3.18)  0 = 1 ,  1 = k − m − 2 , to get that (3.19) σ = X σ ( α 1 , · · · α k − m − 2 ) s α 1 · · · s α k − m − 2 , with σ ( α 1 , · · · α k − m − 2 ) ∈ H 0 ( Y , ( m + 2) L ) and (3.20) ∥ σ ( α 1 , · · · α k − m − 2 ) ∥ 2 Hilb m +2 ⩽ ( k − 2)! m !( k − m − 2)! ( m + 2) m k m A k + m ⩽ C ( m, A ) A k and then Theorem 3.2 giv es (3.21) ∥ σ ( α 1 , · · · α k − m − 2 ) ∥ 2 L ∞ ( Y reg ,ω Y ,h m +2 Y ) ⩽ C ( m, A ) A k +1 . Then b y the non-effectiv e b ound (3.16), we know that (3.22) |∇ σ ( α 1 , · · · α k − m − 2 ) | ⩽ C ( m, A ) A k/ 2 . The k ey p oin t here is that the constan t C ( m, A ) is indep enden t of k . Then using (3.15), (3.17), (3.19), (3.21) and (3.22), and b eing wasteful with the 10 MAN-CHUN LEE, V ALENTINO TOSA TTI, AND JUNSHENG ZHANG p o w ers of k , we can estimate (3.23) ∥∇ σ ∥ L ∞ ( Y reg ,ω Y ,h k Y ) ⩽ C ( m, A ) N k − m − 2 1 ( A k · A k − m − 2 + A k +1 · A k − m − 2 · C 1 ( k − m − 2)) ⩽ C ( m, A ) N k 1 ( A + 1) 2 k ⩽ B k , b y c ho osing B large. □ Lemma 3.5. Ther e exists K > 0 such that for any r ∈ N ⩾ 2 , k = ( m + 2) r and U ∈ H 0 ( Y , k L ) , we have (3.24) ∥∇ U ∥ L ∞ ( Y reg ,ω Y ,h k Y ) ⩽ k K ∥ U ∥ Hilb k Pr o of. F or r ⩾ 2, we tak e  0 = ( m + 2) r − 2 and  1 = m 2 + 3 m + 2 and k = ( m + 2) r , whic h satisfies (3.25) k = ( m + 2 +  1 )  0 . Let s 0 , · · · , s N ℓ 0 b e an orthonormal basis of H 0 ( Y ,  0 L ) with resp ect to Hilb ℓ 0 . F or a giv en p oin t x ∈ Y reg , by considering the kernel of the ev al- uation map at x and the 1-jet map at x , together with their orthogonal complemen ts, we may choose the basis { s i } suc h that s 0 ( x )  = 0, the sec- tions { s i | 1 ⩽ i ⩽ m } v anish at x to first order, and the remaining sections v anish at x to at least second order. W e can then apply Theorem 3.3, and write (3.26) U = m X j =0 U j s ℓ 1 − 1 0 s j , and so (3.27) ∇ U ( x ) = ∇   m X j =0 U j s ℓ 1 − 1 0 s j   ( x ) with (3.28) ∥ U j ∥ Hilb ( m +2) r − 1 ⩽  ( m 2 + 4 m + 2)! m !( m 2 + 3 m + 2)! A m 2 +5 m +4  1 2 ∥ U ∥ Hilb ( m +2) r =: C ( m, A ) ∥ U ∥ Hilb ( m +2) r . Expanding (3.27), we get that at the p oin t x , w e ha ve (3.29) ∇ U = s ℓ 1 0 ∇ U 0 + U 0  1 s ℓ 1 − 1 0 ∇ s 0 + m X j =1 U j s ℓ 1 − 1 0 ∇ s j . F or simplicity of notation in the following, we omit the metrics, but it will b e understo o d that all norms are tak en with resp ect to ω Y and h Y , unless GROMO V-HAUSDORFF LIMITS OF IMMOR T AL K ¨ AHLER-RICCI FLOWS 11 sp ecified otherwise. W e can estimate (3.30) |∇ U | ( x ) ⩽ A ℓ 1 |∇ U 0 | + C ( m, A )(  1 + 1) A ℓ 1 − 1 m X j =0 |∇ s j |∥ U j ∥ Hilb ( m +2) r . Then com bining Lemma 3.4 with (3.28), w e get that (3.31) |∇ U | ( x ) ⩽ D B ( m +2) r − 1 ∥ U ∥ Hilb ( m +2) r , where the constan t (3.32) D = D ( m, A ) = C ( m, A ) A ℓ 1 (1 + m (  1 + 1) A − 1 ) ⩾ 1 . Then we prov e b y induction on t that for any 1 ⩽ t ⩽ r − 2 and U ∈ H 0 ( Y , ( m + 2) r L ), w e ha ve (3.33) ∥∇ U ∥ L ∞ ( Y reg ,ω Y ,h ( m +2) r ) Y ) ⩽ D t B ( m +2) r − t ∥ U ∥ Hilb ( m +2) r . The case t = 1 w as just prov ed by the previous argument. W e assume the claim is true for a given t ⩾ 1, and then prov e it holds for t + 1. W e consider an r with r − 2 ⩾ t + 1 and U ∈ H 0 ( Y , ( m + 2) r L ). W e hav e (3.27) and (3.28) as b efore with U j ∈ H 0 ( Y , ( m + 2) r − 1 L ) and with the normalization (3.34) ∥ U ∥ Hilb ( m +2) r = 1 . Since r − 1 − 2 ⩾ t , w e can apply the inductiv e assumption to U j to get (3.35) ∥∇ U j ∥ L ∞ ( Y reg ,ω Y ,h ( m +2) r − 1 Y ) ⩽ D t B ( m +2) r − 1 − t ∥ U j ∥ Hilb ( m +2) r − 1 . Similarly we hav e s j ∈ H 0 ( Y , ( m + 2) r − 2 ), r − 2 − 2 ⩾ t − 1 and hence by the inductiv e assumption w e obtain that (3.36) ∥∇ s j ∥ L ∞ ( Y reg ,ω Y ,h ( m +2) r − 2 Y ) ⩽ D t − 1 B ( m +2) r − 1 − t ∥ s j ∥ Hilb ( m +2) r − 2 . Then combining (3.30), (3.35) and (3.36), we obtain that any given p oin t x ∈ Y reg , w e ha ve (3.37) |∇ U | ( x ) ⩽ C ( m, A )  A ℓ 1 D t B ( m +2) r − 1 − t + m (  1 + 1) A ℓ 1 − 1 D t − 1 B ( m +2) r − 1 − t  ⩽ C ( m, A ) A ℓ 1 (1 + A − 1 D − 1 m (  1 + 1)) D t B ( m +2) r − 1 − t ⩽ D t +1 B ( m +2) r − 1 − t . This prov es (3.33). T aking t = r − 2, w e obtain the desired inequality stated in the Lemma. □ Pr o of of The or em 3.1. W e take k = ( m + 2) r for some r ⩾ 2 to b e sp ecified b elo w. As a consequence of Lemma 3.5 and the definition of ϕ k , we obtain that (3.38) ∥∇ ϕ k ∥ L ∞ ( Y reg ) ⩽ Ak K − 1 . 12 MAN-CHUN LEE, V ALENTINO TOSA TTI, AND JUNSHENG ZHANG Using (3.13) and (3.38), giv en an y tw o distinct p oin ts x, y ∈ Y reg w e can then estimate (3.39) | ϕ ( x ) − ϕ ( y ) | ⩽ | ϕ ( x ) − ϕ k ( x ) | + | ϕ k ( x ) − ϕ k ( y ) | + | ϕ k ( y ) − ϕ ( y ) | ⩽ 2 log A k + Ak K − 1 d Y ( x, y ) . Cho osing (3.40) r =     log  d Y ( x, y ) − 1 2 K  log( m + 2)     , then giv es (3.41) | ϕ ( x ) − ϕ ( y ) | ⩽ C 0  d Y ( x, y ) 1 2 K + d Y ( x, y ) 1 2 + 1 2 K  , for some uniform constan t C 0 , whic h pro ves (3.1). □ 3.2. H¨ older b ound for d can . In this subsection, we use the H¨ older b ound in Theorem 3.1 and the metho d of Li [28] to prov e the following: Theorem 3.6. Then ther e exist c onstants C > 0 and α ∈ (0 , 1] such that for al l x, y ∈ Y , (3.42) C − 1 d Y ( x, y ) ⩽ d can ( x, y ) ⩽ C d Y ( x, y ) α . In particular, the tw o metric spaces ( Y , d Y ) and ( Y , d can ) are bi-H¨ older equiv alent. Pr o of. The first inequality in (3.42) is a well-kno wn and simple consequence of the Sch warz Lemma estimate (2.11) together with the con v ergence in (2.6), so it suffices to pro v e the second inequalit y . Recall that we hav e an embedding Y  → P N and w e hav e defined ω Y := ω FS | Y , and d Y is the in trinsic distance function on Y defined by the metric ω Y . There is also an “extrinsic” distance function d ext on Y , whic h is defined b y restricting to Y the distance function d FS of the F ubini-Study metric on P N . Clearly w e hav e d ext ⩽ d Y , and as a simple consequence of the Lo jasiewicz inequality one has (3.43) d Y ⩽ C d α ext , for some C, α > 0, see [16, Prop osition 4.6]. Thanks to this and Theorem 3.1, w e see that ϕ is also H¨ older con tin uous with respect to d ext , with H¨ older exp onen t that we still denote by α (up to shrinking it). In the following, metric balls are all taken with resp ect to the distance function d ext , which means that they are equal to F ubini-Study metric balls in tersected with Y . W e then follo w closely [28]. F or each fixed x 0 ∈ Y , w e denote u ( y ) := d can ( x 0 , y ) so that (3.44) |∇ u | 2 g Y ⩽ tr ω Y ω can . GROMO V-HAUSDORFF LIMITS OF IMMOR T AL K ¨ AHLER-RICCI FLOWS 13 It follows from [38, Theorem B, Prop osition 1, Chapter I I] that there is C > 0 such that for all 0 < r < 1 and all x ∈ Y we hav e (3.45) C − 1 r 2 m ⩽ V ol( B ( x, r ) , ω m Y ) ⩽ C r 2 m . Then, for r < 1, restricting to Y a cutoff function on P N , w e obtain a smo oth function χ on Y whic h is supp orted on B ( x, 2 r ) and identically equal to 1 on B ( x, r ), such that (3.46) √ − 1 ∂ ∂ χ ⩽ 10 r 2 ω Y . Recalling that ω can = ω Y + i∂ ∂ ϕ , integrating b y parts, and using (3.44), (3.45), (3.46) and the d ext -H¨ older b ound for ϕ , w e get that (3.47) ˆ B ( x,r ) |∇ u | 2 g Y ω m Y = ˆ B ( x,r ) (tr ω Y ω can ) ω m Y = m ˆ B ( x,r ) ω can ∧ ω m − 1 Y ⩽ m ˆ Y χω can ∧ ω m − 1 Y ⩽ m V ol( B ( x, 2 r ) , ω m Y ) + m ˆ Y χ √ − 1 ∂ ∂ ϕ ∧ ω m − 1 Y = C r 2 m + m ˆ Y  ϕ − inf B ( x, 2 r ) ϕ  √ − 1 ∂ ∂ χ ∧ ω m − 1 Y ⩽ C r 2 m + C  osc B ( x, 2 r ) ϕ  r − 2 V ol( B ( x, 2 r ) , ω m Y ) ⩽ C r 2 m − 2+ α . By [2, Theorem 3] and the triangle inequality , w e kno w there exists C > 0 and 0 < r 0 < 1 suc h that for an y x ∈ Y and any r ⩽ r 0 w e ha ve (3.48) ˆ B ( x,r ) | u − u x,r | 2 m 2 m − 1 ω m Y ! 2 m − 1 2 m ⩽ C ˆ B ( x,C r ) |∇ u | g Y ω 2 m Y , where (3.49) u x,r := 1 V ol( B ( x, r ) , ω m Y ) ˆ B ( x,r ) uω m Y . 14 MAN-CHUN LEE, V ALENTINO TOSA TTI, AND JUNSHENG ZHANG Then, giv en an y x ∈ Y reg and 2 r ⩽ r 0 , using (3.45), (3.47) and (3.48), w e can estimate (3.50) | u x,r − u x, 2 r | ⩽ 1 V ol( B ( x, r ) , ω m Y ) ˆ B ( x,r ) | u − u x, 2 r | ω m Y ⩽ 1 V ol( B ( x, r ) , ω m Y ) ˆ B ( x, 2 r ) | u − u x, 2 r | ω m Y ⩽ V ol( B ( x, 2 r ) , ω m Y ) 1 2 m V ol( B ( x, r ) , ω m Y ) ˆ B ( x, 2 r ) | u − u x, 2 r | 2 m 2 m − 1 ω m Y ! 2 m − 1 2 m ⩽ C r 1 − 2 m ˆ B ( x,C r ) |∇ u | g Y ω 2 m Y ⩽ C r 1 − m ˆ B ( x,C r ) |∇ u | 2 g Y ω 2 m Y ! 1 2 ⩽ C r α 2 . F or i ⩾ 1 let u i := u x, 2 − i r , so that (3.50) giv es (3.51) | u i +1 − u i | ⩽ C r α/ 2 2 − iα/ 2 , and since lim i →∞ u i = u ( x ) (using that u is contin uous and x ∈ Y reg ), we obtain | u ( x ) − u x,r | ⩽ ∞ X i =1 | u i +1 − u i | ⩽ C r α/ 2 ∞ X i =1 2 − iα/ 2 = C ′ r α/ 2 . (3.52) Giv en then x, y ∈ Y reg with d ext ( x, y ) =: r/ 2 and 0 < r ⩽ r 0 , w e obtain | u ( x ) − u ( y ) | ⩽ | u ( x ) − u x,r | + | u y ,r − u ( y ) | + | u x,r − u y ,r | ⩽ C r α/ 2 + | u x,r − u y ,r | , (3.53) and to estimate the last term observe that B ( x, r / 2) ⊂ B ( x, r ) ∩ B ( y , r ), and use (3.45) and (3.50) again to estimate | u x,r − u y ,r | ⩽ C r − 2 m ˆ B ( x,r/ 2) | u − u x,r | ω 2 m Y + C r − 2 m ˆ B ( x,r/ 2) | u − u y ,r | ω 2 m Y ⩽ C r − 2 m ˆ B ( x,r ) | u − u x,r | ω 2 m Y + C r − 2 m ˆ B ( y,r ) | u − u y ,r | ω 2 m Y ⩽ C r α/ 2 , (3.54) whic h shows that | u ( x ) − u ( y ) | ⩽ C d ext ( x, y ) α/ 2 for all x, y ∈ Y reg , and by con tin uity for all x, y ∈ Y . Since u ( x 0 ) = 0, this pro v es (3.42). □ GROMO V-HAUSDORFF LIMITS OF IMMOR T AL K ¨ AHLER-RICCI FLOWS 15 4. Reduction to the key estima te In this section, we reduce the pro of of Theorem 1.2 to establishing a k ey estimate, stated in Prop osition 4.4, which is a sharp upp er b ound for d can in terms of Perelman’s L -distance. This reduction follo ws closely the argumen ts in [29], with some ca veats due to the fact that in [29] the space Y w as assumed to b e smo oth, while here we allow it to b e singular. The pro of of Prop osition 4.4 will then b e given in the next section. 4.1. Reduction to an upp er of d can b y d t . In the follo wing, let d t denote the distance function on X defined b y the metric ω ( t ). T o pro ve Theorem 1.2 w e need to sho w that ( X , d t ) conv erges in the Gromov-Hausdorff top ology to ( Y , d can ). Recall that we hav e the analytic subv ariet y D ⊂ Y so that Y ◦ = Y \ D is smo oth and f is a submersion ov er Y ◦ . Giv en ε > 0 we define (4.1) V ε := { y ∈ Y | d can ( D , y ) < ε } , ˜ V ε := f − 1 ( V ε ) . and let ( Y \ V ε , d can ) denote the restriction of the metric d can from Y to Y \ V ε , and similarly let ( X \ ˜ V ε , d t ) b e restriction of d t from X to X \ ˜ V ε . W e b egin with the follo wing simple observ ation: Lemma 4.1. We have (4.2) d GH (( Y , d can ) , ( Y \ V ε , d can )) = Ψ( ε ) . (4.3) d GH (( X , d t ) , ( X \ ˜ V ε , d t ))) = Ψ( ε, t − 1 ) . Pr o of. W e wan t to sho w the natural inclusion map giv es the desired Gromo v- Hausdorff approximations. F or (4.2), it is enough to sho w that Y \ V ε is Ψ( ε )-dense inside ( Y , d can ), i.e. (4.4) d can ( y , ∂ V ε ) = Ψ( ε ) , for an y y ∈ V ε . This follo ws from the v olume non-collapsing of d can pro v ed in (2.15), and the fact that the v olume of V ε go es to 0, see (2.16). F or (4.3), it is enough to sho w that X \ ˜ V ε is Ψ( ε, t − 1 )-dense inside ( X, d t ), i.e. (4.5) d t ( x, ∂ ˜ V ε ) = Ψ( ε, t − 1 ) , for an y x ∈ ˜ V ε . Using (2.10), (2.3) (and the b oundedness of the function ϕ that app ears there) w e can then estimate (4.6) V ol( ˜ V ε , ω ( t ) n ) V ol( X , ω ( t ) n ) ⩽ C ˆ ˜ V ε ω n 0 ⩽ C ˆ ˜ V ε M = C ˆ V ε f ∗ ( M ) ⩽ C ˆ V ε ω m can , for a constan t C indep enden t of t and ε . Then (4.5) follo ws from this volume upp er b ound for ˜ V ε together with the v olume non-collapsing estimate for ω ( t ) in (2.13). □ The next lemma is also elemen tary: 16 MAN-CHUN LEE, V ALENTINO TOSA TTI, AND JUNSHENG ZHANG Lemma 4.2. Given ε, δ > 0 , ther e exists ε ′ = ε ′ ( ε, δ ) such that for any x, y ∈ Y \ V ε , ther e exists a smo oth curve γ c ontaine d in Y \ V ε ′ , c onne cting x and y such that (4.7) length ω can ( γ ) ⩽ d can ( x, y ) + δ. Pr o of. Given δ > 0, by the definition of metric completion, we know that there exists δ ′ > 0 suc h that giv en p oin ts x, y ∈ Y \ V ε/ 2 , there exists ε ′ = ε ′ ( x, y ) > 0 suc h that for any p oin ts x ′ ∈ B d can ( x, δ ′ ) and y ′ ∈ B d can ( y , δ ′ ), there exists a smo oth curve γ x ′ ,y ′ con tained in Y \ V ε ′ , connecting x ′ and y ′ suc h that (4.8) length( γ x ′ ,y ′ ) ⩽ d can ( x ′ , y ′ ) + δ Since Y \ V ε × Y \ V ε is compact, we can co v er it b y finitely many balls B d can ( x i , δ ′ ) × B d can ( y i , δ ′ ) for i = 1 , · · · , N . W e then define (4.9) ε ′ := min { ε ( x i , y i ) | i = 1 , . . . , N } . It is clear that this c hoice satisfies the desired prop ert y . □ Then w e sho w that the Theorem 1.2 can b e reduced to show an upp er b ound of the d can -distance. Lemma 4.3. Supp ose the fol lowing is true: given any smal l ε > 0 and two p oints p, q ∈ X \ ˜ V ε , we have (4.10) d can ( f ( p ) , f ( q )) ⩽ d t ( p, q ) + Ψ( t − 1 | ε ) . In this c ase, we then have (4.11) lim t →∞ d GH (( X , d t ) , ( Y , d can )) = 0 Pr o of. Thanks to Lemma 4.1, to prov e (4.11) is enough to show that (4.12) d GH (( X \ ˜ V ε , d t ) , ( Y \ V ε , d can )) = Ψ( t − 1 | ε ) . T o show this, we will show that the fibration map (4.13) f : X \ ˜ V ε → Y \ V ε is the Gromov-Hausdorff approximation that w e wan t. Indeed, by Lemma 4.2, and the C 0 loc -con v ergence of ω ( t ) to ω can on f − 1 ( Y ◦ ) in (2.6), w e know that for an y p, q ∈ X \ ˜ V ε , w e ha ve (4.14) d t ( p, q ) ⩽ d can ( f ( p ) , f ( q )) + Ψ( t − 1 | ε ) . Com bining this with (4.10) then giv es (4.15) | d t ( p, q ) − d can ( f ( p ) , f ( q )) | = Ψ( t − 1 | ε ) , and since clearly f is surjectiv e, w e obtain the desired Gromo v-Hausdorff appro ximation, whic h shows (4.12). □ GROMO V-HAUSDORFF LIMITS OF IMMOR T AL K ¨ AHLER-RICCI FLOWS 17 4.2. F urther reduction to an upp er b ound of d can b y the L -distance. In the previous subsection, Theorem 1.2 has b een reduced to proving (4.10). In this section, which still follows closely [29], w e show that (4.10) w ould follo w from the “key estimate” in (4.23), whic h is a sharp upp er b ound for d can in terms of Perelman’s L -distance. The key estimate will b e pro v ed in the next section. As in [29], we reparametrize the flow in a standard w ay . Giv en T ≫ 1, let g ( t ) b e the Riemannian metric defined by ω ( t ), and define (4.16) ˜ g ( s ) := e t − T g ( t ) , s := 1 2  e t − T − 1  , whic h solv e the unnormalized Ricci flo w (4.17) ∂ ∂ s ˜ g ( s ) = − 2 Ric( ˜ g ( s )) , s ⩾ s T := 1 2  e − T − 1  , with ˜ g (0) = g ( T ). W e clearly hav e (4.18) g ( t ) = ˜ g ( s ) 1 + 2 s , t = T + log(1 + 2 s ) , and (2.9) b ecomes (4.19) sup X | R ( ˜ g ( s )) | ⩽ C 1 + 2 s , s ⩾ s T . W e let τ = − s , so that the metrics ˜ g ( τ ) solv e the backw ards Ricci flow (4.20) ∂ ∂ τ ˜ g = 2 Ric( ˜ g ) , ˜ g | τ =0 = g ( T ) . F ollowing Perelman, one defines the L -length of a curve γ ( τ ) in X (which, follo wing P erelman, we think of as the curv e ( γ ( τ ) , τ ) in space-time) by (4.21) L ( γ ) = ˆ √ τ ( R ( ˜ g ( τ )) + | ∂ τ γ | 2 ˜ g ( τ ) ) dτ , and the L -distance for tw o p oints in space-time as the infim um of such. W e will use the follo wing con ven tion. Fix tw o parameters (4.22) T ≫ 1 ≫ τ > 0 . By an L -geo desic γ in space-time from ( p, 0) T to ( q, τ ) T , w e mean that, after p erforming the reparametrization (4.16), the curve γ is an L -geo desic from ( p, 0) to ( q , − τ ) in the space-time of the standard Ricci flo w (4.17). W e will denote b y L T ( q , τ ) the L -distance b et w een ( p, 0) T and ( q , τ ) T . The follo wing is then the k ey estimate that w e shall prov e: Prop osition 4.4 (Key estimate) . F or any two p oints p, q ∈ X \ ˜ V ε , we have (4.23) L T ( q , τ ) ⩾ 1 2 √ τ d can ( f ( p ) , f ( q )) 2 + Ψ( T − 1 | ε, τ ) + Ψ( ¯ τ | ε ) . F ollowing the argument in [29], we no w show that Prop osition 4.4 implies Theorem 1.2: 18 MAN-CHUN LEE, V ALENTINO TOSA TTI, AND JUNSHENG ZHANG Pr o of of The or em 1.2, assuming Pr op osition 4.4. Thanks to Lemma 4.3, it suffices to show that (4.10) holds. F or this, w e first observe the follo wing easy upp er b ound of the L -distance. By Lemma 4.2, given p, q ∈ X \ ˜ V ε , we can find a curv e γ contained in Y \ V ε ′ connecting f ( p ) and f ( q ) such that (4.24) length ω can ( γ ) ⩽ d can ( f ( p ) , f ( q )) + Ψ( ε ′ | ε ) . Then using the fib er bundle structure of f and the C 0 loc con v ergence of ω ( t ), arguing as in [45, Proof of Lemma 9.1], we can construct a curv e ˜ γ contained in X ◦ , connecting p and q , such that (4.25) length ω ( T ) ( ˜ γ ) ⩽ d can ( f ( p ) , f ( q )) + Ψ( ε ′ | ε ) + Ψ( T − 1 | ε ′ ) . Note that this curv e ˜ γ dep ends on the parameter ε ′ and for simplicity of notations, in the follo wing, w e omit the dep endence on ε ′ . W e parametrize ˜ γ by τ ∈ [0 , τ ] suc h that | ∂ τ ˜ γ | ˜ g ( τ ) = A 2 √ τ τ for all 0 < τ ⩽ τ , where (4.26) A = ˆ τ 0 | ∂ τ ˜ γ | ˜ g ( τ ) dτ . By the definition of ˜ g ( τ ) and the asymptotics (2.8), w e kno w that for τ ∈ [0 , ¯ τ ], (4.27) ˜ g ( τ ) = (1 − 2 τ ) g ( T + log(1 − 2 τ )) ⩽ (1 + Ψ( T − 1 | ε ))( f ∗ ω can + e − T (1 − 2 τ ) − 1 ω SRF ) ⩽ (1 + 4 τ )(1 + Ψ( T − 1 | ε )) g ( T ) = (1 + 4 τ )(1 + Ψ( T − 1 | ε )) ˜ g (0) Com bining this with (4.25), w e see that (4.28) A ⩽ (1 + 4 τ + Ψ( T − 1 | ε )) d can ( f ( p ) , f ( q )) + Ψ( T − 1 | τ , ε ) . Using the scalar curv ature b ound (4.19), we can then estimate (4.29) L T ( q , τ ) ⩽ L ( ˜ γ ) ⩽ C τ 3 2 + ˆ τ 0 √ τ | ∂ τ ˜ γ | 2 ˜ g ( τ ) dτ ⩽ C τ 3 2 + A 2 2 √ τ ⩽ 1 2 √ τ d can ( f ( p ) , f ( q )) 2 + Ψ( T − 1 | τ , ε ) + Ψ( τ | ε ) . No w, using the key estimate (4.23) and the triangle inequalit y , we see that giv en any small δ > 0 and an y p oin t q ′ ∈ X with f ( q ′ ) ∈ B d can ( f ( q ) , δ ), we ha v e (4.30) L T ( q ′ , τ ) ⩾ 1 2 √ τ  d can ( f ( p ) , f ( q )) 2 − Ψ( δ | ε )  + Ψ( T − 1 | ε, τ ) + Ψ( ¯ τ | ε ) . W e consider (4.31) L ( q ′ , τ ) := 2 √ τ L T ( q ′ , τ ) , GROMO V-HAUSDORFF LIMITS OF IMMOR T AL K ¨ AHLER-RICCI FLOWS 19 whic h satisfies (4.32)  ∂ ∂ τ + ∆ ˜ g ( τ )  L ⩽ 4 n, (4.33) L ( q ′ , τ ) ⩾ d can ( f ( p ) , f ( q )) 2 + Ψ( T − 1 | ε, τ ) + Ψ( ¯ τ | ε ) + Ψ( δ | ε ) , and (4.34) lim τ → 0 + L ( q ′ , τ ) = d ˜ g (0) ( p, q ′ ) = d T ( p, q ′ ) . Let χ b e a smo oth (time-indep endent) cutoff function on Y supp orted in B d can ( q , 2 δ ) and equal to 1 on B d can ( q , δ ), and denote by the same sym b ol its pullback to X via f . W e know that − C ( ε ) ω Y ⩽ δ 2 √ − 1 ∂ ∂ χ ⩽ C ( ε ) ω Y , hence b y (2.11) we hav e (4.35) sup X | ∆ ˜ g ( τ ) χ | ⩽ C ( ε ) δ − 2 . Using (4.32), we obtain ˆ X χL ( · , 0) ˜ ω n (0) ⩾ ˆ X χL ( · , τ ) ˜ ω n ( τ ) + ˆ τ 0 ˆ X L ( · , τ )∆ ˜ g ( τ ) χ ˜ ω n ( τ ) dτ − 4 n ˆ τ 0 ˆ X L ( · , τ ) χ ˜ ω n ( τ ) dτ − 2 ˆ τ 0 ˆ X χL ( · , τ ) R ( ˜ g ( τ )) ˜ ω n ( τ ) dτ , (4.36) Then using the upp er b ound on L in (4.29), the scalar curv ature b ound (4.19), (4.33) and (4.35), arguing as in [29] we can obtain that (4.37) ˆ X χL ( · , 0) ˜ ω n (0) ⩾  d can ( f ( p ) , f ( q )) 2 + Ψ( T − 1 | ε, τ ) + Ψ( τ | δ, ε )  ˆ X χ ˜ ω n ( τ ) . Note that for q ′ ∈ X with f ( q ′ ) ∈ B d can ( f ( q ) , δ ), w e ha v e (4.38) d T ( p, q ) ⩾ d T ( p, q ′ ) + Ψ( T − 1 , δ | ε ) , then using (4.34), we obtain (4.39) ˆ X χL ( · , 0) ˜ ω n (0) = ˆ X χd T ( p, · ) ˜ ω n (0) ⩽ ( d T ( p, q ) + Ψ( T − 1 | δ, ε )) 2 ˆ X χ ˜ ω n (0) . and using the scalar curv ature low er b ound, w e ha ve (4.40) ˆ X χ ˜ ω n ( τ ) ⩾ (1 − Ψ( τ | δ, ε )) ˆ X χ ˜ ω n (0) Com bining (4.37), (4.39), (4.38), and (4.40), and choosing δ sufficiently small, then τ sufficiently small, and finally T sufficiently large, w e obtain the desired estimate in (4.10): (4.41) d T ( p, q ) ⩾ d can ( f ( p ) , f ( q )) + Ψ( T − 1 | ε ) . 20 MAN-CHUN LEE, V ALENTINO TOSA TTI, AND JUNSHENG ZHANG □ 5. Proof of the main theorem In this Section w e giv e the pro of of Prop osition 4.4, whic h implies Theo- rem 1.2 by the discussion in Section 4. In the following, we alwa ys consider p oin ts p, q ∈ X \ ˜ V ε and parameters δ satisfying (5.1) δ ≪ ε. W e note that by (2.7), for the giv en parameters ε and δ , and for all suffi- cien tly large T , we hav e (5.2) f − 1  B d can ( f ( q ) , δ / 2)  ⊂ B d T ( p, δ ) ⊂ f − 1  B d can ( f ( q ) , 2 δ )  . Therefore, in the following w e will not distinguish b et ween f − 1  B d can ( f ( q ) , δ )  and B d T ( p, δ ). F or simplicity of notation, we will also omit the dep endence on ε in Prop osition 4.4 throughout this section. 5.1. An almost-av oidance principle. In this subsection, w e show that for a family of sets { U η } η > 0 satisfying certain prop erties, namely small volume and a separation b et w een ∂ U η and ∂ U η / 2 , a typical L -geo desic connecting p and q intersects f − 1 ( U η \ U η / 2 ) at most η − ε times. Since w e will apply this result t wice to t w o differen t families of sets, w e find it con venien t to abstract the prop erties that we use, as follows. Throughout this section, w e will consider { U η } η ∈ (0 ,η 0 ] a family of subsets of Y satisfying the following conditions (1) Nestedness: U η ′ ⊂ U η for η ′ ⩽ η . (2) Minko wski conten t b ound: there exists ρ ∈ (0 , 1), such that (5.3) V ol( U η , ω m can ) ⩽ C ρ η 2 − ρ . (3) Separation: there exists c > 0 such that (5.4) d can ( x, ∂ U η ) ⩾ cη , for an y 0 < η ⩽ η 0 and x ∈ U η / 2 . (4) Normalization:  B d can ( f ( p ) , δ ) ∪ B d can ( f ( q ) , δ )  ∩ U η 0 = ∅ . W e set (5.5) ˜ U η = f − 1 ( U η ) ⊂ X. Definition 5.1. F or a pie c ewise differ entiable curve γ inside X , we way γ has an η -event with r esp e ct to { U η } if f ( γ ) enters U η and r e aches U η 2 b efor e r eturning to the b oundary of U η . As explained in Section 4.2, we res cale and reparametrize the flow to obtain an unnormalized Ricci flo w and study its L -geo desics. W e use the notation introduced there throughout the following discussion. The follow- ing Lemma is analogous to [29, (4.31)]: GROMO V-HAUSDORFF LIMITS OF IMMOR T AL K ¨ AHLER-RICCI FLOWS 21 Lemma 5.2. L et γ b e a minimizing L -ge o desic fr om ( p, 0) T to ( q ′ , τ ) T , wher e q ′ satisfies f ( q ′ ) ∈ B d can ( f ( q ) , δ ) . L et τ ′ denote the first time at which f ( γ ) exits the b al l B d can ( f ( p ) , δ ) . Then, for τ sufficiently smal l and T sufficiently lar ge, ther e exists a c onstant C = C ( δ ) such that (5.6) τ ′ ⩾ C − 1 τ . Pr o of. Using (2.8), (4.19) and the upp er b ound of L -geo desics (4.29), we ha v e δ ⩽ (1 + Ψ( T − 1 , τ )) ˆ τ ′ 0 | ∂ τ γ | ˜ g ( τ ) dτ ⩽ C ˆ τ ′ 0 √ τ | ∂ τ γ | 2 ˜ g ( τ ) dτ ! 1 2 ˆ τ ′ 0 1 √ τ dτ ! 1 2 ⩽ C τ ′ 1 4 C τ ′ 3 2 + ˆ τ ′ 0 √ τ ( R ( ˜ g ( τ )) + | ∂ τ γ | 2 ˜ g ( τ ) ) dτ ! 1 2 ⩽ C τ ′ 1 4  C τ ′ 3 2 + C τ − 1 2  1 2 ⩽ C τ ′ 1 4 τ − 1 4 . (5.7) □ The follo wing observ ation will b e crucial on our argumen ts: Prop osition 5.3. L et W ⊂ f − 1 ( B d can ( f ( q ) , 2 δ )) b e a Bor el subset with p ositive volume, i.e. (5.8) V ol( W, ω n 0 ) > 0 . and let { U η } b e a family of subsets of Y satisfying (1)-(4) ab ove and fix ε > ρ/ 2 . Then for (5.9) T − 1 ≪ η ≪ τ ≪ δ, ther e exists a lo c al ly close d subset Ω = Ω( T , η , ¯ τ , δ ) ⊂ W with volume (5.10) V ol(Ω , ω ( T ) n ) ⩾ 1 2 V ol( W, ω ( T ) n ) such that for any q ′ ∈ Ω , ther e exists a minimizing L -ge o desic fr om ( p, 0) T to ( q ′ , τ ) T for which the numb er of η -events with r esp e ct to { U η } is at most η − ε . Pr o of. Let Ω τ ⊂ T p X b e the op en set such that the L -exp onen tial map L exp p,τ restricted to Ω τ giv es a diffeomorphism on to W reg , which is an op en subset of W such that W \ W reg has measure zero. Moreov er we know that for a p oin t q ′ ∈ W reg , there exists a unique L -geo desic from ( p, 0) T to ( q ′ , τ ) T . Let Ω ′ τ ⊂ Ω τ denote the subset consisting of those initial tangent v ectors for whic h the associated L -geo desic has more than η − ε η -ev en ts with 22 MAN-CHUN LEE, V ALENTINO TOSA TTI, AND JUNSHENG ZHANG resp ect to the family { U η } . By the contin uous dep endence of L -geo desics on their initial data, Ω ′ τ is an op en subset of Ω τ . F or any v ∈ Ω ′ τ , there exists a union of op en interv als I v ∈ [0 , τ ] such that L exp p,τ ( v ) ∈ ˜ U η exactly when τ ∈ I v . By Lemma 5.2, for any v ∈ Ω ′ τ , w e ha v e (5.11) I v ⊂ [ C − 1 τ , τ ] . W e consider (5.12) T = { ( v , τ ) | v ∈ Ω ′ τ , τ ∈ I v } ⊂ T p X × [0 , τ ] . Since I v dep ends contin uous on v , the set T is a measurable set and in the follo wing, we can use F ubini’s theorem on T . W e hav e a smo oth injective map (5.13) L exp p : T → ˜ U η × [ C − 1 τ , τ ] , whic h maps ( v , τ ) to ( L exp p,τ ( v ) , τ ). Recall Perelman’s monotonicity [31], (5.14) J ( v , τ ) ⩾  τ τ  m e ℓ ( v ,τ ) − ℓ ( v ,τ ) J ( v , τ ) , where J ( v , τ ) denotes the Jacobian of L exp p,τ and (5.15)  ( v , τ ) := 1 2 √ τ L T ( L exp p,τ ( v ) , τ ) , is P erelman’s reduced length. Then combining this with the upp er b ound on the L -distance (4.29) and Lemma 5.2, w e obtain (5.16) J ( v , τ ) ⩾ C − 1 e − C τ J ( v , τ ) . W e compute (5.17) ˆ τ C − 1 τ V ol( ˜ U η , ˜ ω ( τ ) n ) dτ ⩾ ˆ Ω ′ τ ˆ I v J ( v , τ ) dτ dv ⩾ C − 1 e − C τ ˆ Ω ′ τ ˆ I v J ( v , τ ) dτ dv ⩾ C − 1 e − C τ min v ∈ Ω ′ τ | I v | V ol( L exp p,τ (Ω ′ τ ) , ˜ ω ( τ ) n ) . W e w ant to hav e a lo w er b ound for | I v | . Since we ha ve at most η − ε η -ev en ts and the d can -length of a curve for each η -ev en t is at least η / 2 (by prop ert y (3)), then by (4.27), we obtain that for T sufficien tly large (dep ending on η ) and ¯ τ ∈ (0 , 1 / 10], w e get the follo wing (5.18) η 1 − ε ⩽ 4 ˆ I v | ∂ τ γ | ˜ g ( τ ) dτ . GROMO V-HAUSDORFF LIMITS OF IMMOR T AL K ¨ AHLER-RICCI FLOWS 23 Using Cauch y-Sc h w arz and the upp er b ound of the L -distance (4.29), we get (5.19) ˆ I v | ∂ τ γ | ˜ g ( τ ) dτ ⩽ C τ − 1 4 ˆ I v τ 1 4 | ∂ τ γ | ˜ g ( τ ) dτ ⩽ C τ − 1 4  ˆ I v √ τ | ∂ τ γ | 2 ˜ g ( τ ) dτ  1 2 | I v | 1 2 ⩽ C ¯ τ − 1 / 2 | I v | 1 2 , and together with (5.18) w e see that (5.20) | I v | ⩾ C − 1 τ η 2(1 − ε ) . Plugging this bac k in to (5.17), w e get (5.21) V ol( L exp p,τ (Ω ′ τ ) , ˜ ω ( τ ) n ) ⩽ C e C τ τ − 1 η − 2(1 − ε ) ˆ τ C − 1 τ V ol( ˜ U η , ˜ ω ( τ ) n ) dτ ⩽ C e C τ η − 2(1 − ε ) V ol( ˜ U η , ω ( T ) n ) . But using (4.6), (2.10) and assumption (2) we can b ound (5.22) V ol( ˜ U η , ω ( T ) n ) ⩽ C V ol( X , ω ( T ) n )V ol( U η , ω m can ) ⩽ C e − ( n − m ) T η 2 − ρ , and inserting this in to (5.21) w e obtain that (5.23) V ol( L exp p,τ (Ω ′ τ ) , ˜ ω ( τ ) n ) < Ψ( η | ¯ τ , δ ) e − ( n − m ) T . By the estimate on the volume form (2.10) and the initial assumption on W (5.8), w e kno w that for some c > 0 indep enden t of T , (5.24) V ol( W, ω ( T ) n ) ⩾ ce − ( n − m ) T . Therefore b y for η s ufficien tly small, W e hav e thus prov ed that there exists a closed subset Ω ⊂ W with v olume (5.25) V ol(Ω , ω ( T ) n ) ⩾ 1 2 V ol( W, ω ( T ) n ) suc h that for any q ′ ∈ Ω, there exists a minimizing L -geo desic from ( p, 0) T to ( q ′ , τ ) T whic h has at most η − ε η -ev en ts with resp ect to { U η } . □ 5.2. In tegrabilit y of ω can on the almost regular set. Recall that Y reg denotes the p oints where the v ariet y Y is smo oth, and by definition we ha ve Y ◦ ⊂ Y reg . F ollo wing [30], we define the almost regular set as follows: given ε > 0, let (5.26) R ε :=  y ∈ Y     lim r → 0 V ol( B d can ( y , r )) ω 2 m r 2 m > 1 − ε  . Clearly R ε is op en and for ε ′ < ε , w e ha v e (5.27) R ε ′ ⊂ R ε . 24 MAN-CHUN LEE, V ALENTINO TOSA TTI, AND JUNSHENG ZHANG The following result is prov ed in [40, 30], building on the earlier work in [10, 4]. Note that as discussed in [40, Section 2], b y using the argument in [4, 30], we kno w that for an y iterated tangent cone V of Y , the singular set V \ R ε ( V ) has capacity zero. Therefore the results in [30, Section 4] can b e applied to ( Y , d can ). Theorem 5.4 ([30, 40]) . Ther e exists ε 1 > 0 such that (5.28) R ε 1 ⊂ Y reg . Mor e over for e ach y ∈ Y reg , ther e exists C = C ( Y , y ) and α = α ( Y , y ) > 0 and holomorphic c o or dinates z = ( z 1 · · · , z m ) ar ound y such that (5.29) C − 1 | z | ⩽ d can ( y , · ) ⩽ C | z | α . The main result in this section is the follo wing. Lemma 5.5. F or any p > 0 , ther e exists ε = ε ( p ) such that (5.30) ω m can ω m Y ∈ L p loc ( R ε , ω m Y ) . Pr o of. This basically is contained in [30, P .268–270]. W e follow their argu- men t closely . F or an y small ε > 0, there exists ε ′ ≪ ε such that for any y ∈ R ε ′ the ball B  y , 1 ε  (with the distance d can ) is ε -Gromov Hausdorff close to a ball in C m , b y scaling. Then w e can find a holomorphic map F = ( f 1 , . . . , f m ) on B ( p, 100) which gives a Ψ( ε )-Gromov Hausdorff approximation onto its image in C m and defines a holomorphic c hart on B ( p, 1). Moreov er, we ma y assume f j ( y ) = 0 , ´ B ( p, 1) f j f k = 0 for j  = k . W e hav e (5.31) f B ( y, 2) | f j | 2 f B ( y, 1) | f j | 2 ⩽ 4 + Ψ( ε ) . Using a three annulus type argument and the fact that the tangent cone at y is close to C m (see [30, Lemma 2.4] and [11, Prop osition 3.7]), w e see that for an y 0 < r < 1, we ha v e (5.32) 4 − Ψ( ε ) ⩽ f B ( y, 2 r ) | f j | 2 f B ( y,r ) | f j | 2 ⩽ 4 + Ψ( ε ) . F or any given r > 0, w e can choose the functions f 1 , . . . , f n so that they are orthogonal simultaneously with resp ect to the L 2 inner pro ducts on B ( y , 1) and B ( y , r ). Now let c j = c j ( r ) b e the constants so that sup B ( y,r ) | c j f j | = r . By (5.32), we know that (5.33) | c j ( r ) | ⩽ C r − Ψ( ε ) . GROMO V-HAUSDORFF LIMITS OF IMMOR T AL K ¨ AHLER-RICCI FLOWS 25 Define f ′ j = c j f j . Then as in [30] we see that F ′ = ( f ′ 1 , · · · f ′ n ) is a Ψ( ε ) r - Gromo v-Hausdorff approximation to a ball in C m and an estimate of Cheeger- Colding implies that (5.34) B ( y,r ) || d f ′ 1 ∧ d f ′ 2 ∧ . . . ∧ d f ′ n | g can − 1 | = Ψ( ε ) , and in particular, w e ha ve (5.35) sup B ( y,r )   d f ′ 1 ∧ d f ′ 2 ∧ . . . ∧ d f ′ n   g can ⩾ c ( n ) > 0 . Therefore w e ha ve (5.36) sup B ( y,r ) | d f 1 ∧ d f 2 ∧ . . . ∧ d f n | g can ⩾ c ( n ) r Ψ( ε ) . Since near any p oin t of Y we can write lo cally ω can = i∂ ∂ ϕ for some con- tin uous p oten tial ϕ , since and Ric( ω can ) ⩾ − ω can on Y ◦ , w e ha ve that (5.37) log | d f 1 ∧ d f 2 ∧ . . . ∧ d f n | g can + ϕ is a psh function on Y ◦ near y . F urthermore, as a consequence of the Sch warz Lemma estimate (2.11) together with the conv ergence in (2.6) we hav e that ω can ⩾ C − 1 ω Y on Y ◦ , hence this psh function is lo cally b ounded ab o v e near D ∩ Y reg , and so it extends to a psh function near y on Y reg , whic h w e denote with the same name. By (5.36) and the lo cal distance estimate (5.29), we kno w that the Lelong n um b er of this psh function at y is b ounded ab o v e b y Ψ( ε ), i.e., (5.38) ν (log | d f 1 ∧ d f 2 ∧ . . . ∧ d f n | g can + ϕ, y ) ⩽ Ψ( ε ) . Then for an y giv en p , w e can c ho ose ε sufficiently small suc h that p Ψ( ε ) ⩽ 1 2 , where Ψ( ε ) denotes the righ t hand side of (5.38). This then determines ε ′ and b y the ab o v e argumen t, w e kno w that for y ∈ R ε ′ , w e ha ve (5.39) ν (log | d f 1 ∧ d f 2 ∧ . . . ∧ d f n | g can + ϕ, y ) ⩽ 1 2 p . Sk o da’s integrabilit y theorem, together with the simple observ ation that (5.40) log ω m can ω m Y ⩽ − (log | d f 1 ∧ d f 2 ∧ . . . ∧ d f n | g can + ϕ ) + C, then sho ws that ω m can ω m Y is in L p ( ω m Y ) in a neigh b orhoo d of y . □ 5.3. The c hoice of the regions. Recall now that thanks to Theorem 3.6, on Y we hav e the estimate (5.41) C − 1 d Y ⩽ d can ⩽ C d α Y , for some α > 0. W e then choose ε 0 , p 0 > 0 suc h that (5.42) ε 0 = α 4(1 − α ) , 2 p 0 < ε 0 . 26 MAN-CHUN LEE, V ALENTINO TOSA TTI, AND JUNSHENG ZHANG W e then fix ε 1 sufficien tly small such that b oth Theorem 5.4 and Lemma 5.5 hold, so that w e ha v e (5.43) R ε 1 ⊂ Y reg , ω m can ω m Y ∈ L p 0 loc ( R ε 1 , ω m Y ) . Then w e decomp ose D ⊂ Y into a disjoint union (5.44) D = ( D ∩ ( Y \ R ε 1 )) ∪ ( D ∩ R ε 1 ) , so that if we define D 1 := D ∩ ( Y \ R ε 1 ) , then we hav e D = D 1 ∪ ( D \ D 1 ) . W e now define a family of op en subsets { U η } η ∈ (0 ,η 0 ] b y (5.45) U η := { y ∈ Y | d can ( y , D 1 ) < η } . W e fix η 0 sufficien tly small so that prop ert y (4) holds. Prop erties (1) and (3) are obvious, and prop ert y (2) follows from this Theorem, which uses the w ork of Cheeger-Nab er [6] and its extension to RCD spaces [1]: Theorem 5.6. F or any ε > 0 and ρ > 0 and 0 < r < 1 , we have (5.46) V ol( T r ( Y \ R ε )) ⩽ C ( Y , ε, ρ ) r 2 − ρ . Pr o of. By the volume conv ergence [8] and the Bishop-Gromov volume com- parison, w e kno w that there exists ρ 0 > 0 such that for any 0 < r < 1 and 0 < ρ < ρ 0 , (5.47) Y \ R ε ⊂ S 2 m − 2 ρ,r , where the latter is the effective singular stratum defined in [6, Definition 1.2]. T aking 0 < ρ < ρ 0 , estimate 5.46 follows from [1, Theorem 2.4], which is the extension to R CD spaces of [6, Theorem 1.3]. □ The family { U η } th us satisfies the hypotheses of Prop osition 5.3, whic h w e apply with the c hoices ε = ε 0 and W = B d T ( q , δ ), and for T − 1 ≪ η ≪ τ ≪ δ w e obtain a subset Ω = Ω( T , η , ¯ τ , δ ) ⊂ B d T ( q , δ ) with volume (5.48) V ol(Ω , ω ( T ) n ) ⩾ 1 2 V ol( B d T ( q , δ ) , ω ( T ) n ) ⩾ cδ 2 m e − ( n − m ) T , suc h that for any q ′ ∈ Ω, there exists a minimizing L -geo desic from ( p, 0) T to ( q ′ , τ ) T whic h has at most η − ε 0 η -ev en ts with resp ect to { U η } . F or this fixed v alue of η , we then define a second family of op en sets { U ′ η ′ } η ′ ∈ (0 ,η ′ 0 ] b y taking η ′ 0 ≪ η and defining (5.49) U ′ η ′ := { y ∈ Y | d can ( y , D \ T η ( D 1 )) < η ′ } , Observ e that R ε 1 is an op en subset of Y reg and D \ T η ( D 1 ) is compact subset of R ε 1 , therefore w e ha ve that for η ′ 0 sufficien tly small, (5.50) U ′ η ′ ⊂ R ε 1 ⊂ Y reg . Then we chec k that the prop erties (1)–(4) needed in Prop osition 5.3 hold for { U ′ η ′ } η ′ ∈ (0 ,η ′ 0 ] . Again, we choose η ′ 0 sufficien tly small so that prop erty (4) GROMO V-HAUSDORFF LIMITS OF IMMOR T AL K ¨ AHLER-RICCI FLOWS 27 holds, and prop erties (1) and (3) are ob vious. F or prop ert y (2), using the first estimate in (5.41) w e see that (5.51) U ′ η ′ ⊂ V ′ C η ′ := { y ∈ Y | d Y ( y , D \ T η ( D 1 )) < C η ′ } . F or η ′ sufficien tly small, V ′ C η ′ ⊂ Y reg is a g Y -tubular neigh b orhoo d of a closed analytic sub v ariety inside a complex manifold (in particular, it is (2 m − 2)-rectifiable), and so the Minko wski con ten t b ound (5.52) V ol( V ′ C η ′ , ω m Y ) ⩽ C ( η ′ ) 2 , 0 < η ′ ⩽ η ′ 0 , is well-kno wn, see e.g. [13, Theorems 3.2.39, 3.4.8]. T o prov e prop ert y (2), w e then use our c hoices of p 0 and ε 1 and the H¨ older inequality to b ound (5.53) V ol( U ′ η ′ , ω m can ) ⩽ ˆ U ′ η ′  ω m can ω m Y  p 0 ! 1 p 0 V ol( V ′ C η ′ , ω m Y ) 1 − 1 p 0 ⩽ C ( Y , η )( η ′ ) 2 − 2 p 0 , Note that the constan t C dep ending on η , which is ho w ev er fixed here. Then w e can apply Prop osition 5.3 to the family { U ′ η ′ } and W = Ω, where Ω here denote the set obtained in the last step. By (5.48) and the estimate on the v olume form (2.10), w e know that (5.54) V ol(Ω , ω n 0 ) ⩾ c > 0 , where v ery imp ortan tly the constant is indep endent of T . Then we obtain Prop osition 5.7. F or T − 1 ≪ η ′ ≪ η ≪ τ ≪ δ ther e exists a subset Ω ′ ⊂ Ω ⊂ B d T ( q , δ ) with (5.55) V ol(Ω ′ , ω ( T ) n ) ⩾ 1 4 V ol( B d T ( q , δ ) , ω ( T ) n ) , such that for any q ′ ∈ Ω ′ ther e exists a minimizing L -ge o desic fr om ( p, 0) T to ( q ′ , τ ) T which has at most ( η ′ ) − ε 0 η ′ -events with r esp e ct to { U ′ η ′ } , and also at most η − ε 0 η -events with r esp e ct to { U η } . 5.4. Completion of the pro of. W e are now ready to complete the pro of of Theorem 1.2, by proving the k ey estimate (4.23). W e adapt an idea from [29, Remark 4.2], but using crucially our results from the previous section. F or this, we apply Prop osition 5.7, and let γ b e an y of the minimizing L -geo desics given b y that, with endp oin ts in Ω ′ . W e consider subsets I ∪ I ′ of [0 , τ ] defined by the prop ert y that τ ∈ I ⇔ γ ( τ ) ∈ ˜ U η = f − 1 ( U η ) and τ ∈ I ′ ⇔ γ ( τ ) ∈ ˜ U ′ η ′ = f − 1 ( U ′ η ′ ), and its complemen t (5.56) J = [0 , τ ] \ ( I ∪ I ′ ) . By Lemma 5.2, w e kno w that ev ery τ ∈ I ∪ I ′ satisfies τ ⩾ C − 1 τ . In the follo wing we estimate ´ I | ∂ τ γ | ˜ g ( τ ) dτ and ´ I ′ | ∂ τ γ | ˜ g ( τ ) dτ . First, as in the pro of of Prop osition 5.3, we call Ω ′ p ⊂ T p X the op en subset which 28 MAN-CHUN LEE, V ALENTINO TOSA TTI, AND JUNSHENG ZHANG under L exp p,τ maps diffeomorphically on to an op en subset of Ω ′ with full measure. As in (5.17) w e obtain (5.57) ˆ τ C − 1 τ V ol( ˜ U η , ˜ ω ( τ ) n ) dτ ⩾ C − 1 e − C τ | I | V ol(Ω ′ , ω ( T ) n ) , and using the volume b ounds in (5.22) and (5.55), and rep eating this ar- gumen t for I ′ , we see that there exists at least one of these L -geo desics γ (obtained through Prop osition 5.7) such that (5.58) | I | ⩽ C τ η 2 − ε 0 δ − 2 m e C /τ , | I ′ | ⩽ C τ η ′ 2 − ε 0 δ − 2 m e C /τ . Then using the scalar curv ature b ound (2.9) and the upp er b ound on the L -distance (4.29), w e can estimate (5.59) ˆ I | ∂ τ γ | ˜ g ( τ ) dτ ⩽ C τ − 1 4  ˆ I √ τ | ∂ τ γ | 2 ˜ g ( τ ) dτ  1 2 | I | 1 2 ⩽ C ( δ, ¯ τ ) η 1 − ε 0  τ 3 2 + ˆ I √ τ  R ( ˜ g ( τ )) + | ∂ τ γ | 2 ˜ g ( τ )  dτ  1 2 ⩽ C ( δ, ¯ τ ) η 1 − ε 0  τ 3 2 + τ − 1 2  1 2 = Ψ( η | δ, ¯ τ ) η 1 2 , Similarly b y taking η ′ ev en smaller, w e hav e (5.60) ˆ I ′ | ∂ τ γ | ˜ g ( τ ) dτ ⩽ Ψ( η ′ | δ, ¯ τ , η )( η ′ ) 1 2 . By construction, there exist at most η − ε 0 in terv als, which w e lab el b y [ τ entry ,i , τ exit ,i ], suc h that (5.61) f  γ ([ τ entry ,i , τ exit ,i ])  ⊂ U η / 2 . Moreo v er, there exist at most η ′− ε 0 in terv als, which we lab el b y (5.62) [ τ ′ entry ,i , τ ′ exit ,i ] , suc h that (5.63) f  γ ([ τ ′ entry ,i , τ ′ exit ,i ])  ⊂ U ′ η ′ / 2 . F or each i , we hav e (5.64) d Y ( f ( γ ( τ entry ,i )) , f ( γ ( τ exit ,i ))) ⩽ ˆ τ exit ,i τ entry ,i | ∂ τ f ( γ ) | g Y dτ , so by the H¨ older estimate (5.41), w e can find a curve in Y joining f ( γ ( τ entry ,i )) and f ( γ ( τ exit ,i )) whose g can -length is at most (5.65) C ˆ τ exit ,i τ entry ,i | ∂ τ f ( γ ) | g Y dτ ! α ⩽ C ˆ τ exit ,i τ entry ,i | ∂ τ γ | ˜ g ( τ ) dτ ! α , GROMO V-HAUSDORFF LIMITS OF IMMOR T AL K ¨ AHLER-RICCI FLOWS 29 where w e used (2.11). W e use this path to replace the p ortion of f ( γ ) with τ entry ,i ⩽ τ ⩽ τ exit ,i . Rep eating this for all even ts, and similarly for the τ ′ i ’s, w e obtain a curv e γ ′ in Y from f ( p ) to f ( q ) satisfying (5.66) d can ( f ( p ) , f ( q )) ⩽ length g can  γ ′  ⩽ ˆ J | ∂ τ f ( γ ) | g can dτ + C η − ε 0 X i =1 ˆ τ exit ,i τ entry ,i | ∂ τ γ | ˜ g ( τ ) dτ ! α + C ( η ′ ) − ε 0 X i =1 ˆ τ ′ exit ,i τ ′ entry ,i | ∂ τ γ | ˜ g ( τ ) dτ ! α + Ψ( T − 1 , δ ) . Then using (2.8),(5.59) and (5.60), we obtain (5.67) d can ( f ( p ) , f ( q )) ⩽ (1 + Ψ( T − 1 | ¯ τ )) ˆ J | ∂ τ γ | ˜ g ( τ ) dτ + Ψ( T − 1 , δ ) + C η − ε 0 (1 − α )  ˆ I | ∂ τ γ | ˜ g ( τ ) dτ  α + C η − ε ′ 0 (1 − α )  ˆ I ′ | ∂ τ γ | ˜ g ( τ ) dτ  α ⩽ (1 + Ψ( T − 1 | ¯ τ )) ˆ τ 0 | ∂ τ γ | ˜ g ( τ ) dτ + Ψ( T − 1 , δ ) + Ψ( η | δ, ¯ τ ) η α 2 − ε 0 (1 − α ) + Ψ( η ′ | δ, ¯ τ , η )( η ′ ) α 2 − ε 0 (1 − α ) ⩽ (1 + Ψ( T − 1 | ¯ τ )) ˆ τ 0 | ∂ τ γ | ˜ g ( τ ) dτ + Ψ( T − 1 , δ ) + Ψ( η , T − 1 | δ, ¯ τ ) + Ψ( η ′ , T − 1 | η , δ . ¯ τ ) , where in the last inequality w e ha ve crucially used our choice of ε 0 in (5.42). Then we use the fact that γ is a minimizing L -geo desic from ( p, 0) T to ( q ′ , τ ) T to estimate (5.68) ˆ τ 0 | ∂ τ γ | ˜ g ( τ ) dτ ⩽  ˆ τ 0 √ τ | ∂ τ γ | 2 ˜ g ( τ ) dτ  1 2  ˆ τ 0 1 √ τ dτ  1 2 ⩽ √ 2 τ 1 4  C τ 3 2 + ˆ τ 0 √ τ  R ( ˜ g ( τ )) + | ∂ τ γ | 2 ˜ g ( τ )  dτ  1 2 = √ 2 τ 1 4  C τ 3 2 + L T  q ′ , τ   1 2 = √ 2 τ 1 4  C τ 3 2 + L T ( q , τ ) + Ψ( δ ) ¯ τ − 1 / 2  1 2 Therefore b y taking δ small, then taking η small (dep ending on τ , δ ) and then η ′ ev en smaller (dep ending on δ , τ and η ) and then T sufficiently large, 30 MAN-CHUN LEE, V ALENTINO TOSA TTI, AND JUNSHENG ZHANG w e ha ve finally shown that (5.69) L T ( q , τ ) ⩾ 1 2 √ τ d can ( f ( p ) , f ( q )) 2 + Ψ( T − 1 | τ ) + Ψ( ¯ τ ) , whic h is precisely (4.23). References [1] G. Antonelli, E. Bru` e, D. Semola, V olume b ounds for the quantitative singular str ata of non-c ol lapse d RCD metric measur e sp ac es , Anal. Geom. Metr. Spaces 7 (2019), no. 1, 158–178. [2] E. Bombieri, E. Giusti, Harnack’s ine quality for el liptic differ ential e quations on min- imal surfaces , Inv en t. Math. 15 (1972), 24–46. [3] H.-D. Cao, Deformation of K¨ ahler metrics to K¨ ahler-Einstein metrics on comp act K¨ ahler manifolds , Inv ent. Math. 81 (1985), no. 2, 359–372. [4] X. Chen, S.K. Donaldson, S. Sun, K¨ ahler-Einstein metrics on F ano manifolds. II: Limits with c one angle less than 2 π , J. Amer. Math. So c. 28 (2015), no. 1, 199–234. [5] J. Cheeger, W. Jiang, A. Nab er, R e ctifiability of singular sets of noncollapsed limit sp aces with Ric ci curvatur e b ounde d b elow , Ann. of Math. (2) 193 (2021), no. 2, 407–538. [6] J. Cheeger, A. Nab er, L ower b ounds on Ric ci curvatur e and quantitative b ehavior of singular sets , Inv ent. Math. 191 (2013), no. 2, 321–339. [7] D. Coman, V. Guedj, A. Zeriahi, Extension of plurisubharmonic functions with gr owth c ontrol , J. Reine Angew. Math. 676 (2013), 33–49. [8] G. De Philippis, N. Gigli, Non-c ol lapse d sp ac es with Ric ci curvatur e b ounde d fr om b elow , J. ´ Ec. p olytec h. Math. 5 (2018), 613–650. [9] S. Dinew, Z. Zhang, On stability and c ontinuity of b ounde d solutions of de gener ate c omplex Monge-Amp ` er e e quations over c omp act K¨ ahler manifolds , Adv. Math. 225 (2010), no. 1, 367–388. [10] S.K. Donaldson, S. Sun, Gr omov-Hausdorff limits of K¨ ahler manifolds and algebr aic ge ometry , Acta Math. 213 (2014), no. 1, 63–106. [11] S.K. Donaldson, S. Sun, Gr omov-Hausdorff limits of K¨ ahler manifolds and algebr aic ge ometry, II , J. Differential Geom. 107 (2017), no. 2, 327–371. [12] P . Eyssidieux, V. Guedj, A. Zeriahi, Singular K¨ ahler-Einstein metrics , J. Amer. Math. So c. 22 (2009), 607–639. [13] H. F ederer, Ge ometric me asur e the ory , Die Grundlehren der mathematischen Wis- sensc haften, Band 153. Springer-V erlag New Y ork, Inc., New Y ork, 1969. [14] M. Gross, V. T osatti, Y. Zhang, Col lapsing of ab elian fiber e d Calabi-Y au manifolds , Duk e Math. J. 162 (2013), no. 3, 517–551. [15] M. Gross, V. T osatti, Y. Zhang, Ge ometry of twiste d K¨ ahler-Einstein metrics and c ol lapsing , Comm. Math. Ph ys. 380 (2020), no.3, 1401–1438. [16] V. Guedj, H. Guenancia, A. Zeriahi, Diameter of K¨ ahler currents , J. Reine Angew. Math. 820 (2025), 115–152. [17] V. Guedj, T.D. Tˆ o, K¨ ahler families of Gr e en ’s functions , J. ´ Ec. p olytec h. Math. 12 (2025), 319–339. [18] B. Guo, On the K¨ ahler Ric ci flow on pr oje ctive manifolds of gener al typ e , In t. Math. Res. Not. IMRN 2017, no. 7, 2139–2171. [19] B. Guo, S. Ko lo dziej, J. Song, J. Sturm, H¨ older estimates for de gener ate complex Monge-Amp ` er e e quations , preprint, [20] B. Guo, D.H. Phong, J. Song, J. Sturm, Diameter estimates in K¨ ahler ge ometry , Comm. Pure Appl. Math. 77 (2024), no. 8, 3520–3556. GROMO V-HAUSDORFF LIMITS OF IMMOR T AL K ¨ AHLER-RICCI FLOWS 31 [21] B. Guo, D.H. Phong, J. Song, J. Sturm, Diameter estimates in K¨ ahler ge ometry II: r emoving the smal l de gener acy assumption , Math. Z. 308 (2024), no. 3, Paper No. 43, 7 pp. [22] B. Guo, D.H. Phong, J. Song, J. Sturm, Sob olev ine qualities on K¨ ahler sp ac es , preprin t, [23] B. Guo, J. Song, B. W einko v e, Ge ometric c onver genc e of the K¨ ahler-R ic ci flow on c omplex surfac es of gener al typ e , Int. Math. Res. Not. IMRN 2016, no. 18, 5652–5669. [24] H.-J. Hein, M.-C. Lee, V. T osatti, Col lapsing immortal K¨ ahler-Ric ci flows , F orum Math. Pi 13 (2025), Paper No. e18. [25] S. Ko lodziej, H¨ older c ontinuity of solutions to the complex Monge-Amp ` er e e quation with the right-hand side in L p : the c ase of c omp act K¨ ahler manifolds , Math. Ann. 342 (2008), no. 2, 379–386. [26] W. Jian, J. Song, Diameter estimates for long-time solutions of the K¨ ahler-Ric ci flow , Geom. F unct. Anal. 32 (2022), no. 6, 1335–1356. [27] C. Li, K¨ ahler-Einstein metrics and K-stability , PhD thesis, Princeton Universit y , 2012. [28] Y. Li, On c ol lapsing Calabi-Y au fibr ations , J. Differential Geom. 117 (2021), no. 3, 451–483. [29] Y. Li, V. T osatti, On the c ol lapsing of Calabi-Y au manifolds and K¨ ahler-Ric ci flows , J. Reine Angew. Math. 800 (2023), 155–192. [30] G. Liu, G. Sz´ ekelyhidi, Gr omov-Hausdorff limits of K¨ ahler manifolds with Ric ci cur- vatur e b ounde d b elow , Geom. F unct. Anal. 32 (2022), 236–279. [31] G. Perelman, The entr opy formula for the Ric ci flow and its ge ometric applic ations , preprin t, arXiv:math/0211159. [32] J. Song, Riemannian ge ometry of K¨ ahler-Einstein curr ents , preprint, [33] J. Song, G. Tian, The K¨ ahler-Ric ci flow on surfac es of p ositive Ko dair a dimension , In ven t. Math. 170 (2007), no. 3, 609–653. [34] J. Song, G. Tian, Canonic al me asur es and K¨ ahler-Ric ci flow , J. Amer. Math. So c. 25 (2012), no. 2, 303–353. [35] J. Song, G. Tian, Bounding sc alar curvatur e for glob al solutions of the K¨ ahler-Ric ci flow , Amer. J. Math. 138 (2016), no. 3, 683–695. [36] J. Song, G. Tian, The K¨ ahler-Ric ci flow thr ough singularities , Inv en t. Math. 207 (2017), no. 2, 519–595. [37] J. Song, G. Tian, Z. Zhang, Col lapsing b ehavior of Ric ci-flat K¨ ahler metrics and long time solutions of the K¨ ahler-Ric ci flow , preprint, [38] G. Stolzenberg, V olumes, limits, and extensions of analytic varieties , Lecture Notes in Mathematics, No. 19. Springer-V erlag, Berlin-New Y ork, 1966. [39] G. Sz´ ekelyhidi, Singular K¨ ahler-Einstein metrics and RCD spac es , F orum Math. Pi 13 (2025), P ap er No. e24, 33 pp. [40] G. Sz ´ ekelyhidi, Gr omov-Hausdorff limits of c ol lapsing Calabi-Y au fibr ations , preprin t, [41] G. Tian, Z. Zhang, On the K¨ ahler-Ric ci flow on proje ctive manifolds of gener al typ e , Chinese Ann. Math. Ser. B 27 (2006), no. 2, 179–192. [42] G. Tian, Z. Zhang, Conver genc e of K¨ ahler-Ric ci flow on lower dimensional algebr aic manifolds of gener al typ e , In t. Math. Res. Not. IMRN 2016, no. 21, 6493–6511. [43] H. Tsuji, Existenc e and de gener ation of K¨ ahler-Einstein metrics on minimal algebr aic varieties of gener al typ e , Math. Ann. 281 (1988), no. 1, 123–133. [44] V. T osatti, Immortal solutions of the K¨ ahler-Ric ci flow , to appear in Contemp. Math. [45] V. T osatti, B. W einko v e, X. Y ang, Col lapsing of the Chern-Ric ci flow on el liptic surfac es , Math. Ann. 362 (2015), no. 3-4, 1223–1271. [46] V. T osatti, B. W einko ve, X. Y ang, The K¨ ahler-Ric ci flow, Ric ci-flat metrics and c ol lapsing limits , Amer. J. Math. 140 (2018), no. 3, 653–698. 32 MAN-CHUN LEE, V ALENTINO TOSA TTI, AND JUNSHENG ZHANG [47] V. T osatti, Y. Zhang, Finite time c ol lapsing of the K¨ ahler-Ric ci flow on thr e efolds , Ann. Sc. Norm. Sup er. Pisa Cl. Sci. 18 (2018), no.1, 105–118. [48] D. V arolin, Division the or ems and twiste d c omplexes , Math. Z. 259 (2008), no. 1, 1–20. [49] D.V. V u, Uniform diameter and non-c ol lapsing estimates for K¨ ahler metrics , J. Geom. Anal. 36 (2026), no. 2, Paper No. 75. [50] B. W ang, The lo c al entr opy along R ic ci flow. Part A: the no-lo c al-c ol lapsing theor ems , Cam b. J. Math. 6 (2018), no.3, 267–346. [51] K. Zhang, Some r efinements of the p artial C 0 estimate , Anal. PDE 14 (2021), no. 7, 2307–2326. [52] Z. Zhang, Sc alar curvatur e b ound for K¨ ahler-Ric ci flows over minimal manifolds of gener al typ e , Int. Math. Res. Not. IMRN 2009, no. 20, 3901–3912. Dep ar tment of Ma thema tics, The Chinese University of Hong Kong, Sha tin, N.T., Hong Kong Email address : mclee@math.cuhk.edu.hk Courant Institute School of Ma thema tics, Computing, and Da t a Science, New York University, 251 Mercer St, New York, NY 10012 Email address : tosatti@cims.nyu.edu Courant Institute School of Ma thema tics, Computing, and Da t a Science, New York University, 251 Mercer St, New York, NY 10012 Email address : jz7561@nyu.edu