ON THE CONSTANT SCALAR CURVATURE KÄHLER METRICS (II) EXISTENCE RESULTS

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1 ON THE CONSTANT SCALAR CURVATURE KÄHLER ETRICS (II) EXISTENCE RESULTS XIUXIONG CHEN, JINGRUI CHENG Abstract. In this paper, we generalize our apriori estimates on csck(constant scalar curvature Kähler) metric equation [5] to more general scalar curvature type equations (e.g., twisted csck metric equation). As applications, under the assumption that the automorphism group is discrete, we prove the celebrated Donaldson s conjecture that the non-existence of csck metric is equivalent to the existence of a destabilized geodesic ray where the K-energy is non-increasing. oreover, we prove that the properness of K-energy in terms of L geodesic distance d in the space of Kähler potentials implies the existence of csck metric. Finally, we prove that weak minimizers of the K-energy in (E, d ) are smooth. The continuity path proposed in [4] is instrumental in this proof. Contents. Introduction 2. preliminaries K-energy and twisted K-energy The complete geodesic metric space (E p, d p ) 2.3. Convexity of K-energy Calabi dream anifolds 2 3. more general csck type equations Reduction of C, estimates to W 2,p estimates W 2,p estimates in terms of entropy bound of ωn ϕ ω n 8 4. K-energy proper implies existence of csck regularity of weak minimizers of K-energy Existence of csck and geodesic stability 35 References 39. Introduction This is the second of a series of papers discussing constant scalar curvature Kähler metrics. In this paper, for simplicity, we will only consider the case Aut (, J) =. Here Aut (, J) denotes the identity component of the automorphism group and Aut (, J) = means the group is discrete. Under this assumption, we prove Donaldson s conjecture (mentioned above in the abstract) as well as the existence part of properness conjecture in this paper. Our main method is to adopt the continuity path Date: January 9, 28.

2 2 XIUXIONG CHEN, JINGRUI CHENG introduced in [4] and we need to prove that the set of parameter t [, ] the continuity path is both open (c.f. [4] ) and closed under suitable geometric constraints. The apriori estimates obtained in [5] and their modifications (where the scalar curvature takes twisted form as in the twisted path introduced in [4]) are the crucial technical ingredients needed in this paper. In the sequel of this paper, we will prove a suitable generalization of both conjectures for general automorphism groups (i.e. no longer assume they are discrete). We will begin with a brief review of history of this problem. In 982 and 985, E. Calabi published two seminal papers [8] [9] on extremal Kähler metrics where he proved some fundamental theorems on extremal Kähler metrics. His initial vision is that there should be a unique canonical metric in each Kähler class. Levine s example(c.f [46]) however shows that there is a Kähler class in iterated blowup of CP 2 which admits no extremal Kähler metrics. ore examples and obstructions are found over the last few decades and huge efforts are devoted to formulate the right conditions (in particular the algebraic conditions) under which we can realize Calabi s original dream in a suitable format. The well known Yau-Tian-Donaldson conjecture is one of the important formulations now which states that on projective manifolds, the csck metrics exist in a polarized Kähler class if and only if this class is K-stable. It is widely expected among experts that the stability condition needs to be strengthened to a stronger notion such as uniform stability or stability through filtrations, in order to imply the existence of csck metrics. We will have more in-depth discussions on this issue in the next paper in this series. In a seminal paper [37], S. K. Donaldson proposed a beautiful program in Kähler geometry, aiming in particular to attack Calabi s renowned problem of existence of csck metrics. In this celebrated program, Donaldson took the point of view that the space of Kähler metrics is formally a symmetric space of non-compact type and the scalar curvature function is the moment map from the space of almost complex structure compatible with a fixed symplectic form to the Lie algebra of certain infinite dimensional sympletic structure group which is exactly the space of all real valued smooth functions in the manifold. With this in mind, Calabi s problem of finding a csck metric is reduced to finding a zero of this moment map in the infinite dimensional space setting. From this beautiful new point of view, S. K. Donaldson proposed a network of problems in Kähler geometry which have inspired many exciting developments over the last two decades, culminating in the recent resolution of Yau s stability conjecture on Kähler-Einstein metrics [8] [9] [2]. Let H denote the space of Kähler potentials in a given Kähler class (, [ω]). T. abuchi[5], S. Semmes [52] and S. K. Donaldson [37] set up an L 2 metric in the space of Kähler potentials: δϕ 2 ϕ = (δϕ) 2 ωϕ, n δϕ T ϕ H. Donaldson [37] conjectured that H is a genuine metric space with the pathwise distance defined by this L 2 inner product. In [], the first named author established the existence of C, geodesic segment between any two smooth Kähler potentials and proved this

3 ON THE CONSTANT SCALAR CURVATURE KÄHLER ETRICS (II) EXISTENCE RESULTS 3 conjecture of S.K. Donaldson. He went on to prove (together with E. Calabi) that such a space is necessarily non-positively curved in the sense of Alexandrov[]. ore importantly, S. K. Donaldson proposed the following conjecture to attack the existence problem: Conjecture.. [37] Assume Aut (, J) =. Then the following statements are equivalent: () There is no constant scalar curvature Kähler metric in H; (2) There is a potential ϕ H and there exists a geodesic ray ρ(t)(t [, )) in H, initiating from ϕ such that the K-energy is non-increasing; (3) For any Kähler potential ψ H, there exists a geodesic ray ρ(t)(t [, )) in H, initiating from ψ such that the K-energy is non-increasing. In the above, H = H {φ : I(φ) = }, where the functional I is defined by (2.7). The reason we need to use H is to preclude the trivial geodesic ρ(t) = ϕ + ct where c is a constant. In the original writing of S. K. Donaldson, he didn t specify the regularity of these geodesic rays in this conjecture. In this paper, we avoid this issue by working in the space E in which the potentials have only very weak regularity but the notion of geodesic still makes sense. oreover, Theorem 4.7 of [6] shows the definition of K-energy can be extended to the space E. The precise version of the result we prove is the following: Theorem.. (Theorem 6.) Assume Aut (, J) =. Then the following statements are equivalent: () There is no constant scalar curvature Kähler metric in H; (2) There is a potential ϕ E and there exists a locally finite energy geodesic ray ρ(t)(t [, )) in E, initiating from ϕ such that the K-energy is non increasing; (3) For any Kähler potential ψ E, there exists a locally finite energy geodesic ray ρ(t)(t [, )) in E, initiating from ψ such that the K-energy is non increasing. In the above, the space E is the abstract metric completion of the space H under the Finsler metric d in H (see section 2 for more details) and the notion of finite energy geodesic segment was introduced in [4] (c.f. [3]). Also E = E {φ : I(φ) = }, where the functional I is defined as in (2.7). The idea of using locally finite energy geodesic ray is inspired by the recent beautiful work of Darvas-He [3] on Donaldson conjecture in Fano manifold where they use Ding functional instead of the K-energy functional. From our point of view, both the restriction to canonical Kähler class and the adoption of Ding functional are more of analytical nature. Inspired by Donaldson s conjecture, the first named author introduced the following notion of geodesic stability [3]. Definition.. (c.f. Definition (3.) in [3]) Let ρ(t) : [, ) E be a locally finite energy geodesic ray with unit speed. One can define an invariant ([ρ]) as ([ρ]) = lim K(ρ(k + )) K(ρ(k)). k One can check that this is well defined, due to the convexity of K-energy along geodesics (c.f. Theorem 2.5). Indeed, from the convexity of K-energy along locally finite energy geodesic ray, one actually has K(ρ(k + )) K(ρ(k)) is increasing in k.

4 4 XIUXIONG CHEN, JINGRUI CHENG Definition.2. (c.f. Definition (3.4) in [3]) Let ϕ E with K(ϕ ) <, (, [ω]) is called geodesic stable at ϕ (resp. geodesic-semistable) if for all locally finite energy geodesic ray initiating from ϕ, their invariant is always strictly positive(resp. nonnegative). (, [ω]) is called geodesic stable(resp. geodesic semistable) if it is geodesic stable(resp. geodesic semistable) at any ϕ E. Remark.3. It is possible to define the invariant for a locally finite energy geodesic ray in E p with p >. Note that a geodesic segment in Ep is automatically a geodesic segment in E q for any q [, p]. Following the preceding definition, one can also define geodesic stability in E p (p > ). Note that for a locally given finite energy geodesic ray in E p (p > ), the actual value of invariant in Ep might differ by a positive multiple from the invariant considered in E. However, it will not affect the sign of the invariant for a particular locally finite energy geodesic ray. On the other hand, the collection of locally finite energy geodesic ray in E p (p > ) might be strictly contained in the collection of geodesic rays in E. Therefore, the notion of geodesic stability in the E is strongest while the notion of geodesic stability in E is the weakest. Without going into technicality, we may define geodesic stability in E as the invariant being strictly positive for any locally finite energy geodesic ray which lies in E p. p An intriguing question motivated from above remark is whether geodesic stability in E (in the sense defined in the above remark) implies geodesic stability in E? The first named author believes the answer is affirmative. We will discuss this question and other stability notions in algebraic manifolds in greater detail in our next paper and refer interested readers to the following works and references therein: J. Ross [5], G. Székelyhidi [57], Berman-Boucksom-Jonsson [3], R. Dervan [36]. Using the notion of geodesic stability, we can re-formulate Theorem. as Theorem.2. Suppose Aut (, J) =. Then (, [ω]) admits a csck metric if and only if it is geodesic stable. Given the central importance of the notion of K-energy in Donaldson s beautiful program, the first named author proposed the following conjecture, shortly after []: Conjecture.2. Assume Aut (, J) =. The existence of constant scalar curvature Kähler metric is equivalent to the properness of K-energy in terms of geodesic distance. Here properness means that the K-energy tends to + whenever the geodesic distance tends to infinity (c.f. Definition 4.). The original conjecture naturally chose the distance introduced in [37] which we now call L 2 distance. After a series of fundamental work of T. Darvas on this subject (c.f [29] [3]), we now learn that the L geodesic distance is a natural choice for the properness conjecture. Indeed, we prove Theorem.3. (Theorem 4. and 4.2) Assume Aut (, J) =. The existence of constant scalar curvature Kähler metric is equivalent to the properness of K-energy in terms of the L geodesic distance. Note that the direction that existence of csck implies properness has been established by Berman-Darvas-Lu[5] recently. For the converse (namely the existence part), Darvas

5 ON THE CONSTANT SCALAR CURVATURE KÄHLER ETRICS (II) EXISTENCE RESULTS 5 and Rubinstein have reduced this problem in [32] to a question of regularity of minimizers. In our paper, we will use continuity method to bypass this question and establish existence of csck metrics. For properness conjecture, we remark that there is a more well known formulation due to G. Tian where he conjectured that the existence of csck metrics is equivalent to the propeness of K-energy in terms of Aubin functional J (c.f. Definition (2.7)). One may say that Tian s conjecture is more of analytical nature while Conjecture.2 above fits into Donaldson s geometry program in the space of Kähler potentials more naturally. According to T. Darvas (c.f. Theorem 5.5 of [29]), Aubin s J functional and the L distance are equivalent. Therefore, these two properness conjectures are equivalent. Nonetheless, the formulation in conjecture.2 is essential to our proof. Theorem.3 also holds for twisted csck metric as well (c.f. Theorem ), which is the solution to the equation t(r ϕ R) = ( t)(tr ϕ χ χ). In the above, < t, χ is a fixed Kähler form, and R, χ are suitable constants determined by the Kähler classes [ω ], [χ]. Now we recall an important notion introduced in [4]: (.) R([ω ], χ) = sup{t [, ] : the above equation can be solved for any t t.} In the same paper, the first named author conjectured that this is an invariant of the Kähler class [χ]. In this paper, as a consequence of Theorem 4. and 4.2, we will show that if χ and χ 2 are two Kähler forms in the same class, then one has R([ω ], χ ) = R([ω ], χ 2 ), so that the quantity R([ω ], [χ]) is well-defined and gives rise to an invariant between two Kähler classes [ω ], [χ]. oreover, when the K-energy is bounded from below, the twisted path (2.9) can be solved for any t <, as long as t = can be solved. Thus in this case we have Theorem.4. Let χ be a Kähler form. If the K-energy is bounded from below on (, [ω]), then R([ω ], [χ]) = if and only if R([ω ], [χ]) >. As noted in [4], it is interesting to understand geometrically for what Kähler classes this invariant is but do not admit constant scalar curvature metrics. ore broadly, it is interesting to estimate the upper and lower bound of this invariant. It is not hard to see the relation between the invariant introduced in [56] and the invariant introduced above when restricted to the canonical Kähler class in Fano manifold, where we take [χ] to be the first Chern class in (.) above. Hopefully, the method used there can be adapted to our setting to get estimate for this new invariant, in particular an upper bound. T. Darvas and Y. Rubinstein conjectured in [32](Conjecture 2.9) that any minimizer of K-energy over the space E is actually a smooth Kähler potential. This is a bold and imaginative conjecture which might be viewed as a natural generalization of an earlier conjecture by the first named author that any C, minimizer of K-energy is smooth (c.f. [2], Conjecture 3). Under an additional assumption that there exists a smooth csck metric in the same Kähler class, Darvas-Rubinstein conjecture is verified in [5]. In

6 6 XIUXIONG CHEN, JINGRUI CHENG this paper, we establish this conjecture as an application of properness theorem. Note that Euler-Lagrange equation is not available apriori in our setting, so that the usual approach to the regularity problem in the calculus of variations does not immediately apply. Instead, we need to use the continuity path to overcome this difficulty. Theorem.5. (Theorem 5.) Let ϕ E be such that K(ϕ ) = inf ϕ E K(ϕ). Then ϕ is smooth and ω ϕ := ω + ϕ is a csck metric. We actually establish a more general result which allows us to consider more general twisted K-energy and we can show the weak minimizers of twisted K-energy are smooth as long as the twisting form is smooth, closed and nonnegative. Remark.4. W. He and Y. Zeng [43] proved Chen s conjecture on the regularity of C, minimizers of K-energy. Their original proof contains an unnecessary assumption that the (, ) current defined by the minimizer has a strictly positive lower bound which can be removed by adopting a weak Kähler-Ricci flow method initiated in Section 7 of Chen-Tian [25]. This will be discussed in an unpublished note [6]. In view of Theorem.3, it is important to study, under what conditions, the K-energy functional is proper in a given Kähler class. In [2], the first named author proposed a decomposition formula for K-energy: ( ω n ) ϕ ω n ϕ (.2) K(ϕ) = log n! + J Ric(ϕ). where the functional J Ric is defined through its derivatives: d J Ric ϕ (.3) = d t t ( Ric ωn ϕ (n )! + Rωn ϕ n! ). ω n One key observation in [2] (based on this decomposition formula) is that K-energy has a lower bound if the corresponding J Ric functional has a lower bound. Note that when the first Chern class is negative, one can choose a background metric such that Ric >. Then, J Ric is convex along C, geodesics in H and is bounded from below if it has a critical point. In [53], Song-Weinkove further pointed out that, J Ric functional being bounded from below is sufficient to imply the properness of K-energy. The research in this direction has been very active and intense (c.f. Chen[2], Fang-Lai- Song-Weinkove [39], Song-Weikove [54], Li-Shi-Yao [47], R. Dervan [35], and references therein). Combining these results with Theorem.3, we have the following corollary. Corollary.5. There exists a csck metric in (, [ω]) if any one of the following conditions holds: () There exists a constant ɛ such that ɛ < n+ n α ([ω]) and πc () < ɛ[ω] such that ( n C () [ω] n ) [ω] n + ɛ [ω] + (n )C () >. Here α (ω) denotes the α-invariant of the Kähler class (, [ω]) (c.f. [58]). (2) If and α ([ω]) > C () [ω] n n [ω] n n + C () C () [ω] n n [ω] n n + [ω].

7 ON THE CONSTANT SCALAR CURVATURE KÄHLER ETRICS (II) EXISTENCE RESULTS 7 Here part (i) of Corollary.5 follows Theorem.3 and Li-Shi-Yao [47] (c.f. Fang-Lai- Song-Weinkove [39] Song-Weinkove [54]), part (ii) of Corollary.5 follows Theorem.3 and R. Dervan [35]. Following Donaldson s observation in [38], if a Kähler surface admits no curve of negative self intersections and has C () <, then the condition 2[ω] [ C ()] [ω] 2 [ω] [ C ()] > is satisfied automatically for any Kähler class [ω] (c.f. Song-Weinkove [53]). Consequently, on any Kähler surface with C () < with no curve of negative selfintersection, the K-energy is proper for any Kähler class (c.f. Song-Weinkove [54]). It follows that on these surfaces, every Kähler class admits a csck metric. Albeit restrictive, this is indeed very close to the original vision of E. Calabi that every Kähler class should have one canonical representative. E. Calabi s vision has inspired generations of Kähler geometers to work on this exciting problem and without it, this very paper will never exist. To celebrate his vision, we propose to call such a manifold a Calabi dream manifold. Definition.6. A Kähler manifold is called Calabi dream manifold if every Kähler class on it admits an extremal Kähler metric. Clearly, all compact Riemann surfaces, complex projective spaces CP n and all compact Calabi-Yau manifolds [63] are Calabi dream manifolds. Our discussion above asserts Corollary.7. Any Kähler surface with C < and no curve of negative self-intersection is a Calabi dream surface. It is fascinating to understand how large this family of Calabi dream surfaces is. We will delay more discussions on Calabi dream manifolds to the end of Section 2. The key technical theorem we prove is the following compactness theorem in the space of Kähler potentials: Theorem.6. (Corollary 3.) The set of Kähler potentials(suitably normalized up to a constant) with bounded scalar curvature and entropy (or geodesic distance) is bounded in W 4,p for any p <, hence precompact in C 3,α for any < α <. This is an improvement from earlier work of first named author, Theorem.4 [3], where he additionally assumed a bound on Ricci curvature. ore recently, Chen-Darvas- He [7] proved that the set of Kähler potentials with uniform Ricci upper bound and L geodesic distance bound is precompact in C,α for any < α < (indeed, the Kähler form is bounded from above). As a corollary of Theorem.6, we prove Theorem.7. The Calabi flow can be extended as long as the scalar curvature is uniformly bounded. Remark.8. This is a surprising development. With completely different motivations in geometry, the first named author has a similar conjecture on Ricci flow which states that the only obstruction to the long time existence of Ricci flow is the L bound of scalar curvature. There has been significant progress in this problem, first by a series of works of B. Wang (c.f. [6], [27]) and more recently by the interesting and important work of Balmer-Zhang [] and. Simons [49] in dimension 4.

8 8 XIUXIONG CHEN, JINGRUI CHENG Theorem.7 is a direct consequence of Theorem.6 and Chen-He short time existence theorem (c.f. Theorem 3.2 in [2]), where the authors proved the life span of the short time solution depends only on C 3,α norm of the initial Kähler potential and lower bound of the initial metric. By assumption, we know that t ϕ remains uniformly bounded, hence ϕ is bounded on every finite time interval. On the other hand, since K-energy is decreasing along the flow, in particular K-energy is bounded from above along the flow. Due to (.2) and that ϕ is bounded, we see that the entropy is bounded as well. Hence the flow remains in a precompact subset of C 3,α () on every finite time interval, hence can be extended. In light of Theorem.7 and a compactness theorem of Chen-Darvas-He [7], a natural question is if one can extend the Calabi flow assuming only an upper bound on Ricci curvature. A more difficult question is whether one-sided bound of the scalar curvature is sufficient for the extension of Calabi flow. Ultimately, the remaining fundamental question is Conjecture.3. (Calabi, Chen) Initiating from any smooth Kähler potential, the Calabi flow always exists globally. Given the recent work by J. Street[55], Berman-Darvas-Lu[6], the weak Calabi flow always exists globally. Perhaps one can prove this conjecture via improving regularity of weak Calabi flow. On the other hand, one may hope to prove this conjecture on Kähler classes which already admit constant scalar curvature Kähler metrics and prove the flow will converges to such a metric as t. An important and deep result in this direction is Li-Wang-Zheng s work [48]. Finally we explain the organization of the paper: In section 2, we recall the necessary preliminaries needed for our proof, including the continuity path we will use to solve the csck equation and the theory of geodesic metric spaces established by Darvas and others. In section 3, we generalize our previous estimates in [5] on csck equation to more general type of equations, so that we can apply these estimates to twisted csck equation and Calabi flow. In section 4, we prove the equivalence between the existence of csck metric and properness of K-energy, namely Theorem.3. In section 5, we prove that a minimizer of K-energy over the space E is smooth. ore general twisted K-energy is also considered and we show its minimizer is smooth as long as the twisting form is nonnegative, closed and smooth. In section 6, we show that the existence of csck metric is equivalent to geodesic stability, In particular, we verify the Donaldson s conjecture, Theorem.. Acknowledgement. Both authors are grateful to the help from the first named author s colleague Professor Jason Starr in the discussions about Calabi dream manifolds. 2. preliminaries In this section, we will review some basic concepts in Kähler geometry as well as some fundamental results involving finite energy currents, which will be needed for our proof of Theorem. and.3. In particular, it includes the characterization of the space

9 ON THE CONSTANT SCALAR CURVATURE KÄHLER ETRICS (II) EXISTENCE RESULTS 9 (E, d ), a compactness result on bounded subsets of E with finite entropy. We also include results on the convexity of K-energy along C, geodesics as well as its extension to the space E. For more detailed account on these topics, we refer to a recent survey paper by D ly [34]. At the end of this section, we will discuss about Calabi dream manifolds. 2.. K-energy and twisted K-energy. Let (, ω ) be a fixed Kähler class on. Then we can define the space H of Kähler metrics cohomologous to ω as: (2.) H = {ϕ C 2 () : ω ϕ := ω + ϕ > }. We can introduce the K-energy in terms of its derivative: dk (2.2) dt (ϕ) = ϕ t (R ϕ R) ωn ϕ n!, ϕ H. Here R ϕ is the scalar curvature of ω ϕ, and R = [C ()] [ω] [n ] [ω] [n] = R ϕω n ϕ ωn. Following [2], we can write down an explicit formula for K(ϕ): ( ω n ) ϕ ωϕ n (2.3) K(ϕ) = log + J Ric (ϕ), n! where for a (, ) form χ, we define J χ (ϕ) = ϕ (χ ωn (2.4) Here = n! ω n λϕ (n )! χωn λϕ n! n ϕ Following formula (.3), we have k= dj χ dt ) dλ χ ω k ω n k ϕ χ = = χ ωn (n )! ω n n! t ϕ(tr ϕ χ χ) ωn ϕ n!.. (n + )! χϕ n k= ω k ω n k ϕ. It is well-known that K-energy is convex along smooth geodesics in the space of Kähler potentials. Let β be a smooth closed (, ) form, we define a twisted K-energy with respect to β by (2.5) K β (ϕ) = K(ϕ) + J β (ϕ). The critical points of K β (ϕ) satisfy the following equations: (2.6) R ϕ R = tr ϕ β β, where β = β ωn (n )! ω n n!.

10 XIUXIONG CHEN, JINGRUI CHENG For later use, we also define the functionals I(ϕ), J(ϕ), given by n (2.7) I(ϕ) = ϕ ω k ωϕ n k, J(ϕ) = ϕ(ω n ω n (n + )! ϕ). k= We also need to consider the more general twisted K-energy, which is defined to be (2.8) K χ,t = tk + ( t)j χ. Following [2], we can write down Euler-Lagrange equation for twisted K-energy: (2.9) t(r ϕ R) = ( t)(tr ϕ χ χ), t [, ]. Following [4], for t >, we can rewrite this into two coupled equations: (2.) det(g i j + ϕ i j) = e F det g i j, (2.) ϕ F = (R t χ) + tr ϕ (Ric t χ). t t In the following, we will assume χ >, that is, χ is a Kähler form. The equation (2.9) with t [, ] is the continuity path proposed in [4] to solve the csck equation. ore generally, one can consider similar twisted paths in order to solve (2.6). Namely we consider (2.2) t(r ϕ R) = t(tr ϕ β β) + ( t)(tr ϕ χ χ). The solution to (2.2) is a critical point of tk β + ( t)j χ. We will see later that it is actually a minimizer. For t >, this again can be equivalently put as (2.3) det(g i j + ϕ i j) = e F det g i j, ϕ F = (R β t ( t (2.4) χ) + tr ϕ Ric β χ ). t t An important question is whether the set of t for which (2.2) can be solved is open. The cited result is only for (2.9), but the same argument would work for (2.2). Lemma 2.. ([4], [62], [44]): Suppose for some t <, (2.2) has a solution ϕ C 4,α () with t = t, then for some δ >, (2.2) has a solution in C 4,α for any t (t δ, t + δ) [, ). We observe that we can always make sure (2.9) or (2.2) can be solved for t = by choosing χ = ω or any Kähler form in [ω ]. Remark 2.2. Clearly if χ is smooth, it is easy to see by bootstrap that a C 4,α solution to (2.9) is actually smooth. Hence Lemma 2. shows the set of t for which (2.9) has a smooth solution is relatively open in [, ) The complete geodesic metric space (E p, d p ). Following abuchi, T. Darvas [3] introduced the notion of d on H. (2.5) ξ ϕ = ξ ωn ϕ n!, ξ T ϕh = C (). Using this, we can define the path-length distance d on the space H, i.e. d (u, u ) equals the infimum of length of all smooth curves in H, with α() = u, α() = u. Following Chen [], T. Darvas proved ([3], Theorem 2) that (H, d ) is a metric space.

11 ON THE CONSTANT SCALAR CURVATURE KÄHLER ETRICS (II) EXISTENCE RESULTS In section 3.3 of [4] introduced the following space for any p : (2.6) E p = {ϕ P SH(, ω ) : ωϕ n = ω n, ϕ p ωϕ n < }. A fundamental conjecture of V. Guedj [4] stated that the completion of the space H of smooth potentials equipped with the L 2 metric is precisely the space E 2 (, ω ) of potentials of finite energy. This has been shown by Darvas [3], [29], in which he has shown similar characterization holds for general L p metric. Note that the extension to the L metric is essential and fundamental to our work. We have the following characterization for (E, d ): Theorem 2.. ([3], Theorem 5.5)Define I (u, v) = u v ωn u n! + u v ωn v, u, v H. n! Then there exists a constant C > depending only on n, such that (2.7) C I (u, v) d (u, v) CI (u, v), for any u, v H. For later use, here we describe how to obtain finite energy geodesics from the C, geodesics between smooth potentials. Theorem 2.2. ([3], Theorem 2) The metric completion of (H, d ) equals (E, d ) where d (u, u ) =: lim k d (u k, u k ), for any smooth decreasing sequence {u k i } k H converging pointwise to u i E. oreover, for each t (, ), define u t := lim k uk t, t (, ), where u k t is the C, geodesic connecting u k and uk (c.f. []). We have u t E, the curve [, ] t u t is independent of the choice of approximating sequences and is a d -geodesic in the sense that for some c >, d (u t, u s ) = c t s, for any s, t [, ]. The above limit is pointwise decreasing limit. Since the sequence {u k i } k is decreasing sequence for i =,, we know {u k t } k is also decreasing for t (, ), by comparison principle. We say u t : [, ] t E connecting u, u is a finite energy geodesic if it is given by the procedure described in Theorem 2.2. The following result shows the limit of finite energy geodesics is again a finite energy geodesic. Proposition 2.3. ([6], Proposition 4.3) Suppose [, ] t u i t E is a sequence of finite energy geodesic segments such that d (u i, u ), d (u i, u ). Then d (u i t, u t ), for any t [, ], where [, ] t u t E is the finite energy geodesic connecting u, u. Finally we record the following compactness result which will be useful later. This result was first established in [4]. The following version is taken from [6], which is the form most convenient to us.

12 2 XIUXIONG CHEN, JINGRUI CHENG Lemma 2.4. ([4], Theorem 2.7, [6], Corollary 4.8) Let {u i } i E be a sequence for which the following condition holds: sup i d (, u i ) <, sup K(u i ) <. i Then {u i } i contains a d -convergent subsequence Convexity of K-energy. In this subsection, we record some known results about the convexity of K-energy and J χ functional along C, geodesics and also finite energy geodesics. In [2], the first named author proved the following result about the convexity of the functional J χ. Theorem 2.3. ([2], Proposition 2) Let χ be a closed (, ) form. Let u, u H. Let {u t } t [,] be the C, geodesic connecting u, u. Then [, ] t J χ (u t ) is convex. The convexity of K-energy is more challenging and the first named author made the following conjecture: Conjecture 2.. (Chen) Let u, u H. Let {u t } t [,] be the C, geodesic connecting u, u. Then [, ] t K(u t ) is convex. This conjecture was verified by the fundamental work of Berman and Berndtsson [2] (c.f. Chen-Li-Paun [23] also). Theorem 2.4. Conjecture 2. is true. It turns out that the K-energy and also the fuctional J χ can be extended to the space (E, d ) and is convex along finite energy geodesics. ore precisely, Theorem 2.5. ([6], Theorem 4.7) The K-energy defined in (2.3) can be extended to a functional K : E R {+ }. Besides, the extended functional K E is the greatest d - lower semi-continuous extension of K H. oreover, K E is convex along finite energy geodesics of E. Theorem 2.6. ([6], Proposition 4.4 and 4.5) The functional J χ as defined by (2.4) can be extended to be a d -continuous functional on E. Besides, J χ is convex along finite energy geodesics Calabi dream anifolds. Every example of a Calabi dream surface that we discusse here is constructed from the outside in. We begin with an ambient manifold that satisfies a weaker hypothesis making it easier to construct. Then we construct as an appropriate complete intersections of ample hypersurfaces inside the ambient manifold and we encourage interested readers to D ly-Peternell-Schneider[33] for further readings on this topic. For a smooth, projective surface, the ample cone equals the big cone if and only if the self-intersection of every irreducible curve is nonnegative. In analytic terms, the ample cone equals the big cone if and only if every holomorphic line bundle admitting a singular Hermitian metric of positive curvature current admits a regular Hermitian metric of positive curvature. () For every smooth, projective variety P of dimension n at least 3 such that the ample cone equals the big cone, for every (n-2)-tuple of divisors D,..., D n 2. If the divisor classes of D i are each globally generated, and if the D i are general in their linear equivalence classes, then the surface = D... D n 2 is smooth

13 ON THE CONSTANT SCALAR CURVATURE KÄHLER ETRICS (II) EXISTENCE RESULTS 3 and connected by Bertini s theorems. If also every D i is ample, if K P + (D +...+D n 2 ) is globally generated, and if the divisors D i are very general in their linear equivalence classes, then the surface has ample cone equal to the big cone, cf. the Noether-Lefschetz article of Ravindra and Srinivas. Finally, if also the divisor class K P +(D +...+D n 2 ) is ample, then K is ample. In that case, the smooth, projective surface has c (T ) negative, and the self-intersection of every irreducible curve is nonnegative, and thus are Calabi dream surfaes. (2) If P and Q are projective manifolds whose ample cones equal the big cones, and if there is no nonconstant morphism from the (pointed) Albanese variety of P to the (pointed) Albanese variety of Q, then also the product P Q is a projective manifold whose ample cone equals the big cone. In particular, if P and Q are compact Riemann surfaces of (respective) genera at least 2, and if there is no nonconstant morphism from the Jacobian of P to the Jacobian of Q, then the product = P Q is a Calabi dream manifold. (3) There are many examples of smooth, projective varieties P as in item. When the closure of the ample cone equals the semiample cone and is finitely generated, then such a variety is precisely a ori dream space that has only one ori chamber, yet there are examples arising from Abelian varieties where the cone is not finitely generated. For instance, all projective varieties of Picard rank trivially satisfy this property. The next simplest class consists of all varieties that are homogeneous under the action of a complex Lie group. This class includes all Abelian varieties. It also includes the projective homogeneous varieties, e.g., projective spaces, quadratic hypersurfaces in projective space, Grassmannians, (classical) flag varieties,etc. This class is also stable for products and is Calabi dream manifolds. (4) The next simplest class consists of every projective manifold P of cohomogeneity one, i.e., those projective manifolds that admit a holomorphic action of a complex Lie group G whose orbit space is a holomorphic map from P to a compact Riemann surface. These are also Calabi dream surfaces. Here is an interesting question about Calabi dream manifolds: how far is the class of Calabi dream surfaces from the class of all smooth minimal surfaces of general type? 3. more general csck type equations First we would like to generalize our estimates on csck to more general type of equations. ore specifically, we consider the following coupled equations: (3.) (3.2) det(g i j + ϕ i j) = e F det g i j, ϕ F = f + tr ϕ η. Here f is a given function(not necessarily a constant) and η is a smooth real valued closed (, ) form on, written as η = η α βdz α dz β. Observe that the equations (3.), (3.2) combined gives (3.3) R ϕ = f + tr ϕ (Ric η). Later on, we wish to apply our estimates to the equation (2.9), with choice f = R t t χ, η = Ric t χ, t

14 4 XIUXIONG CHEN, JINGRUI CHENG and also (2.2), with choice f = R β t t χ, η = Ric β t χ. t The goal of this section is to prove the following apriori estimate: Theorem 3.. Let ϕ be a smooth solution to (3.), (3.2) so that sup ϕ =, then there exists a constant C >, depending only on the backgound metric (, g), f, max η ω, and the upper bound of ef F dvol g such that ϕ C, and C ω ω ϕ C ω. The proof of this theorem is very similar to the case of csck, and we will be suitably brief and only highlight the main differences. Before going into the proof of Theorem 3., first we notice the following corollary: Corollary 3.. Let ϕ be a smooth solution to (3.), (3.2) normalized to be sup ϕ =, then for any p <, there exist a constant C.5, depending only on the background metric (, g), f, max η ω, p, and the upper bound of ef F dvol g such that ϕ W 4,p C.5, F W 2,p C.5. Proof. The proof of this corollary (assuming Theorem 3.) is essentially the combination of several classical elliptic estimates. First we know from Theorem 3. that C ω ω ϕ C ω, where C has the said dependence in this corollary. But this means (3.2) is now uniformly elliptic with bounded right hand side. From this we immediately know F α C., where α and C. has the said dependence. Then we go back to (3.), we can then conclude from Evans-Krylov theorem that ϕ 2,α C.2 for any α < α (see [6] for details on extension of Evans-Krylov to complex setting). Again go back to (3.2) and notice that equation can be put in divergence form: (3.4) Re ( i j i (det(g α + ϕ β α β)g ϕ F j) ) = ( f + tr ϕ η) det(g α β + ϕ α β). Here the coefficients on the left hand side is in C α, while the right hand side is bounded. Hence we may conclude F,α C.3, from [42], Theorem Then from (3.), by differentiating both sides of the equation, we see that the first derivatives of ϕ solves a linear elliptic equation with C α coefficient and right hand side, hence Schauder estimate applies and we conclude ϕ C 3,α ([42], Theorem 6.2). But then we go back to (3.4) one more time, the coefficients are in C α for any < α < with bounded right hand side, hence we conclude F C,α for any < α <. Now the equation solved by the first derivatives of ϕ will have coefficients and right hand side in C α for any < α <. Therefore ϕ C 3,α for any < α <. The second equation (3.2) now has C,α coefficient with bounded right hand side, then the classical L p estimate gives F W 2,p for any finite p([42], Theorem 9.). Then differentiating the first equation (3.) twice, we get a linear elliptic equation in terms of second derivatives of ϕ, which has C α coefficients and L p right hand side(we already have F W 2,p ), it follows that ϕ W 4,p. Remark 3.2. If we assume higher regularity of f and η on the right hand side of (3.2), it is easy to get regularity higher than W 4,p by bootstraping. Now we can focus on proving Theorem 3..

15 ON THE CONSTANT SCALAR CURVATURE KÄHLER ETRICS (II) EXISTENCE RESULTS Reduction of C, estimates to W 2,p estimates. This is the part where the main difference comes up with csck case and we will highlight this difference. We will be brief at places where the proof works in the same way as csck case. The exact result we will prove is the following: Proposition 3.3. Let ϕ be a smooth solution to (3.), (3.2), then there exists p n >, depending only on n, such that (3.5) max ϕf ϕ + max (n + ϕ) C. Here C depends only on (, g), ϕ, F, n + ϕ L pn(), f and max η ω. This corresponds to Theorem 4. in our first paper [5]. Proof. We can choose local coordinates so that at a point p under consideration, we have g i j(p) = δ ij, g i j(p) =, ϕ i j(p) = ϕ iī (p)δ ij. Let B : R R be a smooth function, we have(under above said coordinates at p): (3.6) e B(F ) ϕ (e B(F ) ϕ F 2 ϕ) 2 ϕ F ϕ ϕ ( ϕ F ) + + Ric ϕ,i jf ī F j ( + ϕ iī )( + ϕ j j) F i j 2 ( + ϕ iī )( + ϕ j j) + B F if jī F j + F ī F j F ji ( + ϕ iī )( + ϕ j j) + (B ϕ F 2 ϕ + B ϕ F ) ϕ F 2 ϕ. In the above, ϕ means the inner product is taken under the metric ω ϕ. This calculation corresponds to (4.3) in our first paper [5] and it does not use the equation at all. Note that in the above Ric ϕ,i j = R i j F i j. As in csck case, with the choice of B(λ) = λ 2, we obtain e 2 F ϕ (e 2 F ϕ F 2 R i jf ī F j ϕ) 2 ϕ F ϕ ϕ ( ϕ F ) + ( + ϕ iī )( + ϕ j j) (3.7) + F i j 2 ( + ϕ iī )( + ϕ j j) + 2 ( f + tr ϕη) ϕ F 2 ϕ. In the above, we used the same crucial cancellation as in the csck case. Next we can estimate R i jf ī F j (3.8) ( + ϕ iī )( + ϕ Ric F ī F j g j j) ( + ϕ iī )( + ϕ Ric g ϕ F 2 ϕtr ϕ g. j j) Also (3.9) 2 ( f + tr ϕη) ϕ F 2 ϕ 2 ( f + max η gtr ϕ g) ϕ F 2 ϕ. Finally recall that tr ϕ g e F (n + ϕ) n. Hence we obtain from (3.7): (3.) ϕ (e 2 F ϕ F 2 ϕ) 2e 2 F ϕ F ϕ ϕ ( ϕ F ) C. ( (n + ϕ) n + ) ϕ F 2 ϕ + C. F iᾱ 2 ( + ϕ iī )( + ϕ αᾱ ).

16 6 XIUXIONG CHEN, JINGRUI CHENG Here C. has the dependence stated in the proposition. From (4.2) in our first paper, [5], we have ϕ (n + ϕ) C. (n + ϕ) n + F C. C. (n + ϕ) n C. F iī 2 ( + ϕ iī ) 2 C.(n + ϕ) 2 C. C.2 (n + ϕ) n C. F iī 2 ( + ϕ iī ) 2. In the last line above, we used the fact that n + ϕ ne F n, which is bounded from below, and n 2. By the same calculation as we did for csck, if we denote we obtain u = e 2 F ϕ F 2 ϕ + (n + ϕ) +, (3.) ϕ u 2e 2 F ϕ F ϕ ϕ ( ϕ F ) C.2 (n + ϕ) n u, Here C.2 has the said dependence as in proposition. The main difference from the csck case is that we cannot estimate the term e 2 F ϕ F ϕ ϕ ( ϕ F ) directly as we did for csck, otherwise, f and η will enter into the estimates. For any p >, integrate the equality ϕ (u 2p+ ) = (2p + )2p ϕ u 2 ϕ + (2p + )u 2p ϕ u with respect to dvol ϕ, we have 2pu 2p ϕ u 2 ϕdvol ϕ = u 2p ( ϕ u)dvol ϕ (3.2) C.2 (n + ϕ) n u 2p+ dvol ϕ 2 e 2 F ϕ F ϕ ϕ ( ϕ F )u 2p dvol ϕ. We need to integrate by parts in the last term above, then we have 2e 2 F ϕ F ϕ ϕ ( ϕ F )u 2p dvol ϕ = 4pu 2p e 2 F ϕ F ϕ F ϕ ϕ udvol ϕ (3.3) + 2u 2p e 2 F ( ϕ F ) 2 dvol ϕ + u 2p e 2 F ϕ F 2 ϕ ϕ F dvol ϕ. We wish to estimate the three terms on the right hand side of (3.3) from above. First, 4pu 2p e 2 F ϕ F ϕ F ϕ ϕ udvol ϕ pu 2p ϕ u 2 ϕdvol ϕ (3.4) + 4 pu 2p e F ( ϕ F ) 2 ϕ F 2 ϕdvol ϕ pu 2p ϕ u 2 ϕdvol ϕ + 4 pu 2p e 2 F ( ϕ F ) 2 dvol ϕ. Also it is clear that (3.5) u 2p e 2 F ϕ F 2 ϕ ϕ F dvol ϕ u 2p+ ϕ F dvol ϕ.

17 ON THE CONSTANT SCALAR CURVATURE KÄHLER ETRICS (II) EXISTENCE RESULTS 7 Combining (3.3), (5.5) and (3.5), we see e 2 F ϕ F ϕ ϕ ( ϕ F )u 2p dvol ϕ pu 2p ϕ u 2 ϕdvol ϕ (3.6) + (4p + 2)u 2p e 2 F ( ϕ F ) 2 dvol ϕ + u 2p+ ϕ F dvol ϕ. Combine with (3.2), we obtain pu 2p ϕ u 2 ϕdvol ϕ C.2 (n + ϕ) n u 2p+ dvol ϕ (3.7) + u 2p+ ϕ F dvol ϕ + (4p + 2)u 2p e 2 F ( ϕ F ) 2 dvol ϕ. In the above, we can estimate (3.8) ϕ F f + tr ϕ η ( f + max η ω )( + tr ϕ g) C.3 ( + ne F (n + ϕ) n ). Recall that n + ϕ is bounded from below in terms of F, we obtain from (3.7) that (3.9) pu 2p ϕ u 2 ϕdvol g C.3 (p + )(n + ϕ) 2n 2 u 2p+ dvol g. Here C.3 depends only on F, the background metric (, g), f, and max η ω. Above is equivalent to (3.2) ϕ (u p+ 2 ) 2 ϕ dvol g (p + 2 )2 (p + ) C.3 (n + ϕ) 2n 2 u 2p+ dvol g. p For any < ε < 2, apply Höler s inequality, we obtain for any p 2 (by the same calculation as in csck case): (3.2) Here ( ( K ε = n ε 2 ε (u p+ 2 ) 2 ε dvol g ) 2 2 ε C.4 p 2 K ε ( u (p+ 2 )(2+ε) dvol g ) 2 2+ε. ) ε ( ) ε (n + ϕ) 2 ε 2 ε ε dvolg (n + ϕ) (2n 2)(2+ε) 2+ε ε. The key estimate (3.2) corresponds to (4.27) of our first paper, [5]. The passage from (3.2) to (3.2) follows the calculation from (4.22) to (4.26) of our first paper, [5], almost word-for-word. After this, we choose ε sufficiently small so that θ := 2n(2 ε) 2n 2 + ε > 2 + ε. Then we can apply Sobolev inequality to u p+ 2 with exponent 2 ε and obtain ( ) 2 ( ) 2 (3.22) u (p+ 2 )θ θ dvol g C.5 p 2 u (p+ 2 )(2+ε) 2+ε dvol g. This implies that for p 2, one has (3.23) u L (p+ (C 2 )θ.6 p 2 p+ ) 2 u L (p+. 2 )(2+ε)

18 8 XIUXIONG CHEN, JINGRUI CHENG Denote χ = θ 2+ε >, and choose p + 2 = χi for i, then from above we can conclude u L (2+ε)χ i+ ( C.6 χ 2i) χ i u L (2+ε)χ i. Iterate above estimate, and using the inequality u L 2+ε desired result. +ε 2+ε u u L +ε L gives the 3.2. W 2,p estimates in terms of entropy bound of ωn ϕ ω n. In this subsection, we will state the estimates which ultimately shows that one can estimate the W 2,p norm of the solution ϕ in terms of the entropy bound ef F dvol g. The proof in the csck case carries over almost word for word. As in the csck case, let ψ solves the following problem: (3.24) (3.25) det(g i j + ψ i j) = sup ψ =. e F F 2 + ef F 2 + dvol g det g, Theorem 3.2. Normalize ϕ so that sup ϕ =. Given any < ε <, there exists a constant C 3, depending only on ε, the background metric (, g), the upper bound for ef F dvol g, f, and max η ω, such that (3.26) F + εψ 2( + max η ω )ϕ C 3. This corresponds to Theorem 5. in the first paper. Proof. Let η p : R + be the cut-off function such that η p (p) =, η p θ outside the ball B d 2 (p), with η p 2θ d and 2 η p 4θ. Here d d 2 is sufficiently small depending only on (, g) and θ is a sufficiently small constant to be chosen later. Let λ = 2( + max η ω ), and δ = α 2nλ. Here 2α is the α-invariant of the Kähler class (, [ω ]). Suppose e δ(f +εψ λϕ) has maximum at p. Let us first calculate ϕ (e δ(f +εψ λϕ) η p ). Recall formula (5.8) from our first paper [5], we have (3.27) ϕ ( e δ(f +εψ λϕ) η p ) e δ(f +εψ λϕ) = η p ( δ 2 ϕ (F + εψ λϕ) 2 ϕ + δ ϕ (F + εψ λϕ) ) + ϕ η p + 2δ ϕ (F + εψ λϕ) ϕ ϕ η p. This does not use the equation at all. Now we compute(similar to (5.2) in [5]): (3.28) ϕ (F + εψ λϕ) = (f + λn) + tr ϕ (η + λg) + ε ϕ ψ ( f λn + εna n (F 2 + ) 2n ) + (λ ε max η ω )tr ϕ g. In the above A = ef F 2 + dvol g, which can be bounded in terms of upper bound of ef F dvol g. In the second line above, we used that ϕ ψ n( F 2 + A ) n trϕ g. Then we can estimate the terms in (3.27) involving ϕ η p, ϕ η p in terms of tr ϕ g(see (5.9), (5.2) in [5] for more details.) Hence we conclude the following estimate(similar

19 ON THE CONSTANT SCALAR CURVATURE KÄHLER ETRICS (II) EXISTENCE RESULTS 9 to (5.22) in [5]): (3.29) e δ(f +εψ λϕ) ϕ ( e δ(f +εψ λϕ) η p ) δηp ( f λn + εna n (F 2 + ) 2n ) + ( δη p (λ ε max η ω ) d 2 4θ 4θ ) ( θ) d 2 ( trϕ g. θ)2 Since ε <, and because of our choice of δ and λ, we know λ ε max η ω > λ 2, and δλ = α 2n. Therefore we can choose θ small enough to make the coefficients of tr ϕg in (3.29) positive. Now we drop the term involving tr ϕ g in (3.29), and apply the Alexandrov maximum principle in B d (p ), we obtain (3.3) sup B d (p ) uη p sup B d (p ) uη p ( + C n d ( u 2n ( f λn + εna n (F 2 + ) 2n ) ) 2n B d (p ) e 2F dvol g In the above, u = e δ(f +εψ λϕ). Then the argument proceeds the same way as in the first paper [5]. aking use of the α-invariant, namely the fact that e αψ dvol g C 3. for some α > and C 3. depending only on background Kähler metric (, g), we can deduce: Corollary 3.4. Normalize ϕ so that sup ϕ =. For any < q <, there exists a constant C 3.2, depending only on the background metric (, g), the upper bound for ef F dvol g, f, max η ω, and q, such that (3.3) e qf dvol g C 3.2. In particular, there exists a constant C 3.3, with the same dependence as C 3.2 but not on q, such that (3.32) ϕ C 3.3, ψ C 3.3. This corresponds to Corollary 5.3 in the first paper [5]. As in csck case, the in particular part follows from (3.3) and Kolodziej s result. After this, one can estimate F. Proposition 3.5. There exist constants C 3.4 >, C 3.5 > such that (3.33) F C 3.4, e F C 3.5. Here C 3.4 and C 3.5 depend on ϕ, f and max η ω This corresponds to Proposition 2. and Corollary 5.4 in the first paper. Same as in csck case, we compute ϕ (F + λϕ) to estimate the lower bound of F, and the upper bound follows from Theorem 3.2 and Corollary 3.4. Combining Corollary 3.4 and Proposition 3.5, we obtain an estimate for ϕ and F in terms of the entropy bound ef F dvol g, f, max η ω and the background metric g only. As before, we have the following partial C estimate. Proposition 3.6. There exists a constant C 3.6, depending only on ϕ, f, max η ω, and the background metric g, such that (3.34) ϕ 2 e F C 3.6. ).

20 2 XIUXIONG CHEN, JINGRUI CHENG This corresponds to Theorem 2.2 in the first paper. Proof. Let λ, K > be constants to be determined, denote A(F, ϕ) = (F + λϕ) + 2 ϕ2. Then we have (3.35) ( ϕ e A(F,ϕ) ( ϕ 2 + K) ) e A ( Fi λϕ i + ϕϕ i 2 = ϕ (F + λϕ + ϕ iī 2 ϕ2 ) + ϕ i 2 ) + ϕ iī ( ϕ 2 + K) + ϕ ( ϕ 2 2 ) + Re ( ( F i λϕ i + ϕϕ i )( ϕ 2 ) ) + ϕ ī. iī Exactly the same calculation leading to (2.2) in the first paper [5] now gives: (3.36) ϕ ( e A ( ϕ 2 + K) ) e A ϕ (F + λϕ) ϕ ϕ ϕ 2 ϕ( ϕ 2 + K) + ϕ ϕ 2 ϕ( ϕ 2 + K) + (f λn + nϕ + i + C 3.7 ϕ 2 + ϕ iα 2 + ϕ 2 iī + ϕ iī + 2Re( ( F i λϕ i + ϕϕ i )(ϕ α ϕᾱī + ϕ ī ϕ iī ) ). + ϕ iī λ η iī ϕ )( ϕ 2 + K) + ϕ iī + ( 2λ + 2ϕ) ϕ 2 + 2Re ( ) (F α + λϕ α ϕϕ α )ϕᾱ In the above, C 3.7 is a constant depending only on the background metric g. Following (2.22) in the first paper [5], we drop the complete square in (3.36), and observe the crucial cancellation in the last two terms: (F α + λϕ α ϕϕ α )ϕᾱ + ( F i λϕ i + ϕϕ i )ϕ ī ϕ iī + ϕ iī = (F i + λϕ i ϕϕ i )ϕ ī. + ϕ iī Therefore, we get following estimate similar to (2.24) in the first paper: (3.37) ( ϕ e A ( ϕ 2 + K) ) e A K F i λϕ i + ϕϕ i 2 + ϕ i 2 ( ϕ 2 + K) + ϕ iī + ϕ iī + λ η iī ϕ ( ϕ 2 + K) + ( f λn + ϕ)( ϕ 2 + K) + ϕ i iī C 3.7 ϕ 2 + ϕ2 iī + ( 2λ + 2ϕ) ϕ 2 + ϕ i iī + ϕ iī ( ) (Fi + λϕ i ϕϕ i )ϕ + 2Re ī. + ϕ iī Now we choose K = and λ = (max η ω + ϕ +C 3.7 +), then we can estimate: (3.38) ϕ2 iī +ϕ iī i λ η iī ϕ ( ϕ 2 + K) C 3.7 ϕ 2 + ϕ iī i + ϕ iī ϕ 2 i. + ϕ iī We estimate the terms ( f λn + nϕ)( ϕ 2 + K), ( 2λ + 2ϕ) ϕ 2, (F i+λϕ i ϕϕ i )ϕ ī +ϕ iī and in the same way as we did for csck(see (2.25), (2.26), (2.27) and (2.29) in the

21 ON THE CONSTANT SCALAR CURVATURE KÄHLER ETRICS (II) EXISTENCE RESULTS 2 first paper [5] for details). In the end, we obtain the following estimate: ( (3.39) ϕ e A ( ϕ 2 +K) ) e A ϕ i 2 ϕ 2 +9 ϕ 2 + ϕ iī i +e F n C3.8 ( ϕ 2 +). + ϕ iī Here C 3.8 is a positive constant which has the dependence described in this proposition. This estimate corresponds to (2.3) in our first paper, [5]. From here on, the argument is completely the same as in csck case. As a result of this, we deduce the following W 2,p estimates, which were what we needed for Proposition 3.3. Proposition 3.7. For any p >, there exists a constant α(p) >, depending only on p, and another constant C 3.8, depending on ϕ, f, max η ω, the background metric (, g) and p, such that (3.4) e α(p)f (n + ϕ) p dvol g C 3.8. In particular, (3.4) n + ϕ L p (dvol g) C 3.9. Here C 3.9 has the same dependence as C 3.8 but additionally on F. This corresponds to Theorem 3. in the first paper. Proof. We start by calculating: (3.42) ( ϕ e κ(f +λϕ) (n + ϕ) ) κ(f +λϕ) e = ( κ ϕ (F + λϕ) + κ 2 ϕ (F + λϕ) 2 ϕ) (n + ϕ) ( ) (Fi + λϕ i )( ϕ) + ϕ (n + ϕ) 2κRe ī. + ϕ iī In the above, if we choose λ so that λ > 2 max η ω, then (3.43) κ ϕ (F + λϕ) = κ(r λn tr ϕ η + λtr ϕ g) κ( f λn) + λκ 2 tr ϕg. We calculate ϕ (n + ϕ) in exactly the same way as csck case(see (3.4), (3.5), (3.7) in the first paper for details.) Therefore, (3.44) ϕ ( e κ(f +λϕ) (n + ϕ) ) e κ(f +λϕ) ( λκ 2 C 3.9)(n + ϕ) i + κe κ(f +λϕ) ( f λn)(n + ϕ) + e κ(f +λϕ) ( F R g ). + ϕ iī In the above, R g is the scalar curvature of the background metric ω, and C 3.9 depends only on the curvature bound of ω. This estimate is the analogue of (3.7) of our first paper, [5]. Next we use the estimate (n + ϕ) i + ϕ iī e F n (n + ϕ) + n.