Approximation of weak geodesics and subharmonicity of Mabuchi energy, II: ε-geodesics
aa r X i v : . [ m a t h . DG ] F e b APPROXIMATION OF WEAK GEODESICS ANDSUBHARMONICITY OF MABUCHI ENERGY, II: ε -GEODESICS LONG LI
Abstract.
The purpose of this article is to study the strict convex-ity of the Mabuchi functional along a C , ¯1 -geodesic, with the aid of the ε -geodesics. We proved the L -convergence of the fiberwise volume ele-ment of the ε -geodesic. Moreover, the geodesic is proved to be uniformlyfiberwise non-degenerate if the Mabuchi functional is ε -affine. Introduction
In order to study the uniqueness and existence problems of the K¨ahler-Einstein metrics on a Fano manifold X , Mabuchi ([1], [2]) introduced auseful energy functional M , called the K -energy or Mabuchi functional, onthe space H of all the smooth K¨ahler potentials in a given cohomology class[ ω ]. In other words, we can write the space as H := { ϕ ∈ C ∞ ( X ); ω + dd c ϕ > } . It is observed that this functional M is convex along a smooth geodesic G , connecting with arbitrary two points in H . Moreover, if the Mabuchifunctional is affine along a smooth geodesic G , then G must be generated bya holomorphic vector field. In this case, we say that the Mabuchi functionalis strictly convex along the geodesic G .It turns out that the Mabuchi functional M has also played an importantrole in the study of the constant scalar curvature K¨ahler(cscK) metrics. Inparticular, the convexity of M is crucial in the proof of the uniqueness ofthe cscK metrics ([5], [13], [14]). Moreover, the asymptotic behavior of thisconvex function M ( ϕ t ) along a geodesic ray is an invariant on a K¨ahlermanifold X , and it is proved ([9], [10], [11]) that the manifold X admits acscK metric if and only if this invariant is non-negative.However, one can not expect that there always exists a smooth geodesicconnecting two points in H , due to the example in Darvas-Lempert ([16]).What we can rely on in realty is the so called C , ¯1 -geodesic (as the solutionof certain homogenous complex Monge-Amp`ere equation), whose existencewas guaranteed in the work of Chen ([8]). The difficulty to deal with a C , ¯1 -geodesic G is the lack of the regularities and the possible degeneracy onthe geodesic G .Nevertheless, the Mabuchi functional M was proved to be convex andcontinuous along a C , ¯1 -geodesic, by the work of Berman-Berndtsson ([5])and also Chen-Li-P˘aun ([13]). Then people try to ask the question if theMabuchi functional is also strictly convex along such a geodesic. If we allow that the boundary of G has merely C , ¯1 -regularities, thenthe answer is negative due to the example of Berman ([4]). On the otherhand, the answer is affirmative ([20]), when G is connecting with two non-degenerate energy minimizers of M . In fact, the boundary of G must besmooth in the later case by the work of He-Zeng ([19]). Therefore, a geodesic G will always be assumed to have C , ¯1 -regularities and its boundary belongsto the space H through out this paper.In order to circumvent the difficulties arising from a C , ¯1 -geodesic G , weare appealing to using the so called ε -geodesic G ε instead ([8]). The ε -geodesic G ε is a sequence of smooth approximation of G , satisfying somenon-homogenous complex Monge-Amp`ere equation. This is easier to handlein the practical computation, and we have proved in ([13]) that the Mabuchifunctional is almost convex along the ε -geodesic.However, the new difficulty is that the convergence G ε → G is merelyweakly L p for all p >
1. In other words, it is even not clear to us if theMabuchi functional M ( ϕ ε ) along G ε converges to the Mabuchi functional M ( ϕ ) along G . Therefore, we can not conclude the convexity of M by usingthe ε -geodesic G ε in [13].The new observation is that the convergence M ( ϕ ε ) → M ( ϕ ) is true,provided that the Mabuchi function M is affine along G . This first leadsus to the following L -convergence of the fiberwise volume element of G ε .Write ω ε := G ε | X t and ω ϕ := G| X t on a fiber X t := { t } × X for any t ∈ [0 , Theorem 1.1 (Theorem (4.1)) . Suppose the Mabuchi functional M is affinealong a C , ¯1 -geodesic G . Then the fiber-wise volume element of the ε -geodesicconverges to the volume element of the geodesic in the strong L sense. Inother words, we have on each fiber X t (1.1) ω nε ω n → ω nϕ ω n , ε → , under the L -norm, possibly after passing to a subsequence. We emphasis that the L -convergence of the volume element (equation(1.1)) may not be true in general. Next, a slightly stronger condition thanthe affine Mabuchi functional will be introduced, aiming to resolve the pos-sible degeneracy on the geodesic G . Write the restriction of the Mabuchifunctional along a ε -geodesic G ε as K ε ( t ) := M ( ϕ ε ( t ))for all t ∈ [0 , ε -affine alongthe geodesic G if it satisfies(1.2) d K ε dt | t =1 − d K ε dt | t =0 = O ( ε ) , for all ε > G and G ε are uniquely determined if theboundary of G is given. Thanks to a result in ([5]), the Mabuchi functionalmust be affine along the C , ¯1 -geodesic G if it is ε -affine. Moreover, we provedthe following result. Theorem 1.2 (Theorem 5.4) . Suppose the Mabuchi functional M is ε -affine along a C , ¯1 -geodesic G . Then G is uniformly fiberwise non-degenerate, EODESICS APPROXIMATION 3 namely, there exists a uniform constant κ > such that G| X t > κ ω, for almost everywhere t ∈ [0 , . There are three main steps of the proof for the above Theorem.
Step(1) is to figure out the so called gap phenomenon of the geodesic G , whichis first observed in our previous work ([20]). Step (2) is to establish akind of W , -estimate for the volume element ω nε , which is provided fromthe ε -affine condition. Step (3) (see Proposition (5.7)) is to prove that anon-negative function must have a positive lower bound if it has the gapphenomenon and satisfies a certain partial W , -estimate.As an application of our Theorem (1.1) and (1.2), we can prove (seeTheorem (6.1)) that M ( ϕ ε ) is not only converging to M ( ϕ ) in the pointwisesense, but also in its complex Hessian, if the later is ε -affine. That is to say,we actually have(1.3) d K ε dt | t → , for almost everywhere t ∈ [0 , X satisfies c ( X ) = 0or c ( X ) <
0, then we can utilize Chen’s argument ([8]) to conclude thestrict convexity of the Mabuchi functional, provided the ε -affine condition(see Theorem (6.3)).Therefore, we conjecture that the Mabuchi functional is strictly convexalong a C , ¯1 -geodesic, if it is ε -affine. In fact, the convergence (equation(1.3)) further implies an L -estimate on ¯ ∂v ε for a sequence of smooth vectorfields v ε (see Theorem (6.1)). Unfortunately, there is still some difficultiesto conclude the holomorphicity of v ∞ as the limit of this sequence v ε .Finally, it might be worthy to point out that a geodesic G possibly possessmore regularities than C , ¯1 , when the Mabuchi functional is ε -affine alongit. Hopefully, we will see more examples about this fact, and the regularityproblem will be considered in our following projects. Acknowledgment:
The author is very grateful to Prof. Chen and Prof.P˘aun who introduced this problem, and have given continuous encourage-ment. He also wants to thank Prof. Chengjian Yao, Dr. Jingchen Hu andProf. Wei Sun for lots of useful discussion.2.
Preliminary
Suppose Σ is an annulus in C with boundary, and π is the holomorphicprojection from the product space Y := X × Σ to X . Therefore, Y isa compact complex K¨ahler manifold with boundary. Let Φ be a quasi-plurisubharmonic function on Y continuous up to the boundary. Denote G by the closed positive (1 ,
1) current π ∗ ω + dd c Φ LONG LI on Y . We say that G is a geodesic in the space of K¨ahler potential, if itis S -invariant in the argument direction of Σ, and satisfies the following Homogeneous complex Monge-Amp`ere (HCMA) equation(2.1) G n +1 = ( π ∗ ω + dd c Φ) n +1 = 0 , in a suitable sense on Y .The boundary value of Φ is required to be in the space H of the smoothK¨ahler potentials. We say that G is a geodesic connecting two points ϕ , ϕ ∈H if Φ | X ×{ } = ϕ ; Φ | X ×{ } = ϕ , where we identify the annulus Σ by a cylinder [0 , × S via the standarddiffeomorphism.It is proved by Chen ([8]) that such a geodesic is unique with fixed bound-ary value, and has the so called C , ¯1 -regularities, namely, writing G locallyas g τ ¯ τ dτ ∧ d ¯ τ + n X α,β =1 ( g τ ¯ β dτ ∧ d ¯ z β + g α ¯ τ dz α ∧ d ¯ τ + g α ¯ β dz α ∧ d ¯ z β ) , we have || g τ ¯ τ || L ∞ + n X α,β =1 ( || g τ ¯ β || L ∞ + || g α ¯ τ || L ∞ + || g α ¯ β || L ∞ ) < + ∞ . In other words, there exist a uniform constant
C > ≤ G ≤ C ( π ∗ ω + idτ ∧ d ¯ τ )on Y . Therefore the quasi-plurisubharmonic function Φ is of class C ,α forany α ∈ (0 , G n +1 can be interpreted in the senseof Bedford and Talyor ([3]).There is another way to describe the domain of G , which will be useful inour later consideration on energy functionals. LetΓ := { z ∈ C ; 0 ≤ Rez ≤ } be a strip domain in C . Then there is a holomorphic map from Γ to Σ as τ ( z ) := e z . This is a branched cover of Σ, and the inverse map is multi-valued in general.Fortunately, G is S -invariant in the argument direction of Σ. Therefore,we can select one of the branches as the inverse map of τ . Then the pull backof G under τ is the unique solution of the HCMA equation on Γ × X , whichis independent of the imaginary part of z . For this reason, the z -variable onΓ can be taken as the complex coordinate of the cylinder R := [0 , × S , and then we can view that the solution G is actually defined on R × X .Furthermore, we can identify the complex variable z := t + is with its realpart t for the same reason. EODESICS APPROXIMATION 5
The Mabuchi functional.
It was in introduced in ([2]) by Mabuchithe following functional on the space HM := R E − E
Ricω + H, where the constant R is the average of the scalar curvature R = nc ( X ) · [ ω ] n − [ ω ] n . The energy functional E is defined for any ϕ ∈ H as E ( ϕ ) := 1 n + 1 n X i =0 ˆ X ϕω i ∧ ω n − iϕ . The twisted energy functional E α (by a closed smooth (1 ,
1) form α ) isdefined as E α ( ϕ ) := n − X i =0 ˆ X ϕω i ∧ ω n − i − ϕ ∧ α. Finally, the entropy functional is H ( ϕ ) := ˆ X (cid:18) log ω nϕ ω n (cid:19) ω nϕ . Suppose Φ is a π ∗ ω -plurisubharmonic function on Y , which correspondsto a geodesic G . Then its restriction ϕ τ := Φ | X τ on a fiber is actuallya ω -plurisubharmonic function on X τ and has the C , ¯1 -regularities. It isobserved that M can be defined on such functions. Therefore, we can writethe Mabuchi functional along the geodesic G as K ( τ ) := M ( ϕ τ ) , τ ∈ Σ . Next, we introduce the following modified versions of the Mabuchi func-tional. Suppose Ψ( τ, · ) = ψ τ ( · ) is a locally bounded singular metric onthe relative canonical bundle K Y/ Σ , and then − ψ τ is a metric on the anti-canonical line bundle − K X τ := V n T X τ . Therefore, the following is a mea-sure on X µ := e ψ τ . We note that µ is absolutely continuous with respect to the Lebesgue mea-sure. Now define the following energy function on Σ as K Ψ ( τ ) := R E ( ϕ τ ) − E Ricω ( ϕ τ ) + ˆ X log (cid:18) e ψ τ ω n (cid:19) ω nϕ τ . This energy function will equal to K ( τ ), if Ψ is the (unbounded) metricdefined by ω nϕ τ . For any large constant A , Berman-Berndtsson ([5]) intro-duced the following A-truncated Mabuchi functional along the geodesic G . K Ψ A ( τ ) := R E ( ϕ τ ) − E Ricω ( ϕ τ ) + ˆ X log (cid:18) max (cid:26) ω nϕ τ ω n , h A ω n (cid:27)(cid:19) ω nϕ τ , where h A := e χ − A , and χ is a fixed continuous metric on K Y/ Σ satisfying dd c χ ≥ k ( π ∗ ω + dd c Φ) , for some positive integer k . LONG LI
The values of these energy functions K , K Ψ , K Ψ A do not depend on theargument part of Σ, and hence we can view that they are actually definedon the cylinder R = [0 , × S by our previous discussion. Moreover, itis proved ([5]) that K Ψ A is a convex function, and it converges to K as A → ∞ by the dominated convergence theorem. Eventually, we concludethe following convexity result ([5], [13]). Theorem 2.1.
The Mabuchi functional M is convex and continuous along a C , ¯1 -geodesic G . That is to say, K is an S -invariant, convex and continuousfunction on R . Energies on the ε -geodesics Suppose we have two points ϕ , ϕ ∈ H . For each ε > π ∗ ω -plurisubharmonic function Φ ε on Γ × X satisfying(3.1) G n +1 ε = ( π ∗ ω + dd c Φ ε ) n +1 = ε √− dt ∧ d ¯ t ∧ ω n , with boundary conditionsΦ ε (0 , · ) = ϕ ( · ); Φ ε (1 , · ) = ϕ ( · ) . Then we say that G ε := π ∗ ω + dd c Φ ε is the ε -geodesic connecting with ϕ and ϕ . It is proved ([8]) that the ε -geodesic G ε is uniformly bounded in the C , ¯1 -norm. Moreover, we have known that Φ ε → Φ in C ,α -norm for any α ∈ (0 , W ,p -norm for all 1 < p < ∞ .In general, we write the restriction of the geodesic potential on each fiber X t := { t } × X as ϕ ( t, · ) := Φ | X t ; ω ϕ := G| X t = ω + dd c ϕ. Similarly, we have for the ε -geodesic potential ϕ ε ( t, · ) := Φ ε | X t ; ω ε := G ε | X t = ω + dd cX ϕ ε . Then a standard computation shows the following equation(3.2) ( π ∗ ω + dd c Φ ε ) n +1 = ρ ε √− dt ∧ d ¯ t ∧ ω nε , where ρ ε := ρ ε ( ϕ ε ) = g t ¯ t − g ¯ βα g α ¯ t g t ¯ β , and hence the ε -geodesic equation(3.1) can be re-written as(3.3) ρ ε ( ϕ ε ) = ε ω n ω nε . Energy and entropy.
Denote K ε : Γ → R by the restriction of theMabuchi functional M to the ε -geodesic as K ε ( t ) := R E ( ϕ ε ) − E Ricω ( ϕ ε ) + ˆ X log ω nε ω n ω nε . Then the A -truncated Mabuchi functional along the ε -geodesic can also beintroduced as K ε,A ( t ) := R E ( ϕ ε ) − E Ricω ( ϕ ε ) + ˆ X log max (cid:26) ω nε ω n , h A ω n (cid:27) ω nε , EODESICS APPROXIMATION 7 where h A := h ε,A = e χ ε − A is a smooth volume element whose associatedcurvature is greater than − C G ε for some fixed positive constant C . Moreprecisely, we construct this auxiliary element as follows.Let χ be a smooth metric on the line bundle K X , and k be a positivenumber such that dd c χ + k ω >
0. Then we set χ ε := π ∗ χ − k Φ ε , andhence dd c χ ε = π ∗ χ + k π ∗ ω − k ( π ∗ ω + dd c Φ ε ) ≥ − k G ε . We emphasis that the sub-index ε in the notation h A is omitted, since χ ε is uniformly bounded in its C , ¯1 -norm, and converges uniformly to χ := π ∗ χ − k Φ in C ,α -norm for any α ∈ (0 , K ε,A is continuous in t by the construction. Moreover, itscomplex Hessian can be computed in sense of local currents. Suppose v is alocally compact supported smooth test function on Γ h dd c K ε,A , v i = R ˆ Γ × X v ( π ∗ ω + dd c Φ ε ) n +1 − ˆ Γ × X v ( π ∗ ω + dd c Φ ε ) n ∧ π ∗ Ric ( ω )+ ˆ Γ × X vdd c (cid:18) max (cid:26) log ω nε ω n , log h A ω n (cid:27)(cid:19) ∧ ( π ∗ ω + dd c Φ ε ) n . (3.4)In other words, if we take the fiberwise integral ´ X t as the push forwardoperator acting on the currents from Γ × X to Γ, we obtain the followingequation. dd c K ε,A ( t ) = Rn + 1 ˆ X t G n +1 ε − ˆ X t Ric ( ω ) ∧ G nε + ˆ X t dd c (cid:18) max (cid:26) log ω nε ω n , log h A ω n (cid:27)(cid:19) ∧ G nε . (3.5)On the one hand, we have seen locally(3.6) 1 n + 1 (∆ G ε log h A ) G n +1 ε = dd c log h A ∧ G nε ≥ − k G n +1 ε . Next, we denote the function f ε : Γ × X → R by the equality ω nε ω n (cid:12)(cid:12)(cid:12)(cid:12) X t = e f ε ( t, · ) , and introduce the following sub-level set P ε,A := (cid:26) ( t, z ) ∈ Γ × X ; f ε ( t, z ) > log h A ω n ( t, z ) (cid:27) . It is proved in ([13]) that there exists a positive number c A such thatthe following inequality is satisfied in P ε,A (see more details in the followingsections).(3.7) ( dd c f ε − Ric ( ω )) ∧ G nε ≥ − c A G n +1 ε . LONG LI
We emphasis that c A only depends on the constant A , the background metric ω and the uniform upper bound of ω ε , and it can be assumed to be increasingin A . Hence we have the following estimate locally in P ε,A +1 .(3.8) (∆ G ε log ω nε ) G n +1 ε ≥ − c A +1 G n +1 ε . By a Theorem proved by Greene-Wu (see Lemma (5.2), [13]), we canestablish the following inequality in an open neighbourhood of each pointon Γ × X .(3.9) (∆ G ε max { log ω nε , log h A } ) G n +1 ε ≥ − c ′ A G n +1 ε , where c ′ A := max { c A +1 , ( n + 1) k } . Thus we infer the inequality(3.10) (cid:18) dd c max (cid:26) log ω nε ω n , log h A ω n (cid:27) − Ricω (cid:19) ∧ G nε ≥ − c ′ A G n +1 ε , globally on Γ × X . Combing this estimate with equation (3.5), we have thefollowing result. Theorem 3.1 (Chen-Li-P˘aun) . For each positive number A , there is a uni-form constant C A > such that the function (3.11) e K ε,A := K ε,A ( t ) − εC A t (1 − t ) is convex and continuous on [0 , for each ε > small. One may expect that K ε,A converges to the energy K . If so, then we candirectly conclude the convexity of M by Theorem (3.1).The convergence of the energy parts of M , i.e. E ( ϕ ε ) → E ( ϕ ) and E α ( ϕ ε ) → E α ( ϕ ) as ε →
0, is indeed true ([5], [20]). However, there isa rather severe difficulty: the convergence of the fiber-wise volume element ω nε ω n ⇀ ω nϕ ω n is only known to be in the weakly L p sense for any 1 < p < ∞ . Un-fortunately, this is not enough to conclude the convergence of the entropyfunctional H ( ϕ ε ).Nevertheless, we have the following estimate on the entropy functionals.Recall that the entropy functional along a geodesic G is defined as H ( ϕ ) := ˆ X f ( ϕ ) log f ( ϕ ) · ω n , where f ( ϕ ) := ω nϕ /ω n , and its truncated version is H A ( ϕ ) := ˆ X f ( ϕ ) log f A ( ϕ ) · ω n , where f A ( ϕ ) := max (cid:26) ω nϕ ω n , h A ω n (cid:27) . As before, we omit the sub-index ε in the definition of f A as in h A ’s, andhope that this will be clear from the context. Then the energy function K ε,A can be re-written as(3.12) K ε,A ( t ) = R E ( ϕ ε ) − E Ricω ( ϕ ε ) + H A ( ϕ ε ) . EODESICS APPROXIMATION 9
Finally we state the following lower semi-continuity type property (seeLemma (4.8), [13]) for the truncated entropy functionals.
Lemma 3.2.
There exists a function η : R + → R + with η ( x ) → as x → + ∞ such that we have (3.13) lim inf ε → H A ( ϕ ε ) ≥ H A ( ϕ ) − η ( A ) , for all A large enough. We emphasis that η ( A ) is independent of ε and t .3.2. The affine energy.
From now on, we assume that the Mabuchi func-tional M is affine along G , namely, we have K ( t ) is a linear function on [0 , K A by the following limit for each t ∈ [0 ,
1] and A large K A ( t ) := lim sup ε → K ε,A ( t ) = lim sup ε → e K ε,A ( t ) , where e K ε,A is defined in equation (3.11). Then K A is a convex function onthe unit interval [0 , K A obtained from takingthe limit of K ε,A is exactly equal to K Ψ A (see Corollary (3.4)). First, wenote that K A is actually a decreasing sequence in A . This is because K ε,A is a decreasing sequence in A for each ε fixed. In fact, we have H A ′ ( ϕ ε ) ≤ H A ( ϕ ε ) , for each t ∈ [0 ,
1] and any A ′ ≥ A , since f A ′ ( ϕ ε ) ≤ f A ( ϕ ε ) and f ϕ ≥ Theorem 3.3.
Suppose the Mabuchi functional M is affine along a geodesic G . Then there is a positive number A such that we have K A ( t ) = K ( t ) , for all t ∈ [0 , and A ≥ A .Proof. Up to a linear function on [0 , M is identically zero along G , namely we have on [0 , K ( t ) ≡ . The first observation is that we have(3.14) K A (0) = 0; K A (1) = 0 , since the ε -geodesic potential Φ ε coincides with the geodesic potential Φ foreach ε on the boundary of R × X . Then it is easy to see that K ε,A (0) = K ε,A (1) = 0 for all A large enough.As a convex function on [0 , K A is upper semi-continuous near theboundaries, and then we havelim sup t → , K A ( t ) ≤ . Thanks to the convexity again, K A must be below the line segment joiningits two boundaries. Therefore, it is non-positive under our assumption andwe have(3.15) K A ( t ) ≤ , for all t ∈ [0 , t ∈ (0 , K A ( t ) ≥ K ( t ) − η ( A ) , This directly follows from Lemma (3.2), where we have provedlim sup ε → H A ( ϕ ε ) ≥ H A ( ϕ ) − η ( A ) . Next define a new function on [0 ,
1] as e K ( t ) := lim sup A → + ∞ K A ( t ) . For the same reason, e K is a convex function which verifies the boundarycondition e K (0) = e K (1) = 0 by equation (3.14). Moreover, inequality (3.16)implies that e K ( t ) ≥ t ∈ (0 , η ( A ) → A → + ∞ byLemma (3.2). Therefore, we conclude that e K must be identically equal tozero for all t ∈ [0 , K A is actually a decreasing sequence in A ,namely, we have K A ( t ) ց e K ( t ) for each t ∈ [0 , K A ( t ) ≥ , for each t ∈ [0 , A large enough K A ( t ) ≡ , (cid:3) In other words, the limit of the A -truncated Mabuchi functional on the ε -geodesics will coincide with M for all A large enough, provided the linearityof the Mabuchi functional along G .3.3. Gap phenomenon.
When the Mabuchi functional is affine along G ,we have seen from ([20]) that the A -truncated Mabuchi functional K Ψ A willalso coincide with K for all A large. This implies(3.17) H A ( ϕ ) = H ( ϕ ) , for all A large enough, and then we have the following corollary. Corollary 3.4.
Suppose the Mabuchi functional M is affine along a geodesic G . Then for all A large enough, the following limit exists for each t ∈ [0 , and satisfies (3.18) lim ε → K ε,A ( t ) = K Ψ A ( t ) = K ( t ) . Proof.
For each t ∈ [0 ,
1] fixed, we define e H A ( t ) := lim inf ε → H A ( ϕ ε ) . Thanks to Lemma (3.2), we have(3.19) lim A → + ∞ e H A ( t ) ≥ H ( ϕ ) , EODESICS APPROXIMATION 11 by equation (3.17). However, e H A is actually a decreasing sequence in A .This follows from its construction and the fact that H A ( ϕ ε ) is decreasingfor each ε fixed. Hence, we conclude(3.20) e H A ( t ) ≥ H ( ϕ ) . On the other hand, Theorem (3.3) implies for all A large(3.21) lim sup ε → H A ( ϕ ε ) = H ( ϕ ) = H A ( ϕ ) . Combined the two inequalities above together, we have(3.22) H ( ϕ ) = lim sup ε → H A ( ϕ ε ) ≥ lim inf ε → H A ( ϕ ε ) ≥ H ( ϕ ) , at each t ∈ [0 , (cid:3) Moreover, equation (3.17) implies the so called “gap phenomenon” ([20])for the fiber-wise volume element of the geodesic G . Denote P by the fol-lowing measurable subset P := { ( t, z ) ∈ Γ × X ; f ( ϕ ) > } , and P A by the following sub-level set P A := (cid:26) ( t, z ) ∈ Γ × X ; f ( ϕ ) > e χ − A ω n (cid:27) . For each t ∈ [0 , P t := P \ X t ; P A,t := P A \ X t . Then there is a large constant A such that for all A ≥ A , and each t ∈ [0 , X t = P A,t [ P ct up to a set of measure zero. In other words, there exists a uniform constant κ > X t (3.24) f ( ϕ ) > κ or f ( ϕ ) = 0almost everywhere. We emphasis that κ does not depend on t or x . This iscalled the gap phenomenon for the fiberwise volume element of the geodesic G . 4. Convergence of the volume element
Recall that the ε -geodesic potential Φ ε converges to the geodesic potentialΦ in the weak C , ¯1 -norm. Fixing a fiber X τ , we can pick up a convergentsubsequence of the volume elements on this fiber f ℓ ( x ) := f ( ϕ ε ℓ )( x ); f ( x ) := f ( ϕ )( x ) , satisfying f ℓ → f weakly in L p for all p ≥ X τ . It is interesting to know whether we havestrong convergence in L p for this sequence or not, and it turns out that thisis indeed the case if the Mabuchi functional is affine. Theorem 4.1.
Suppose the Mabuchi functional M is affine along a geodesic G . Then the fiber-wise volume element of the ε -geodesic converges to thevolume element of the geodesic in the strong L sense as ε → . Moreprecisely, we have lim ℓ → + ∞ || f ℓ − f || L = 0 , on each fiber X τ . In the following, we will provide two different proves of Theorem (4.1).The first one is more complicated, but it will give an accurate estimate forthe L -norm of the difference f ℓ − f directly from the convergence of thetruncated entropies. We expect that this estimate will be useful for someindependent interests.For the beginning, denote κ by the auxiliary function on the fiber X τ (4.1) κ ( x, A ) := e χ − A ω n ( x ) . As before, we omit the sub-index ε ℓ in χ due to the uniform control on their C , ¯1 -norms. Then the following result holds. Lemma 4.2.
Suppose we have lim ℓ →∞ ˆ X f ( ϕ ε ℓ ) log f A ( ϕ ε ℓ ) = ˆ X f ( ϕ ) log f A ( ϕ ) . Then there exists a uniform constant C , only depending on the upper boundof f ℓ and f , satisfying (4.2) lim ℓ →∞ ˆ X ( f ℓ − f ) ω n ≤ (cid:18) Cκ (cid:19) max X τ κ, for all κ small enough (or A large enough). Here κ is the gap of the volume element f ( ϕ ) (equation (3.24)), whichis a fixed constant. Therefore, Theorem (4.1) directly follows from Lemma(4.2) if we take A → ∞ .4.1. The maximum function.
In order to prove this lemma, the firststep is to investigate the following maximum function. Define a function F : [0 , + ∞ ) → R as F ( x ) := x log x, and F (0) = 0. Then F is a convex continuous function on its domain. Infact, it is smooth in R + , and its first and second derivatives are F ′ ( x ) = log x + 1; F ′′ ( x ) = x − Moreover, we can truncate F by a small number κ > h κ ( x ) := max { x log x, x log κ } . This is also a convex and continuous function on [0 , + ∞ ), and it is piecewisesmooth in this domain. Its first derivative exists everywhere on R + exceptat the point x = κ , and we have(4.3) h ′ κ ( x ) = (cid:26) log κ, for x < κ x, for x > κ. EODESICS APPROXIMATION 13
Furthermore, we can also compute its second derivative on R + as(4.4) h ′′ κ ( x ) = , for x < κδ ( κ ) , for x = κ x , for x > κ, where δ ( κ ) is the Dirac-delta function at the point x = κ . We note that theFundamental Theorem of Calculus is still satisfied for h κ , h ′ κ , h ′′ κ on the inter-val [0 , x on the fiber, we introduce another variable t ∈ [0 ,
1] andtake u t := tf ℓ + (1 − t ) f = at + b, where a := ( f ℓ − f )( x ) and b := f ( x ). We note that u t is strictly positiveand has a uniform upper bound for all t , ℓ and x . Define a new compositionfunction F ℓ,κ ( t ) := h κ ( u t ) , and then its derivatives can be written as(4.5) F ′ ℓ,κ ( t ) = (cid:26) a log κ, for at + b < κa log( at + b ) + a, for at + b > κ ;(4.6) F ′′ ℓ,κ ( t ) = , for at + b < κaδ ( t ) , for at + b = κ a at + b , for at + b > κ, where t is determined by the equation at + b = κ. In particular, we have F ′ ℓ,κ (0) = a log κ if f ( x ) < κ , and F ′ ℓ,κ (0) = a log b + a if f ( x ) > κ . Then the following convergence holds. Lemma 4.3.
For all A large enough, we have lim ℓ → + ∞ ˆ X F ′ ℓ,κ (0) ω n = 0 . Proof.
When the constant A is large enough, we can assume κ < κ / X τ , where κ is the gap of the fiber-wise volume element of G definedin equation (3.24). Then the fiber can be completely decomposed into twoparts as in equation (3.23) X τ = P A,τ [ P cτ , up to a set of measure zero. Recall that the two sets can be re-written asfollows P A,τ = { x ∈ X τ ; f ( x ) > κ ( x ) } ; P cτ := { x ∈ X τ ; f ( x ) = 0 } . Then we have ˆ X F ′ ℓ,κ (0) = ˆ P A,τ F ′ ℓ,κ (0) + ˆ P cτ F ′ ℓ,κ (0)= ˆ P A,τ ( f ℓ − f ) log f + ˆ P A,τ ( f ℓ − f ) + ˆ P cτ f ℓ log κ = ˆ P A,τ ( f ℓ − f ) log f + ˆ P cτ ( f ℓ − f ) log κ + ˆ P cτ f log κ + ˆ X ( f ℓ − f ) − ˆ P cτ ( f ℓ − f )= ˆ X ( f ℓ − f ) log max { f, κ } + ˆ X ( f ℓ − f ) − ˆ P cτ f ℓ . (4.7)The three terms on the RHS of equation (4.7) will all converge to zero as ℓ → + ∞ , since || f || L ∞ , || f ℓ || L ∞ are uniformly bounded and f ℓ → f weaklyin L p for any p ≥
1, and then our result follows. (cid:3)
The four cases.
Next we will apply the Fundamental Theorem ofCalculus on the function F ℓ,κ ( t ) and its first derivative, namely, we have F ℓ,κ (1) − F ℓ,κ (0) = h κ ( f ℓ ) − h κ ( f )= ˆ t F ′ ℓ,κ ( t ) dt + ˆ t F ′ ℓ,κ ( t ) dt. (4.8)A first observation is that the point t may not be in the integrationdomain above. Suppose the point x is in the subset P cτ , and then we have b = f ( x ) = 0, a = f ℓ ( x ) >
0. Therefore, the point t belongs to the interval(0 ,
1) if and only if 0 < κ < a .Otherwise, we have t ≥ f ℓ ( x ) ≤ κ , but t ≤ κ ≤ x is in the subset P A,τ . Then wehave b = f ( x ) > κ and a = f ℓ ( x ) − f ( x ). Hence t ∈ (0 ,
1) if and only if a < b > κ > a + b = f ℓ ( x ).Otherwise, when a ≥
0, we have f l ( x ) ≥ f ( x ) > κ for t ≤
0; or when a <
0, we have κ ≤ f ℓ ( x ) for t ≥ x ∈ P A,τ , f ℓ ( x ) ≥ f ( x ) > κ or f ( x ) > f ℓ ( x ) ≥ κ ;(ii) x ∈ P cτ , f ( x ) = 0 and f ℓ ( x ) ≤ κ ;(iii) x ∈ P cτ , f ( x ) = 0 and f ℓ ( x ) > κ ;(iv) x ∈ P A,τ , f ℓ ( x ) < κ < f ( x ).We note that these four cases are disjoint from each other. Then wewill discuss case by case. For Case (i) , we note that u t > κ and then EODESICS APPROXIMATION 15 h κ ( u t ) = u t log u t for all t ∈ [0 , F ℓ,κ (1) − F ℓ,κ (0) = F ′ ℓ,κ (0) + ˆ ˆ t F ′′ ℓ,κ ( s ) dsdt = F ′ ℓ,κ (0) + ˆ ˆ t a dsas + b dt ≥ F ′ ℓ,κ (0) + a C , (4.9)where the constant C is the uniform upper bound of u t .For Case ( ii ), we note u t ≤ κ and then h κ ( u t ) = u t log κ . Hence we have F ℓ,κ (1) − F ℓ,κ (0) = F ′ ℓ,κ (0) + ˆ ˆ t F ′′ ℓ,κ ( s ) dsdt = F ′ ℓ,κ (0) . (4.10)The two cases above are the easy ones. For the remaining cases, we willutilise equations (4.5) and (4.6) in the computation. In Case ( iii ), we notethat h κ ( u t ) = u t log κ for t ≤ t and h κ ( u t ) = u t log u t for t > t . Recallthat t = κ/a in this case, and hence the computation follows. F ℓ,κ (1) − F ℓ,κ (0) = ˆ κ/a F ′ ℓ,κ ( t ) dt + ˆ κ/a F ′ ℓ,κ ( t ) dt = κa F ′ ℓ,κ (0) + ˆ κ/a ˆ t F ′′ ℓ,τ ( s ) dsdt + ˆ κ/a F ′ ℓ,κ ( t ) dt = F ′ ℓ,κ (0) + ˆ κ/a ( a + ˆ tκ/a F ′′ ℓ,κ ( s ) ds ) dt = F ′ ℓ,κ (0) + ( a − κ ) + ˆ κ/a ˆ tκ/a a dsas + b dt ≥ F ′ ℓ,κ (0) + ( a − κ ) + a C (1 − κ/a ) ≥ F ′ ℓ,κ (0) + a C + a (1 − κ/C ) − κ. (4.11)Recall that a > F ℓ,κ (1) − F ℓ,κ (0) ≥ F ′ ℓ,κ (0) + a C − κ, for all κ small enough. Finally, the most difficult one is Case ( iv ). As before,we first note that h κ ( u t ) = u t log u t for t ≤ t and h κ ( u t ) and h κ ( u t ) = u t log κ for t > t . Then we compute in a similar way. F ℓ,κ (1) − F ℓ,κ (0) = ˆ κ − ba F ′ ℓ,κ ( t ) dt + ˆ κ − ba F ′ ℓ,κ ( t ) dt = κ − ba F ′ ℓ,κ (0) + ˆ κ − ba ˆ t F ′′ ℓ,τ ( s ) dsdt + ˆ κ − ba F ′ ℓ,κ ( t ) dt = F ′ ℓ,κ (0) + ˆ κ − ba ˆ t a dsas + b dt + ˆ κ − ba ( ˆ κ − ba F ′′ ℓ,κ ( s ) ds + a ) ≥ F ′ ℓ,κ (0) + a + b + ( κ − b ) C + ( κ − b )( a + b − κ ) C − κ. (4.13)Recall that we have a + b = f ℓ ( x ) > κ − b = κ − f ( x ) < a + b − κ = f ℓ ( x ) − κ <
0. Hence the following estimate holds. F ℓ,κ (1) − F ℓ,κ (0) ≥ F ′ ℓ,κ (0) + ( κ − b ) C − κ ≥ F ′ ℓ,κ (0) + κ C − κ. (4.14)The last inequality in equation (4.14) is because that we have picked up κ < κ /
2, and P A,τ is actually the set where f ( x ) > { x ∈ X τ ; f ( x ) > κ } up to a set of measure zero by the gap phenomenon.Combining with equations (4.9) - (4.14) above, we conclude the followinginequality after taking the integral on X τ . ˆ X ( h κ ( f ℓ ) − h κ ( f )) ω n ≥ ˆ X F ′ ℓ,κ (0) ω n − max X τ κ + 12 C ˆ ( P cτ T { f ℓ ≤ κ } ) c T ( P A,τ T { f l <κ By our choice on the function κ and the maximumfunction h κ , it follows(4.16) H A ( ϕ ε ℓ ) − H A ( ϕ ) = ˆ X ( h κ ( f ℓ ) − h κ ( f )) ω n . Thanks to Corollary (3.4), the LHS of equation (4.15) converges to zero as ℓ → + ∞ . Meanwhile, our Lemma (4.3) implies the first term on the RHSof equation (4.15) also converges to zero. Therefore, it implies(4.17) 8 Cκ (max X τ κ ) ≥ lim ℓ → + ∞ µ (cid:16) P A,τ \ { f l < κ < f } (cid:17) . EODESICS APPROXIMATION 17 Furthermore, we note that the two subsets P A,τ T { f l < κ < f } and P cτ T { f ℓ ≤ κ } are mutually disjoint. Then the third term on the RHS ofequation (4.15) can be decomposed into the following three parts. ˆ X ( f ℓ − f ) − ˆ P cτ T { f ℓ ≤ κ } ( f ℓ − f ) − ˆ P A,τ T { f l <κ From the previous argument, we can see that the L -convergence of the difference ( f ℓ − f ) actually follows from the conver-gence of the entropy as(4.21) lim ℓ →∞ ˆ X f ( ϕ ε ℓ ) log f ( ϕ ε ℓ ) = ˆ X f ( ϕ ) log f ( ϕ ) , but this is a direct consequence of our Corollary (3.4) as follows. Corollary 4.4. Suppose the Mabuchi functional is affine along a geodesic G . Then on each fiber X τ , τ ∈ [0 , , we have lim ε → H ( ϕ ε ) = H ( ϕ ) . Proof. During the proof of Corollary (3.4), we have seen that H A ( ϕ ε ) isdecreasing in A for each ε fixed. Moreover, Theorem (3.3) and the gapphenomenon imply thatlim sup ε → H A ( ϕ ε ) = H A ( ϕ ) = H ( ϕ ) , for all A large enough. Therefore, we conclude the following inequality(4.22) lim sup ε → H ( ϕ ε ) ≤ lim sup ε → H A ( ϕ ε ) = H ( ϕ ) . On the other hand, by the lower semi-continuity property of the entropyfunctional, we have(4.23) lim inf ε → H ( ϕ ε ) ≥ H ( ϕ ) , and then our result follows. (cid:3) In particular, the above corollary implies the convergence of the Mabuchifunctional along the ε -geodesic, namely, we havelim ε → K ε ( t ) = K ( t ) , for each t ∈ [0 , f ℓ of the ε -geodesic converges to the volume element f of the geodesic inmeasure. Moreover, thanks to the Riesz-Lebesgue Theorem, we have f ℓ → f almost everywhere on each fiber, possibly after passing to a subsequence.5. The ε -affine energy and non-degneracy Suppose the Mabuchi functional M is affine along a geodesic G . For theconvergence e K ε,A → K A of a sequence of convex functions, the first derivative e K ′ ε,A ( t ) also converges uniformly to the slope k := K ′ A ( t ) on the closedinterval [ δ, − δ ] for any δ > e K ′ ε,A ( t ) is unclear in general. Therefore, we impose the following conditionon the boundary of G . Definition 5.1. The Mabuchi functional M is essentially affine along ageodesic G , if K ( t ) is a linear function on [0 , with slope k and we have (5.1) − ˆ X ˙ ϕ t ( R ϕ t − R ) ω nϕ t = k, at t = 0 , . It is proved in Berman-Berndtsson ([5]) that the one side inequality ofequation (5.1) always holds at the two boundaries. − ˆ X ˙ ϕ ( R ϕ − R ) ω nϕ ≤ k, and − ˆ X ˙ ϕ ( R ϕ − R ) ω nϕ ≥ k. As we have seen before, the potential ϕ ε ( t, · ) of the ε -geodesic convergesto the potential ϕ ( t, · ) of the geodesic G in C ,α -norm for each α ∈ (0 , 1) onΓ × X . Therefore, we have(5.2) lim ε → ˆ X ˙ ϕ ε ( R ϕ ε − R ) ω nϕ ε = ˆ X ˙ ϕ ( R ϕ − R ) ω nϕ , at the boundaries t = 0 , 1, since ω ε coincides with ω ϕ at these two bound-aries. Moreover, we note that the volume element ω nε varies smoothly nearthe boundaries, and then f A ( ϕ ε ) will keep to be the same as f ( ϕ ε ) for all A large in a small neighbourhood near the boundaries.Therefore, the A -truncated Mabuchi functional along G ε is equal to theMabuchi functional along G ε in a small neighbourhood of the two boundaries.That is to say, there exist a small number δ > 0, possibly depends on ε and A , such that we have EODESICS APPROXIMATION 19 (5.3) K ε,A ( t ) = K ε ( t ) , for all t ∈ [0 , δ ) S (1 − δ, t in this small interval(5.4) K ′ ε,A ( t ) = K ′ ε ( t ) = − ˆ X ˙ ϕ ε ( R ϕ ε − R ) ω nϕ ε . In particular, we conclude the following result. Lemma 5.2. Suppose the Mabuchi functional M is essentially affine alonga geodesic G with slope k . Then K ′ ε,A converges uniformly to k on [0 , forall A large enough.Proof. Combing with equation (5.2) and (5.4), we have for each A large(5.5) lim ε → e K ′ ε,A ( t ) = lim ε → K ′ ε,A ( t ) = k, for t = 0 , 1. Moreover, the function e K ε,A ( t ) is convex on the interval [0 , e K ′ ε,A (0) ≤ e K ′ ε,A ( t ) ≤ e K ′ ε,A (1) , for all t ∈ (0 , e K ′ ε,A ( t ) → k uniformly as ε → (cid:3) However, this condition is still too difficult to handle in the application.Therefore, we will introduce an even stronger one based on the ε -geodesicas follows. Definition 5.3. The Mabuchi functional M is ε -affine along a geodesic G ,if K ( t ) is a linear function on [0 , and we have K ′ ε (1) − K ′ ε (0) = O ( ε ) , where K ε is the Mabuchi functional along the ε -geodesic G ε . For the Mabuchi functional M along G , it is easy to see that the ε -affineis a stronger condition than the essentially affine. Therefore, Lemma (5.2)implies that K ′ ε,A converges to the constant slope k uniformly on [0 , 1] if M is ε -affine. Moreover, we have the following result under this assumption. Theorem 5.4. Suppose the Mabuchi functional M is ε -affine along a geo-desic G . Then G is uniformly fiberwise non-degenerate, namely, there existsa uniform constant κ > such that G| X t > κ ω, for almost everywhere t ∈ [0 , . We emphasis that the constant κ does not depend on t . Before movingon, we need to recall and improve some computations in [13]. Computations. We will take a closer look at equation (3.7), and tryto evaluate the lower bound of the following ( n + 1 , n + 1) form on P ε,A ( dd c f ε − Ric ( ω )) ∧ G nε . Locally near a point p ∈ Γ × X , we write the ε -geodesic as follows G ε := g t ¯ t dt ∧ d ¯ t + n X α,β =1 (cid:16) g t ¯ β dt ∧ d ¯ z β + g α ¯ t dz α ∧ d ¯ t + g α ¯ β dz α ∧ d ¯ z β (cid:17) . This is a K¨ahler metric on Γ × X , and its restriction on the fiber X t := { t }× X can be written as ω ε ( t, · ) := G ε | X t = n X α,β =1 g α ¯ β dz α ∧ d ¯ z β . Up to a change of holomorphic coordinates on X t , we can assume(5.6) g α ¯ β = δ αβ ; dg = 0at this particular point p . Near this point, the ε -geodesic equation can bere-written as(5.7) ρ ε := g t ¯ t − g ¯ βα g α ¯ t g t ¯ β = εe − f ε . Then we introduce another (1 , 1) form χ ε defined by the following equation χ ε := G ε − ρ ε √− dt ∧ d ¯ t. This (1 , X t , and satisfies χ n +1 ε = 0 . Hence we have G nε = χ nε + nρ ε √− dt ∧ d ¯ t ∧ χ n − ε = χ nε + nρ ε √− dt ∧ d ¯ t ∧ G n − ε (5.8)The first factor nρ ε ( dd c f ε − Ric ( ω )) ∧ √− dt ∧ d ¯ t ∧ G n − ε can be computed as follows.(5.9) nρ ε dd c f ε ∧ √− dt ∧ d ¯ t ∧ G n − ε = ε (∆ ε f ε ) √− dt ∧ d ¯ t ∧ ω n , and(5.10) − nρ ε Ric ( ω ) ∧ √− dt ∧ d ¯ t ∧ G n − ε = − ε (tr ω ε Ricω ) √− dt ∧ d ¯ t ∧ ω n . We note that there exist a constant c A such that we havetr ω ε Ricω < c A , on P ε,A , since the eigenvalues of ω ε are bounded from below (and above) bya uniform constant on this set. Therefore, we conclude the inequality nρ ε ( dd c f ε − Ric ( ω )) ∧ √− dt ∧ d ¯ t ∧ G n − ε ≥ (∆ ε f ε ) G n +1 ε − c A G n +1 ε . (5.11) EODESICS APPROXIMATION 21 Next, we compute the second factor as follows. Introduce the followingvector field on Γ × X as v := ∂∂t − g ¯ βα g t ¯ β ∂∂z α . Then one observes that this vector field v generates the kernel of the (1 , χ ε . Hence we have(5.12) ( dd c f ε − Ric ( ω )) ∧ χ nε = ( dd c f ε − Ric ( ω ))( v, ¯ v ) √− dt ∧ d ¯ t ∧ χ nε Then the goal is to compute the lower bound of the following term ∂ ¯ ∂ log det( g α ¯ β )( v, ¯ v ) . At this point p , a standard computation shows the following equation(here we are using the Einstein summation convention) ∂ ¯ ∂ log det( g α ¯ β )( v, ¯ v ) = g t ¯ t,α ¯ α − X α,β | g t ¯ β,α | − g t ¯ γ g γ ¯ t,α ¯ α − g γ ¯ t g t ¯ γ,α ¯ α + R α ¯ γ g t ¯ α g γ ¯ t (5.13)Taking the Laplacian ∆ ε with respect to the metric ω ε on the both sidesof equation (5.7), we have g t ¯ t,α ¯ α − ε ∆ ε ( e − f ε ) = X α,p | g p ¯ t, ¯ α | + X β,q | g t ¯ q, ¯ β | − R q ¯ p g p ¯ t g t ¯ q + g t ¯ p g p ¯ t,α ¯ α + g p ¯ t g t ¯ p,α ¯ α . (5.14)Combing with the two equations above, it turns out that(5.15) ∂ ¯ ∂ log det( g α ¯ β )( v, ¯ v ) = || ¯ ∂ X v || ε + εe − f ε |∇ ε f ε | − εe − f ε (∆ ε f ε ) . Furthermore, we can improve the above equality as follows. First, weobserve ¯ ∂ X v = − g ¯ βα g t ¯ β, ¯ λ d ¯ z λ ⊗ ∂∂z α . In other words, it can be written in tensors as(5.16) ¯ ∂ λ v α = − g t ¯ α, ¯ λ ; ¯ ∂ λ v t = 0 , at the point p . Moreover, we have in the time direction(5.17) ¯ ∂ t v γ = − g t ¯ t, ¯ γ + g t ¯ β g β ¯ γ, ¯ t ; ¯ ∂ t v t = 0 . By differentiating equation (5.7) once we get(5.18) εe − f ε ( ∂ α f ε ) = g t ¯ t,α − g t ¯ p g p ¯ t,α − g p ¯ t g t ¯ p,α , and similarly(5.19) εe − f ε ( ∂ ¯ β f ε ) = g t ¯ t, ¯ β − g t ¯ q g q ¯ t, ¯ β − g q ¯ t g t ¯ q, ¯ β . Hence we have( εe − f ε ) |∇ ε f ε | = g ¯ βα (cid:8) ( g t ¯ t,α − g p ¯ t g α ¯ p,t ) − g t ¯ p g p ¯ t,α (cid:9) · (cid:8) ( g t ¯ t, ¯ β − g t ¯ q g q ¯ β, ¯ t ) − g q ¯ t g t ¯ q, ¯ β (cid:9) = g ¯ βα ( ∂ t v ¯ α − g t ¯ p ∂ p v ¯ α )( ¯ ∂ t v β − g q ¯ t ¯ ∂ q v β )= X β | ¯ ∂ t v β | + g t ¯ µ g λ ¯ t ∂ µ v ¯ β ¯ ∂ λ v β − g µ ¯ t ∂ t v ¯ β ¯ ∂ µ v β − g t ¯ λ ¯ ∂ t v β ∂ λ v ¯ β . (5.20)On the other side, we can compute the inverse matrix of G ε at the point p . First notice that det G ε ( p ) = εe − f ε ( p ) , since g α ¯ β = δ αβ at this point. Then a standard calculation shows the fol-lowing equations. G ¯ ttε = ( εe − f ε ) − ; G ¯ tpε = − ( εe − f ε ) − g t ¯ p , and also(5.21) G ¯ pqε = δ pq + ( εe − f ε ) − g p ¯ t g t ¯ q Therefore, the four terms on the RHS of equation (5.20) can be re-writtenas(5.22) ( εe − f ε ) − X β | ¯ ∂ t v β | = G ¯ ttε g α ¯ β ¯ ∂ t v α ∂ t v ¯ β ;(5.23) − ( εe − f ε ) − g µ ¯ t ∂ t v ¯ β ¯ ∂ µ v β = G ¯ µtε g α ¯ β ¯ ∂ µ v α ∂ t v ¯ β ;(5.24) ( εe − f ε ) − g t ¯ µ g λ ¯ t ∂ µ v ¯ β ¯ ∂ λ v β = G ¯ λµε g α ¯ β ¯ ∂ λ v α ∂ µ v ¯ β − g α ¯ β ¯ ∂ λ v α ∂ λ v ¯ β . Combining with equations (5.15) - (5.24) above, we eventually derive thefollowing equality. Lemma 5.5. We have ∂ ¯ ∂ log det( g α ¯ β )( v, ¯ v ) = || ¯ ∂ X v || ω ε + εe − f ε |∇ ε f ε | − εe − f ε (∆ ε f ε )= || ¯ ∂v || G ε − εe − f ε (∆ ε f ε ) . (5.25)Combing with Lemma (5.5) and equations (5.11), (5.12), we infer as equa-tion (3.10) the following (cid:18) dd c max (cid:26) log ω nε ω n , log h A ω n (cid:27) − Ricω (cid:19) ∧ G nε + c ′ A G n +1 ε ≥ χ P ε,A (cid:16) || ¯ ∂ X v || ω ε + εe − f ε |∇ ε f ε | (cid:17) idt ∧ d ¯ t ∧ ω nε , (5.26)where χ P ε,A is the characteristic function of the set P ε,A . Finally, by usingthe modified energy e K ε,A (defined in the Theorem (3.1)), we conclude thefollowing integral estimate. EODESICS APPROXIMATION 23 e K ′ ε,A (1) − e K ′ ε,A (0) ≥ ˆ P ε,A || ¯ ∂v || G ε idt ∧ d ¯ t ∧ ω nε ≥ ˆ P ε,A || ¯ ∂ X v || ω ε idt ∧ d ¯ t ∧ ω nε + ε ˆ P ε,A |∇ ε f ε | idt ∧ d ¯ t ∧ ω n . (5.27)If we assume that the Mabuchi functional is essentially affine along G ,then the RHS of equation (5.27) converges to zero as ε → 0. Moreover,suppose the Mabuchi functional is ε -affine along the geodesic. Then wehave e K ′ ε,A (1) − e K ′ ε,A (0) = K ′ ε (1) − K ′ ε (0) + 2 εC A ≤ εC ′ A , (5.28)for some constant C ′ A . Therefore, we conclude the following estimate fromequations (5.27) and (5.28). ˆ P ε,A |∇ f ( ϕ ε ) | idt ∧ d ¯ t ∧ ω n ≤ C ˆ P ε,A |∇ f ( ϕ ε ) | f ( ϕ ε ) idt ∧ d ¯ t ∧ ω n ≤ C ˆ P ε,A |∇ log f ( ϕ ε ) | idt ∧ d ¯ t ∧ ω n ≤ C ′ ˆ P ε,A |∇ ε log f ( ϕ ε ) | idt ∧ d ¯ t ∧ ω n ≤ C ′′ A , (5.29)where we used the fact ω ε ≤ cω for a uniform constant c . Moreover, if wetake F ε,A ( t ) := ˆ P ε,A T X t |∇ f ( ϕ ε ) | ω n , then Fatou’s lemma implies(5.30) ˆ lim inf ε F ε,A ( t ) dt ≤ lim inf ε ˆ F ε,A ( t ) dt ≤ C ′′ A . Therefore, for almost everywhere t ∈ [0 , C t and asubsequence F ε ℓ ,A such that lim ℓ → + ∞ F ε ℓ ,A ( t ) ≤ C t . Hence we have the following result. Lemma 5.6. Suppose the Mabuchi functional is ε -affine along the geodesic.Then there exists a constant C (possibly depending on t and A ), and asubsequence of volume elements f ( ϕ ε ℓ ) satisfying (5.31) ˆ P εℓ,A T X t |∇ f ( ϕ ε ℓ ) | ω n ≤ C, for almost everywhere t ∈ [0 , and any ℓ large enough. Thank to Theorem (4.1), we can further assume that f ( ϕ ε ℓ ) → f ( ϕ ) in L on X t , possibly after passing to a subsequence. Positive lower bound. To deal with the non-degeneracy of the fiber-wise volume element of G , we would like to utilise the partial W , -estimateobtained in equation (5.31) and the L convergence of the volume elements.However, the difficulty is that the integral on the LHS of this equation isnot taken on the whole manifold X , and the integration domain varies withrespect to ε and A . In order to overcome this difficulty, we first investigatea local model as follows.Suppose f ℓ is a sequence of positive smooth functions on the domain D := [0 , m ⊂ R m with uniformly bounded L ∞ -norm, and f is an L ∞ non-negative function on D such that f ℓ converges to f in L -norm. Wefurther assume that the function f satisfies the gap phenomenon, namely,there exists a constant κ > { f > κ } [ { f = 0 } = D, up to a set of measure zero. In the following, we set P := { f > κ } ; P c := { f = 0 } , and assume µ ( P ) > 0. Let κ denote a continuous function on D and we set P ℓ,κ := { x ∈ D ; f ℓ ( x ) > κ } . Then the following result is crucial. Proposition 5.7. Assume max κ < κ / on D . Suppose there exists aconstant C > , such that the following estimate holds (5.32) ˆ P ℓ,κ |∇ f ℓ | < C, for a fixed κ and all ℓ large enough. Then f > κ almost everywhere on D . First we will prove that Proposition (5.7) holds in R , namely, we assume D = [0 , Lemma 5.8. Suppose u is a smooth positive function on [0 , , and weassume that u ( a ) > k and u ( b ) < k for some ≤ a < b ≤ and a constant k > . Then we have | b − a | > k ´ ba ( f ′ ( t )) dt . Suppose E , E are two subsets of D , and we denote d ( E , E ) by thedistance between them d ( E , E ) := inf x ∈ E ,x ∈ E d ( x , x ) . Then the following observation is crucial. Lemma 5.9. Suppose P, Q are two non-empty disjoint subsets of the inter-val [0 , . Assume that the union of P, Q is the whole interval [0 , up to aset with measure zero. Then we have d ( P, Q ) = 0 .Proof. We will prove by contradiction. Suppose the distance between thetwo sets is positive as d ( P, Q ) > δ > . EODESICS APPROXIMATION 25 Take a large number m ∈ Z satisfying m < δ . For each point p ∈ P , wedefine the following open interval as I p,m := ( p − m , p + 1 m ) , and similarly for each point q ∈ Q I q,m := ( q − m , q + 1 m ) . We note that I p,m is disjoint from the set Q for each p ∈ P , and I q,m isdisjoint from the set P for each q ∈ Q by our assumptions. Therefore, thefollowing two unions U := [ p ∈ P I p,m , V := [ q ∈ Q I q,m are mutually disjoint open subsets of the interval [0 , E of [0 , 1] with measure zero satisfying P [ Q [ E = [0 , . We claim that E is contained in the union U S V . Otherwise, there is apoint a ∈ E such that we have d ( { a } , P ) > m , d ( { a } , Q ) > m . Then the open interval ( a − m , a + m ) must be contained in E , but this isimpossible since E has measure zero.Therefore, we have proved that the union U S V is exactly the interval[0 , , 1] is a connected set. (cid:3) After passing to a subsequence, we can assume that f ℓ converges to f almost everywhere on [0 , 1] due to the L convergence. Therefore, there is asubset E ⊂ [0 , 1] with measure zero such that f ℓ → f in the poinwise senseoutside of E . Then we are ready to prove the 1-dimensional case as follows. Lemma 5.10. Proposition (5.7) holds in one dimension.Proof. Thanks to Lemma (5.9), everything boils down to prove d ( P − E, P c − E ) > , if P c is non-empty.We will prove by contradiction again. If not, then there exists a sequenceof pairs ( a j , b j ) ∈ ( P − E ) × ( P c − E ) such that | a j − b j | → j → ∞ .Without loss of generality, we assume a j < b j in the following.Fix one interval [ a j , b j ]. For all ℓ large enough, we have f ℓ ( a j ) > κ f ℓ ( b j ) < κ . Therefore, there exists a point c j ∈ [ a j , b j ] such that f ℓ ( c j ) = κ by thecontinuity of f ℓ . Moreover, the point c j can be chosen close enough to a j such that we have(5.33) f ℓ | [ a j ,c j ] > κ , since the first derivative f ′ ℓ is bounded. Hence we have [ a j , c j ] ⊂ P ℓ,κ asmax κ < κ / 4. Then the following inequality holds by Lemma (5.8) andequation (5.32)(5.34) | b j − a j | > | c j − a j | ≥ κ C . However, this contradicts to the fact that d ( a j , b j ) → j → ∞ , andthen our result follows. (cid:3) Proof of Proposition (5.7). We will use induction on the dimension m . Lemma(5.10) implies that the result is true for m = 1, and we assume that Propo-sition (5.7) holds on R m − for some integer m ≥ x ∈ [0 , m as ( x , · · · , x m − , x m ) = ( x ′ , y ), where x ′ = ( x , · · · , x m − ) and y = x m . Denote X y the ( m − X y := { x ∈ [0 , m ; x = ( x ′ , y ) } . Consider the L difference of f ℓ and f on each slice as F ℓ ( y ) := ˆ X y | f ℓ − f | dx ′ Then the L -convergence of f ℓ − f on [0 , m implies that F ℓ converges tozero as L functions on [0 , F ℓ k suchthat lim ℓ k →∞ F ℓ k ( y ) → , for almost everywhere y ∈ [0 , y . Therefore, after re-writing the sub-index, we can assume(5.35) || f ℓ − f || L ( X y ) → , ℓ → ∞ for almost everywhere y ∈ [0 , ˆ ˆ X y T P ℓ,κ |∇ f ℓ | dx ′ ! dy < C Then by Fatou’s lemma (as we argued in equation (5.30)), there exists aconstant C y > f ℓ , possibly depending on y , satisfying(5.37) ˆ X y T P ℓ,κ |∇ x ′ f ℓ | ≤ ˆ X y T P ℓ,κ |∇ f ℓ | dx ′ < C y , for almost everywhere y ∈ [0 , 1] and all ℓ large.Furthermore, the gap phenomenon (either we have f > κ or f = 0 almosteverywhere ) must be satisfied on X y for almost everywhere y ∈ [0 , µ ( P T X y ) > 0, are satisfied on such a slice X y and a subsequence f ℓ . Then EODESICS APPROXIMATION 27 for almost everywhere y ∈ [0 , f > κ almost everywhere on X y ;(2) f = 0 almost everywhere on X y .In fact, we claim that the Case ( ) occurs for almost everywhere y ∈ [0 , S := { y ∈ [0 , f = 0 a.e. on X y } , will have positive measure on [0 , S := [ y ∈ S X y has positive measure on [0 , m .On the other hand, switch the direction and take another slicing as x := ( x , x , · · · , x m ) = ( y ′ , x ′′ ) , where x ′′ = ( x , · · · , x m ). Repeating our previous argument on this newslicing, we can also conclude that for almost everywhere y ′ ∈ [0 , f > κ or f = 0 as an L ∞ function on the slice X y ′ .However, the measure of the following set on each slice X y ′ ≈ [0 , m − S y ′ := [ y ∈ S (cid:16) X y ′ \ X y (cid:17) is the same for different y ′ , and then µ ( S y ′ ) must be positive by Fubini’sTheorem. In other words, the set P c T X y ′ has positive measure for each y ′ ∈ [0 , f = 0 on X y ′ for almost everywhere y ′ ∈ [0 , µ ( P ) > 0, and our claim follows. (cid:3) Now we are going to prove the main theorem in this section. Proof of Theorem (5.4). For almost everywhere t ∈ [0 , X t such that the estimate in Lemma (5.6) holds for a sequence f ( ϕ ε ℓ ). Then wewill prove that the volume element f ( ϕ ) = ω nϕ /ω n is bounded below by thegap κ on X t as an L ∞ function. Hence the restriction of the geodesic G| X t ,as a metric on the fiber, must have a lower bound determined by κ and itsuniform upper bound.We will prove by contradiction. Suppose the set P c = { f ( ϕ ) = 0 } has apositive measure. As before, we denote P by the set P = { f ( ϕ ) > κ } , and it also has a positive measure. Recall that P ε,A is the subset of Γ × X where f ε is larger than the auxiliary function log e χ − A ω n , or equivalently, P ε,A \ X t = { x ∈ X t ; f ( ϕ ε ( t, x )) > κ } , where κ = e χ − A /ω n is the auxiliary function (equation (4.1)). When theconstant A is large enough, we can assume max X t κ < κ / t ∈ [0 , As a compact connected K¨ahler manifold, the fiber X t has an open cov-ering by holomorphic coordinate charts, namely, we have X t = N [ j =1 U j , and the local trivialisation map π j : B j → U j , where B j ⊂ C n is an openball. Without loss of generality, we assume that each ball B j is centred atthe origin of C n and has radius larger than 2. In fact, we can further assumethat the manifold X t is covered by the union of V j ⊂ U j , where V j denotesthe open set V j := π j ( B ) , for the unit ball B ⊂ C n . Therefore, there exists at least one j , such thatwe have on the coordinate U j (5.38) µ (cid:16) { f ( ϕ ) = 0 } \ V j (cid:17) > . Then we claim that among all these j ’s, there exists at least one j ′ suchthat the set P c does not cover the whole unit ball, namely, we have on U j ′ (5.39) a n > µ (cid:16) { f ( ϕ ) = 0 } \ V j ′ (cid:17) > . If not, then we can find a non-empty subset Λ $ { , · · · , N } such that wehave f = 0 pointwise a.e. on V j for each j ∈ Λ and f > κ pointwise a.e. on V k for each k ∈ Λ c . Take two open sets as U := [ j ∈ Λ V j ; V := [ k ∈ Λ c V k . Then we note that f = 0 pointwise a.e. on U and f > κ pointwise a.e. on V . This implies that the intersection of the two open sets U T V is empty.Otherwise, the intersection is an open set with positive measure, and weboth have f = 0 and f > κ pointwise a.e. on U T V , which is not possible.Therefore, we have(5.40) U \ V = ∅ ; U [ V = X t , but this is also impossible, since X t is a compact connected manifold and U , V are both non-empty finite union of coordinate charts. Hence our claimfollows.Pick up such a j ′ that equation (5.39) holds, and then we take f ℓ := f ( ϕ ε ℓ ) | U j ′ , f := f ( ϕ ) | U j ′ , and consider the m th -cube D := [ − , m in U j ′ . Combined with equation(5.31) and (5.39), all the conditions in Proposition (5.7) are satisfied on thisdomain D for the sequence f ℓ . Therefore, we conclude that f > κ almosteverywhere on D . However, this contradicts to the fact that { f = 0 } haspositive measure on V j ′ ⊂ D , and our main result follows. (cid:3) EODESICS APPROXIMATION 29 Applications As one application of the main result (Theorem (5.4)), we will prove thatthe Mabuchi functional K ε := M ( ϕ ε ) along the ε -geodesic converges to theMabuchi functional M ( ϕ ) along the geodesic, not only pointwise on eachfiber (Corollary (3.4)), but also in its complex Hessian.For the first step, we will take a closer look at equations (5.7) as follows.Recall that the ε -geodesic G ε is uniformly C , ¯1 on Γ × X . Therefore, we have(6.1) C > g t ¯ t − g ¯ βα g α ¯ t g t ¯ β = εω n ω nε , for some uniform constant C . For the same reason, the eigenvalues of ω ε are bounded from above by a uniform constant C > 0. Moreover, at aparticular point p ∈ Γ × X , we can write ω = n X j =1 dz j ∧ d ¯ z j ; ω ε = n X j =1 λ j dz j ∧ d ¯ z j . where 0 < λ ≤ · · · ≤ λ n are the n eigenvalues of the metric ω ε at this point.Hence we have the following inequality.tr ω ε ω = n X k =1 λ i ≤ nλ · · · λ n λ · · · λ n ≤ nC n − det( g α ¯ β ) ≤ nC C n − ε . (6.2)Equipped equation (6.2) into the RHS of equation (5.10), we have − ε ˆ X t (tr ω ε Ricω ) ω n ≥ − εC ˆ X t (tr ω ε ω ) ω n ≥ − εC ˆ P ε,A T X t (tr ω ε ω ) ω n − C µ ( { f ( ϕ ε ) ≤ κ } ) ≥ − εC A − C µ ( { f ( ϕ ε ) ≤ κ } ) , (6.3)where we fix a constant A large enough such thatmax X t κ < κ / . Then the first factor on the RHS of equation (6.3) converges to zero. More-over, Theorem (5.4) implies that the measure µ ( { f ( ϕ ε ) ≤ κ } ) convergesto zero as ε → 0, possibly after passing to a subsequence. Therefore, weconclude the following result. Theorem 6.1. Suppose the Mabuchi functional is ε -affine along a geodesic.Then for almost everywhere t ∈ [0 , , we have on the fiber X t dd c K ε | t → , as ε → , possible after passing to a subsequence. In particular, we have thefollowing convergence of the L -norms (6.4) ˆ X t || ¯ ∂ X v || ω ε ω nε , ˆ X t || ¯ ∂v || G ε ω nε → , as ε → for this subsequence.Proof. The complex Hessian of K ε can be computed as(6.5) dd c K ε ( t ) = Rn + 1 ˆ X t G n +1 ε + ˆ X t ( dd c f ε − Ricω ) ∧ G nε . Repeat the previous computation on the factor( dd c f ε − Ricω ) ∧ G nε . Equipped the estimate (6.3) into (5.10), and then we have . (cid:18) dd c log ω nε ω n − Ricω (cid:19) ∧ G nε + C ′ A G n +1 ε ≥ (cid:16) || ¯ ∂ X v || ω ε + εe − f ε |∇ ε f ε | (cid:17) idt ∧ d ¯ t ∧ ω nε , (6.6)where C ′ A := C A + χ { f ( ϕ ε ) ≤ κ } ε − C is a function on Γ × X . Then our previous argument shows that the followingenergy K ε ( t ) − ε (cid:18) ˆ X t C ′ A (cid:19) t (1 − t )is convex on the unit interval, and it converges to an affine function point-wise as ε → 0. Therefore, its second derivative converges to zero for almosteverywhere t ∈ [0 , µ { f ( ϕ ε ) ≤ κ } → , ε → , possibly after passing to a subsequence on the fiber X t , and then our resultfollows. (cid:3) By utilizing the estimate in the above Theorem, i.e. ¯ ∂v ε → L sense, one may expect that v ε would converge to a vector field v ∞ , and thislimit v ∞ is holomorphic on Γ × X . However, this is still unclear to us sincethe fiberwise volume element ω nε may not converge uniformly to the volumeelement ω nϕ of the geodesic G . Up to this stage, we can only conclude theholomorphicity of v ∞ under some special cases.6.1. Special cases. Suppose the two boundaries ϕ , ϕ of G are both non-degenerate energy minimizers of M . Then M ( ϕ t ) keeps to be a constantalong this geodesic, and it satisfies the ε -affine condition automatically.Therefore, our Theorem (5.4) implies that the geodesic G is fiberwise uni-formly non-degenerate, and then the regularities of G can be improved bythe work of He-Zeng([19]). Then we recover one of our result in [20]. Theorem 6.2 (L.) . Suppose the two boundaries of a C , ¯1 -geodesic G are bothnon-degenerate energy minimizers of M . Then the geodesic G is generatedby a holomorphic vector field. EODESICS APPROXIMATION 31 On the other hand, we can assume that the K¨ahler manifold X satisfies c ( X ) < c ( X ) = 0. Then our computation in Section (4) would recoverChen’s estimates in [8], under the ε -affine condition. In conclusion, we caninfer the following result by a similar argument as in Section (6), [8]. Theorem 6.3. Suppose the manifold X satisfies c ( X ) < or c ( X ) = 0 .Assume that the Mabuchi functional M is ε -affine along a C , ¯1 -geodesic G .Then G is generated by a holomorphic vector field. Finally, we would like to emphasis that the boundaries of the geodesic G are assumed to be smooth and non-degenerate in our set up. Therefore,one possible way to utilize the L -convergence of ¯ ∂v ε is to consider theirbehavior close enough to the boundary. Hence we will end up with thefollowing observation, which may be useful in our later consideration. Proposition 6.4. Suppose G is a C , ¯1 -geodesic connecting two K¨ahler po-tentials ϕ , ϕ ∈ H . Then its fiberwise volume element has the followingconvergence near the boundaries f ( ϕ t ) → f ( ϕ ); f ( ϕ s ) → f ( ϕ ) , as t → and s → in the L -sense.Proof. By a Theorem proved in Chen-Tian (Theorem 7.1.1, [15]), it is enoughto show lim t → ˆ X f ( ϕ t ) log f ( ϕ t ) → ˆ X f ( ϕ ) log f ( ϕ ) , and lim s → ˆ X f ( ϕ s ) log f ( ϕ s ) → ˆ X f ( ϕ ) log f ( ϕ ) . In other words, the entropy H ( ϕ t )( H ( ϕ s )) converges to H ( ϕ )( H ( ϕ )) as t → s → 1. This is true because the Mabuchi functional M is convexand continuous up to the boundaries of G . (cid:3) References [1] T. Mabuchi, A functional integrating Futaki’s invariant , Proc. Japan. Acad 61 Ser.A (1985), 119-120.[2] S.H. Bando and T. Mabuchi, Uniqueness of Einstein K¨ahler metrics modulo con-nected group actions , Algebraic geometry, Sendai, 1985, 11-40, Adv. Stud. Pure Math. (1987), 11-40.[3] E. Bedford and A. Talyor, A new capacity for plurisubharmonic functions , Acta Math. (1982), 1-41.[4] R. Berman, On the strict convexity of the K-energy , arXiv:1710.09075.[5] R. Berman and B. 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Trudinger, Elliptic partial differential equations of second order ,Springer (2001).[19] Weiyong He and Yu Zeng, Constant scalar curvature equation and the regularityof its weak solution , Communications on Pure and Applied mathematics (2019),no. 2, 422-448.[20] Long Li, The strict convexity of the Mabuchi functional for energy minimizers , toappear in Annales de la faculte des sciences de Toulouse. Mathematics Institute of ShanghaiTech University, 393 Middle Huaxia Road,Pudong 201210, Shanghai, China Email address : [email protected] Science Institute, University of Iceland, Reykjavik, Iceland. Email address ::