Gauduchon metrics with prescribed volume form

c

2017 by Institut Mittag-Leffler. All rights reserved

Gauduchon metrics with prescribed volume form

by

G´abor Sz´ekelyhidi

University of Notre Dame Notre Dame, IN, U.S.A.

Valentino Tosatti

Northwestern University Evanston, IL, U.S.A.

Ben Weinkove

Northwestern University Evanston, IL, U.S.A.

1. Introduction

LetMbe a compact complex manifold of complex dimensionn. Suppose thatM admits a metricα=√

−1α_i¯jdzⁱ∧d¯z^j>0 which is K¨ahler (that is,dα=0). Yau’s celebrated solution [43] of the Calabi conjecture says that given any smooth positive volume formσonM withR

Mσ=R

Mαⁿ, we can find a K¨ahler metricω with this prescribed volume form:

ωⁿ=σ. (1.1)

Moreover, there exists such a metric so that [ω]=[α] in H²(M,R), and with this coho- mological constraint the metricωis unique.

Furthermore, Yau’s theorem is equivalent to a statement about the first Chern class c₁(M). Namely, given any smooth representative Ψ of c₁(M), there exists a unique K¨ahler metricω cohomologous toαsuch that

Ricci(ω) = Ψ, (1.2)

where Ricci(ω) is the Ricci form of the K¨ahler metricω. Indeed, this follows immediately from the definition ofc1(M) and by applying the operator−√

−1∂∂¯log to (1.1).

It is natural to investigate whether similar results hold when M does not admit a K¨ahler metric, but only a Hermitian metric α. If we do not impose any constraint on

Supported in part by National Science Foundation grants DMS-1306298 and DMS-1350696 (G. Sz.), DMS-1308988 (V. T.) and DMS-1406164 (B. W.). The second-named author is supported in part by a Sloan Research Fellowship.

the class of Hermitian metrics that we consider, then (1.1) can be trivially solved by a conformal change of metric. However, there is a natural class of Hermitian metrics which exist on all compact complex manifolds, namely Gauduchon metrics. A Hermitian metricαis calledGauduchon if

∂∂(α¯ ⁿ⁻¹) = 0,

and a classical result of Gauduchon [12] says that every Hermitian metric is conformal to a Gauduchon metric (uniquely up to scaling, whenn>2). In particular, if we restrict our attention to Gauduchon metrics, then we cannot use non-trivial conformal changes.

Motivated by Yau’s theorem, in 1984 Gauduchon [13, §IV.5] posed the following conjecture.

Conjecture1.1. LetM be a compact complex manifold and Ψ be a closed real (1,1)- form onM with [Ψ]=c^BC₁ (M)∈H_BC^1,1(M,R). Then there is a Gauduchon metricω onM with

Ricci(ω) = Ψ. (1.3)

To explain our notation here,

H_BC^1,1(M,R) ={d-closed real (1,1)-forms}

{√

−1∂∂ψ¯ :ψ∈C^∞(M,R)}

denotes the (finite-dimensional)Bott–Chern cohomology group, and Ricci(ω) is the Chern–

Ricci form ofω, which is locally given by Ricci(ω) =−√

−1∂∂¯log detg, where we writeω=√

−1g_i¯jdzⁱ∧d¯z^j. It is a closed real (1,1)-form and its first Bott–Chern cohomology classc^BC₁ (M)=[Ricci(ω)]∈H_BC^1,1(M,R) is immediately seen to be independent of the choice ofω.

In the spirit of Yau’s theorem, we restate Conjecture1.1as an equivalent statement about the existence of Gauduchon metrics with prescribed volume form.

Conjecture 1.2. LetM be a compact complex manifold andσbe a smooth positive volume form. Then, there is a Gauduchon metricω onM with

ωⁿ=σ. (1.4)

The equivalence with Conjecture1.1follows by applying the operator−√

−1∂∂¯log to (1.4).

Our result, Theorem 1.3 below, gives a proof of Conjecture 1.1 (and hence also of Conjecture1.2). Moreover, our result strengthens the conjecture by imposing a cohomo- logical constraint on the solutionω. Before we state our results, we make some remarks about Conjecture1.1.

(1) When M is K¨ahler, this conjecture follows from Yau’s theorem.

(2) When n=2, the conjecture was proved by Cherrier [4] in 1987 by solving a complex Monge–Amp`ere equation (see also [38] and [17] for different proofs).

(3) More recently, the second- and third-named authors [40] proved Conjecture1.1 whenM admits an astheno-K¨ahler metric, i.e. a Hermitian metricαwith ∂∂(α¯ ⁿ⁻²)=0 (a condition introduced in [21]).

(4) Clearly, there can be no uniqueness in Conjecture1.1as stated.

(5) In [37] the second- and third-named authors proved that given a Hermitian metricα, one can always find another Hermitian metricωof the formω=α+√

−1∂∂u, for¯ u∈C^∞(M,R), solving (1.3). Ifn=2, thenαGauduchon implies thatωis also Gauduchon (and this equation was solved in [4]), but this is no longer the case when n>3. Hence, the result of [37] does not help to solve Conjecture1.1in dimension 3 or higher.

(6) A consequence of Conjecture 1.1 is that c^BC₁ (M)=0 holds if and only if there exist Chern–Ricci-flat Gauduchon metrics onM. More information about these “non- K¨ahler Calabi–Yau” manifolds can be found in [35].

We now state our main results. We first introduce some terminology concerning cohomology classes of (n−1, n−1)-forms. Define the Aeppli cohomology group

H_A^n−1,n−1(M,R) ={∂∂-closed real (n−1, n−1)-forms}¯ {∂γ+∂γ:γ∈Λ^n−2,n−1(M)} .

This space is naturally in duality with the Bott–Chern cohomology group we consid- ered earlier, with the non-degenerate pairingH_A^n−1,n−1(M,R)⊗H_BC^1,1(M,R)!Rgiven by wedge product and integration overM (see e.g. [1]). Ifα0 is a Gauduchon metric, then αⁿ⁻¹₀ defines a class [αⁿ⁻¹₀ ]∈H_A^n−1,n−1(M,R).

We prove the following result.

Theorem1.3. Let M be a compact complex manifold with a Gauduchon metricα0, and Ψ be a closed real (1,1)-form on M with [Ψ]=c^BC₁ (M)∈H_BC^1,1(M,R). Then, there exists a Gauduchon metric ω satisfying [ωⁿ⁻¹]=[αⁿ⁻¹₀ ] in H_A^n−1,n−1(M,R)and

Ricci(ω) = Ψ. (1.5)

This result immediately implies Conjectures 1.1and1.2.

In [40], the second- and third-named authors observed that to solve Theorem 1.3it is enough to solve a certain partial differential equation, which was also independently introduced by Popovici [30]. This equation is a variant of one introduced by Fu–Wang–

Wu [10] and related to Harvey–Lawson’s notion of (n−1)-plurisubharmonic functions [18], [19].

Namely, we seek a Hermitian metricω onM with the property that ωⁿ⁻¹=αⁿ⁻¹₀ +∂γ+∂γ,

where

γ=¹₂√

−1 ¯∂u∧αⁿ⁻²,

u∈C^∞(M,R) andα is a background Gauduchon metric. Clearly, by construction, the metricω is Gauduchon assuming thatα₀ is Gauduchon. Substituting, we see that

ωⁿ⁻¹=αⁿ⁻¹₀ +√

−1∂∂u∧α¯ ⁿ⁻²+Re(√

−1∂u∧∂(α¯ ⁿ⁻²)), (1.6) while if we write

Ricci(ω) = Ψ+√

−1∂∂F,¯ then (1.5) is equivalent to

ωⁿ=e^F+bαⁿ,

for some constant b∈R. This is exactly the equation that we solve, thus resolving [40, Conjecture 1.5] and [30, Question 1.2].

Theorem1.4. Let M be a compact complex manifold with dim_CM=n>2,equipped with a Hermitian metricα₀ and a Gauduchon metric α. Given a smooth function F on M,we can find a unique u∈C^∞(M,R)with sup_Mu=0,and a uniqueb∈Rsuch that the Hermitian metric ω defined by

ωⁿ⁻¹:=αⁿ⁻¹₀ +√

−1∂∂u∧α¯ ⁿ⁻²+Re(√

−1∂u∧∂(α¯ ⁿ⁻²))>0 satisfies

ωⁿ=e^F+bαⁿ. (1.7)

Clearly, as we just described, Theorem1.3follows from this result if we takeα0=α Gauduchon. We make some remarks about Theorem1.4.

(1) In the case when α is K¨ahler, or more generally if the linear term involving

∂u is removed, the equation (1.7) reduces to the Monge–Amp`ere equation for (n−1)- plurisubharmonic functions, solved by the second- and third-named authors [39], [40]

(see also [11] for earlier partial results).

(2) In the case when n=2, this equation reduces to the complex Monge–Amp`ere equation solved in [4] (see also [38]).

(3) It was shown in [40] that Theorem1.4can be reduced to a second-order a-priori estimate of the form (cf. [20])

sup

M

|√

−1∂∂u|¯ α6C 1+sup

M

|∇u|²_α ,

for solutionsuof (1.7). This is precisely the estimate we prove in this paper.

(4) If in Theorem 1.4 we assume that α₀ is strongly Gauduchon in the sense of Popovici [29], namely that ¯∂(αⁿ⁻¹₀ ) is∂-exact, then, by construction, so is the solutionω.

Thus, we also get a Calabi–Yau-type theorem for strongly Gauduchon metrics. More applications of this theorem can be found in [30].

(5) Our method of proof of Theorem 1.4 can also be used to solve an equation introduced by Fu–Wang–Wu [10] in certain cases. Suppose we have a compact Hermitian manifold (M, α0) and we seek a Hermitian metricω solving (1.7) with the property that

ωⁿ⁻¹=αⁿ⁻¹₀ +√

−1∂∂(u α¯ ⁿ⁻²)

for someu∈C^∞(M,R) and some Hermitian metric α. This setup is particularly inter- esting because ifα₀ is balanced (i.e.d(αⁿ⁻¹₀ )=0, see [26]), then so isω, and one obtains a Calabi–Yau theorem for balanced metrics (see also [35, §4]). Whenα is K¨ahler, this setup reduces to the setting of item (1). If we instead assume thatαis astheno-K¨ahler, then we see that

ωⁿ⁻¹=α₀ⁿ⁻¹+√

−1∂∂u∧α¯ ⁿ⁻²+2 Re(√

−1∂u∧∂(α¯ ⁿ⁻²)), (1.8) which differs from (1.6) just for a factor of 2. Therefore, this problem falls into our general framework (see Theorem 2.2 below), and we conclude that we have uniform a-priori estimates for solutions of this equation. The exact same argument as in [40, Theorem 1.7] using the continuity method then shows that the equation is indeed solvable.

In the case when we chooseα0to be balanced, this gives a proof of [35, Conjectures 4.1 and 4.2], assuming thatM admits astheno-K¨ahler metrics. However, we should remark that we are not aware of any example of a non-K¨ahler compact complex manifold which admits both balanced and astheno-K¨ahler metrics.(¹)

(6) The same argument as the proof of Theorem1.4also allows us to find a Gaudu- chon metricω solving the “complex-Hessian” equation

ω^k∧α^n−k=e^F+bαⁿ

(¹) After this paper was posted, and prompted by our remark, explicit examples were constructed in [9] and [22] in all complex dimensions>4.

for any 16k6n; see also [34, Proposition 24] for the case of (n−1)-plurisubharmonic functions, and [8], [20] for the standard K¨ahler case whereω=α+√

−1∂∂u.¯

(7) The complex setting is very different from the real analogue of (1.7), treated for example in more generality in [16]. The underlying reason is the fact that there are two different types of complex derivatives. In our case, the special structure of the gradient term in (1.7) plays a key role.

In fact, Theorem 1.4 follows from a much more general result, where we consider a large class of fully non-linear second-order elliptic equations on Hermitian manifolds.

This result is analogous to the main result in [34], giving a-priori estimates in the presence of a suitable subsolution. We will state this as Theorem2.2 in §2. This result fits into a large body of work on fully non-linear second-order elliptic equations, going back to the work of Caffarelli–Nirenberg–Spruck [3] on the Dirichlet problem on domains inRⁿ. Some other works on this topic include [6], [7], [14]–[17], [23]–[25], [27], [28], [32], [33], [41], [42], [44]–[46].

In our proof of Theorem 2.2, we use some of the language and approaches of the recent paper of the first-named author [34]. However, if one is only interested in a direct proof of Theorem1.4, one can equally well use the language of [40]. In any case, the key new ingredient is an understanding of the structure of the term Re(√

−1∂u∧∂(α¯ ⁿ⁻²)).

The paper is organized as follows. In§2 we will introduce some notation and state our main technical result (Theorem2.2). The proof of this theorem will be given in §3, and in§4we show how this implies Theorem 1.4.

As the present work neared completion, we were informed that Bo Guan and Xiaolan Nie have a work in progress on related results.

Acknowledgments. We thank the referee for useful remarks.

2. Background and the general setting

Let (M, α) be a compact Hermitian manifold of complex dimensionnand write α=√

−1α_i¯jdzⁱ∧d¯z^j>0.

Fix a background (1,1)-formχ=√

−1χ_i¯jdzⁱ∧d¯z^j which is not necessarily positive defi- nite. LetW_i¯j(∇u) be a Hermitian tensor which depends linearly on∇u. For u:M!R, define a new tensorg_i¯j by

g_i¯j:=χ_i¯j+u_i¯j+W_i¯j. (2.1) Note that we do not assume that (g_i¯j) is positive definite. We will study equations for g, whereW has a special structure related to the equation (1.7). To define this, let us

write

˜

g_i¯j:=Pα(g_i¯j) = 1

n−1((trαg)α_i¯j−g_i¯j), (2.2) where Pα is an operator on tensors, depending on the fixed metric α, defined by the second equality in (2.2). Here trαg is the trace of g with respect toα. As an aside, if αis the Euclidean metric onCⁿ, then the conditionPα(u_i¯j)>0 is equivalent to saying thatuis (n−1)-plurisubharmonic, in the sense of Harvey–Lawson [18].

Observe that, writing ∆=α<sup>k`¯</sup>∂k∂`¯,

˜

g_i¯j=χe_i¯j+ 1

n−1((∆u)α_i¯j−u_i¯j)+Z_i¯j (2.3) forZ given by

Z_i¯j:=P_α(W_i¯j) = 1

n−1((tr_αW)α_i¯j−W_i¯j), (2.4) and similarly,χe_i¯j=P_α(χ_i¯j). Note that we can also writeW explicitly in terms ofZ:

W_i¯j= (trαZ)α_i¯j−(n−1)Z_i¯j. (2.5) A crucial assumption we make is thatW depends on∇uin the following way: we assume that the tensorZ has the form

Z_i¯j=Z_i^p_¯jup+Z_j^p_¯iup, (2.6) for some tensorZ_i^p_¯j independent ofu. In addition we have the following.

Assumption for W. In orthonormal coordinates forαat any given point, the com- ponentZ_i¯j is independent ofu¯ianduj (in other words,Z_i^j_¯j=0 for alliandj), and∇iZ_i¯i

is independent ofu¯i (in other words,∇¯iZ_iⁱ_¯i=0 for alli).

Here, ∇ is the Chern connection of α. This assumption expresses a certain skew- symmetry requirement for the tensor W. This assumption is satisfied for the (n−1)- plurisubharmonic Monge–Amp`ere equation, the case of most interest to us, see (4.3) below, the key reason being that the torsion tensor is skew-symmetric.

Let us record here a few consequences of this assumption, which will be used later.

Taking ∇p of (2.6), setting i=j, evaluating at that point and using that Z_iⁱ_¯i=0, we see that∇pZ_i¯i is independent ofu_ii andu_i¯i (at that point, in orthonormal coordinates forα). Here, the subscripts ofudenote ordinary partial derivatives. Similarly,∇iZ_p¯i is independent ofu_ii. Taking two covariant derivatives, we have

∇¯i∇_iZ_i¯i=∇¯i∇_iZ_i^p_¯iu_p+Z_i^p_¯i∇¯i∇_iu_p+∇_iZ_i^p_¯i∇¯iu_p+∇¯iZ_i^p_¯i∇_iu_p +∇¯i∇_iZ_i^p_¯iu_p¯+Z_i^p_¯i∇¯i∇_iu_p¯+∇_iZ_i^p_¯i∇¯iu_p¯+∇¯iZ_i^p_¯i∇_iu_p¯,

and evaluating at our point and using the assumptions∇¯iZ_iⁱ_¯i=0 andZ_iⁱ_¯i=0, we see that

∇¯i∇iZ_i¯i is independent ofuii,u¯i¯i,u_ii¯i andu_i¯i¯i.

Given a smooth functionh, we study equations of the form F(A) =h,

where A is the endomorphismAⁱ_j=α^ip¯g_j¯p of the holomorphic tangent bundle, which is Hermitian with respect to the inner product defined by α, and F(A) is a symmetric function of the eigenvaluesλ1, ..., λn ofA:

F(A) =f(λ₁, ..., λ_n). (2.7) We assume that our operatorF has the special form F(M)=Fe(P(M)), where

P(M) = (n−1)⁻¹(Tr(M)I−M),

analogous toPαabove (here writing Tr(M) for the trace of the matrixM), and Fe(B) = ˜f(µ₁, ..., µ_n),

whereµ₁, ..., µ_n are the eigenvalues ofB, and ˜f is another symmetric function. In terms of eigenvalues, this means that

f(λ1, ..., λn) = ( ˜fP)(λ1, ..., λn), (2.8) where we are writingP for the mapRⁿ!Rⁿinduced on diagonal matrices by the matrix map P above. Explicitly, writing µ=P(λ) for the correspondingn-tuples λ, µ∈Rⁿ, we have

f(λ1, ..., λn) = ˜f(µ1, ..., µn), for µk= 1 n−1

X

i6=k

λi. (2.9)

Assumptions for f˜and h. We make the following assumptions on ˜f, and the func- tionhin our equation:

(i) ˜f is defined on an open symmetric convex coneΓe Rⁿ, containing the positive orthant Γn={(x1, ..., xn)∈Rⁿ:xi>0, i=1, ..., n}.

(ii) ˜f is symmetric, smooth, concave and increasing, i.e. its partials satisfy ˜fi>0 for alli.

(iii) sup_∂

Γef <˜ infMh.

(iv) For allµ∈Γ, we havee

tlim!∞f˜(tµ) = sup

Γe

f ,˜ where both sides are allowed to be∞.

(v) his a smooth function onM.

Define the cone Γ⊂Rⁿ by Γ=P⁻¹(eΓ). Observe that P maps Γn into Γn. It is then easy to see that the functionf= ˜fP: Γ!Rsatisfies exactly the same conditions as ˜f. In particular, some of the results of [34] can be applied to the equationF(A)=h. We need the following definition (see [34, Remark 8] to see the equivalence with the definition there), which is a modification of a notion introduced by Guan [14].

Definition 2.1. We say that u is a C-subsolution for the equation F(A)=h if the following holds. Let g_i¯j be defined as in (2.1). We require that for every pointx∈M, ifλ=(λ1, ..., λn) denote the eigenvalues of the endomorphism α^ip¯gjp¯ at x, then for all i=1, ..., nwe have

lim

t!∞f(λ+te_i)> h(x).

Heree_idenotes theith standard basis vector. Note that part of the requirement is that λ+te_i∈Γ for sufficiently larget, for the limit to be defined.

With this background, our main estimate is the following, analogous to the main result in [34]. We will give the proof in§3.

Theorem 2.2. Suppose that u is a C-subsolution for the equation F(A)=h, and u is a smooth solution, normalized by sup_Mu=0. Suppose that F and h satisfy the assumptions above, including the assumption for the gradient term W. Then, for each k=0,1,2, ..., we have an estimate kuk_Ck(M,α)6Ck, with constant Ck depending on k, on the background data M, α, χ, F, h, the coefficients of W and the subsolution u.

The case of primary interest for us is equation (1.7), which corresponds to the symmetric function

f˜(µ1, ..., µn) = log(µ1... µn) (2.10) on the positive orthanteΓ=Γn. It is straightforward to check that ˜fsatisfies the conditions above. Indeed, ˜f converges to−∞on the boundary∂Γn, so (iii) is satisfied, and for (iv) it is enough to note that ˜f(tµ)= ˜f(µ)+nlogt, which converges to∞ast!∞.

In addition, if µ∈Γn, then we also have

tlim!∞

f˜(µ+tei) =∞

for all i. This means that for a function uto be a C-subsolution for this equation, the only requirement is that, at each point, the eigenvaluesλofα^i¯pgj¯p satisfyP(λ)∈Γn. In other words, the requirement is that ˜g_i¯j, defined in (2.2), is positive definite.

Note that, if uis a C-subsolution, then replacingχby χ⁰_i¯j=χ_i¯j+u_i¯j+W_i¯j(∇u),

we may assume thatu=0. The important consequence of 0 being aC-subsolution is the following, which follows from [34, Proposition 6 and Lemma 9].

Proposition2.3. Suppose that 0is a C-subsolution for the equation F(A)=h,and uis a solution. Define g_i¯j as in (2.1). There are constants R,>0,independent of u, with the following property. Let x∈M, and choose orthonormal coordinates for α at x, such that g is diagonal, with eigenvalues λ=(λ1, ..., λn). If |λ|>R, then there are two possibilities:

(a) we either have

X

k

fk(λ)[χ_k¯k−λk]>X

k

fk(λ), (b) or fk(λ)>P

ifi(λ)for all k.

In addition P

kfk(λ)>.

We collect some other basic properties of the functionsf and ˜f. Suppose thatλ∈Γ withλ₁>...>λ_n. Thenµ₁6...6µ_n, and so, by property (ii), ˜f₁>...>f˜_n>0 (see e.g. [34, p. 12]). We have

fk= 1 n−1

X

i6=k

f˜i, (2.11)

which implies that 0<f₁6...6f_n. Also, fork>1, 0<

f˜1

n−16fk6f˜1, (2.12)

i.e. the f_k for k>1 are all comparable, while f₁ may be relatively small. In addition, from (2.11) withk=1, we obtain

f˜i6(n−1)f1, fori >1. (2.13) Proposition2.3is easy to verify directly in the case of equation (1.7), where

f(λ) = log(µ1... µn), withµk defined as in (2.9). Indeed, in this case

f˜i(µ) = 1 µ_i and

f_k(λ) = 1 n−1

X

i6=k

1 µi

.

The function 0 being aC-subsolution means that χein (2.3) is positive definite. We have

X

k

f_k(λ)χ_k¯k=X

i

1

µ_iχe_i¯i> τX

i

1

µ_i=τX

k

f_k(λ),

for someτ >0 depending on a lower bound forχ. We also havee X

k

λ_kf_k(λ) =n.

It follows that we have the alternative (a) in Proposition 2.3 whenever there is one sufficiently smallµi, which by the equationf(λ)=his equivalent to having at least one largeµi, i.e. at least one large λi. In addition,

X

k

fk(λ) =

n

X

i=1

1

µi >n(µ1... µn)^−1/n=ne^−h/n, so that the final claim in Proposition2.3also holds.

3. Proof of the main estimate In this section, we give the proof of Theorem2.2.

First of all, a uniform boundkukL^∞(M)6Ccan be obtained by a simple modification of the argument in [34, Proposition 10 and Remark 12], which is itself inspired by B locki’s proof of theL^∞ estimate in Yau’s theorem [2]. In the setting of equation (1.7), theL^∞ estimate ofu was first proved in [40], using a different method more analogous to the arguments in [4], [37], [39], [43].

Our main goal is the following estimate:

sup

M

|√

−1∂∂u|¯ α6C sup

M

|∇u|²_α+1

, (3.1)

for a constantC depending only on the fixed data of Theorem2.2. We remark that an estimate of this form was proved in the context of the complex Hessian equations by Hou–

Ma–Wu [20], making use of ideas of Chou–Wang [5]. For the (n−1)-plurisubharmonic equation (namely, equation (1.7) without the linear term in ∂u), an estimate of this type was proved in [39], [40]. This was then generalized much further in [34], where the estimate was shown to hold for a large class of equations. Our proof begins along similar lines to these papers. The new difficulty comes from the linear term in∂u, which, fortunately, has a special structure that we can exploit.

In fact, the estimate (3.1) is equivalent to the bound λ16CK,

whereK=1+sup_M|∇u|²_αandλ₁is the largest eigenvalue ofA=(Aⁱ_j)=(α^i¯pg_j¯p). Indeed, our assumption on the cone Γ implies thatP

iλ_i>0 (see Caffarelli–Nirenberg–Spruck [3]).

Then, ifλ1 is bounded from above byCK, so is|λi|for alli, giving the same bound for sup_M|√

−1∂∂u|¯ α.

We consider the function

H= logλ₁+φ(|∇u|²_α)+ψ(u), whereφis defined by

φ(t) =−1 2log

1− t

2K

,

so thatφ(|∇u|²_α)∈

0,¹₂log 2

satisfies 1

4K< φ⁰< 1

2K, φ⁰⁰= 2(φ⁰)²>0, andψis defined by

ψ(t) =D1e^−D2t, (3.2) for sufficiently large uniform constantsD1, D2>0 to be chosen later. By theL^∞ bound onu, the quantityψ(u) is uniformly bounded.

We remark that we follow [34] by computing with the largest eigenvalue λ₁instead of the analogous quantity in [40], but in fact either quantity works, at least in the case of equation (1.7). Also note that, while the functionφ here coincides with that in [20]

(and also in [39], [40]), our choice ofψis crucially different.

We work at a point whereH achieves its maximum, in orthonormal complex coordi- nates forαcentered at this point, such thatg is diagonal andλ₁=g_1¯1. The quantityH need not be smooth at this maximum point, because the largest eigenvalue ofAmay have eigenspace of dimension larger than 1. To take care of this, we carry out a perturbation argument as in [34], choosing local coordinates such thatH achieves its maximum at the origin, whereA is diagonal with eigenvalues λ₁>...>λ_n, as before. We fix a diag- onal matrix B, with B₁¹=0 and 0<B₂²<...<Bⁿ_n, and we define ˜A=A−B, denoting its eigenvalues by ˜λ1, ...,λ˜n.

At the origin, we have

λ˜1=λ1 and ˜λi=λi−B_iⁱ, i >1,

and ˜λ1>λ˜2>...>˜λn. As discussed above, our assumption on the cone Γ implies that P

iλi>0, and we fix the matrixB small enough so that X

i

λ˜i>−1.

We can choose this matrixB in such a way that, in addition, X

p>1

1 λ1−˜λp

6C, (3.3)

for some fixed constantC depending on the dimensionn. Now, after possibly shrinking the chart, the quantity

He= log ˜λ1+φ(|∇u|²_α)+ψ(u)

is smooth on the chart and achieves its maximum at the origin. We will apply the maximum principle toH. Our goal is to obtain the bound ˜e λ16CK at the origin, which will give us the required estimate (3.1). Hence, we may and do assume that ˜λ1K at this point.

We now differentiate He at the origin and, as before, we use subscripts k and ¯` to denote the partial derivatives∂/∂z<sup>k</sup> and∂/∂z¯<sup>`</sup>. We have

Hek= λ˜1,k

λ˜1

+φ⁰(α^p¯qupuqk¯ +α^p¯qupkuq¯+(α^p¯q)kupuq¯)+ψ⁰uk

= λ˜1,k

λ˜₁ +φ⁰(u_pu_pk¯ +u_pku_p¯+(α^p¯q)_ku_pu_¯q)+ψ⁰u_k= λ˜1,k

λ˜₁ +φ⁰V_k+ψ⁰u_k,

(3.4)

whereV_k:=u_pu_pk¯ +u_pku_p¯+(α^p¯q)_ku_pu_q¯. Differentiating once more,

He_kk¯= λ˜_1,k¯k

λ1

−|λ˜_1,k|² λ²₁ +φ⁰

u_pu_pk¯ ¯k+u_p¯u_pk¯k+X

p

|upk|²+X

p

|upk¯ |²

+(α^p¯q)¯kupuqk¯ +(α^p¯q)¯kupku¯q+(α^p¯q)_k¯kupuq¯+(α^p¯q)k(upu_q¯k¯+u_p¯ku¯q)

+φ⁰⁰|Vk|²+ψ⁰⁰|uk|²+ψ⁰u_kk¯,

where we use the convention that we sum in all repeated indices except the free indexk.

Since 1/4K <φ⁰<1/2K, we can absorb all the terms involvingα into the squared terms up to a constant, i.e. we have

He_k¯k>

˜λ_1,k¯k

λ₁ −|λ˜1,k|²

λ²₁ +φ⁰(u_pu_pk¯k¯+u_p¯u_pk¯k)+ 1 5K

X

p

(|u_pk|²+|u_pk¯ |²) +φ⁰⁰|Vk|²+ψ⁰⁰|uk|²+ψ⁰u_kk¯−C.

(3.5)

The constantC denotes a constant that may change from line to line, but it does not depend on the parametersD₁andD₂that we are yet to choose.

Calculation of ˜λ_1,k¯k

Let us now compute the derivatives of ˜λ1. We have the following general formulas for the derivatives of the eigenvalueλi of complexn×nmatrices at a diagonal matrix with distinct real eigenvalues (see for instance Spruck [31] in the case of matrices with real entries):

λ^pq_i =δpiδqi

λ^pq,rs_i = (1−δip)δiqδirδps

λ_i−λ_p +(1−δir)δisδipδrq

λ_i−λ_r , (3.6)

whereλ^pq_i denotes the derivative with respect to the (p, q)-entryA^p_q of the matrixA, as a complex variable.

Denoting by ˜λ₁the largest eigenvalue of the endomorphism ˜Aagain, we have, using (3.6),

˜λ_1,k= ˜λ^pq₁ ∇k( ˜A^p_q) =∇k( ˜A¹₁) =∇kg_1¯1−∇kB₁¹=g_1¯1k+(α^1¯1)_kg_1¯1, (3.7) since∇kB₁¹=0 at the origin. Here we computed using covariant derivatives with respect to the Chern connection ofα, which makes the positivity of certain terms more apparent when we take second derivatives:

˜λ_1,k¯k= ˜λ^pq₁ ∇¯k∇_kA˜^p_q+ ˜λ^pq,rs₁ (∇_kA˜^p_q)(∇k¯A˜^r_s) =∇k¯∇_kg_1¯1+ ˜λ^pq,rs₁ (∇_kA˜^p_q)(∇¯kA˜^r_s), (3.8) where we used (3.6) and the fact that∇¯k∇kB₁¹=0 at the origin. To rewrite this in terms of partial derivatives, note first that

∇kg_1¯1=g_1¯1k−Γ^m_k1g_m¯1

∇¯k∇_kg_1¯1=g_1¯1kk¯−(∂¯kΓ^m_k1)g_m¯1−Γ^m_k1g_m¯1¯k−Γ^q_k1g_1¯qk+Γ^q_k1Γ^m_k1g_m¯q

=g_1¯1kk¯+O

X

m

|g_{1 ¯mk}|+λ₁

.

(3.9)

In addition, we have

˜λ^pq,rs₁ (∇kA˜^p_q)(∇¯kA˜^r_s) =X

p>1

(∇kA˜^p₁)(∇k¯A˜¹_p)+(∇kA˜¹_p)(∇¯kA˜^p₁) λ1−˜λp

=X

p>1

(∇kg1 ¯p+Γ^q_k1B_q^p)(∇k¯g_p¯1)+(∇kg_p¯1−Γ¹_kqB_p^q)(∇¯kg1 ¯p) λ1−λ˜p

>1 2

X

p>1

|∇kg1 ¯p|²+|∇kg_p¯1|² λ1−λ˜p

−C,

(3.10)

where we used (3.3) and (3.6). Recall that, due to our choice ofB, we haveP

iλ˜i>−1, which implies 1/(λ1−˜λp)>1/(nλ1+1) forp>1, and so

λ˜^pq,rs₁ (∇_kA˜^p_q)(∇¯kA˜^r_s)> 1 2(nλ1+1)

X

p>1

(|∇_kg_{1 ¯p}|²+|∇_kg_p¯1|²)−C.

To rewrite this in terms of partial derivatives, note that

∇_kg_{1 ¯p}=g_{1 ¯pk}−Γ<sup>`</sup><sub>k1</sub>g<sub>`¯p</sub>=g_{1 ¯pk}+O(λ₁), where we made use of the fact thatP

iλ_i>0 to conclude that|λ_i|6(n−1)λ₁for all i. It follows, since we assumeλ₁>1, that

λ˜^pq,rs₁ (∇kA˜^p_q)(∇¯kA˜^r_s)> 1 4nλ1

X

p>1

(|g1 ¯pk|²+|g_p¯1k|²)−Cλ1.

Combining this with (3.8) and (3.9), we obtain

˜λ_1,kk¯>g_1¯1k¯k+ 1 4nλ1

X

p>1

(|g1 ¯pk|²+|g_p¯1k|²)−C

X

m

|g1 ¯mk|+λ1

>g_1¯1k¯k+ 1 8nλ₁

X

p>1

(|g_{1 ¯pk}|²+|g_p¯1k|²)−C(|g_1¯1k|+λ₁).

(3.11)

Rewriting gin terms ofu, we have

g_1¯1k¯k=χ_1¯1k¯k+u_1¯1kk¯+W_1¯1k¯k=χ_1¯1k¯k+u_kk1¯¯ 1+W_1¯1k¯k

=χ_1¯1k¯k−χ_kk1¯¯ 1+g_k¯k1¯1−W_k¯k1¯1+W_1¯1k¯k,

(3.12)

and so

F^kk˜λ_1,k¯k>F^kkg_k¯k1¯1+F^kk(W_1¯1kk¯−W_k¯k1¯1) + 1

8nλ1

X

p>1

F^kk(|g1 ¯pk|²+|g_p¯1k|²)−C(F^kk|g_1¯1k|+λ1F), (3.13)

whereF^pq denotes the partial derivative of the functionF(A) with respect to the (p, q)- entry of the matrixA(as explained earlier), and we have setF=P

kF^kk. Observe that, due to (2.7) and (3.6), at the origin we have thatF^pq vanishes wheneverp6=q, while on the other hand F^kk=f_k, using the notation from §2. Recall from the last assertion of Proposition2.3that

F>>0, (3.14)

for a uniform>0.

The term F^kkg_kk1¯¯ 1

We now differentiate the equationF(A)=h, using covariant derivatives to simplify a term that appears below. Applying∇_i, we obtain

F^pq∇ig_qp¯=h_i, namely,

F^kkg_kki¯ +F^kk(α^k¯k)_ig_k¯k=h_i. (3.15) Applying∇¯i and settingi=1,

F^pq,rs∇1g_qp¯∇¯1g_s¯r+F^kk∇¯1∇1g_kk¯=h_1¯1. (3.16) To rewrite this using partial derivatives, note that

∇¯1∇₁g_k¯k=g_kk1¯¯ 1−(∂¯1Γ^m_1k)g_m¯k−Γ^m_1kg_m¯k¯1−Γ^q_1kg_k¯q1+Γ^q_1kΓ^m_1kg_m¯q

=g_kk1¯¯ 1−2 Re(Γ^q_1kgkq1¯)+O(λ1).

(3.17) By rewritingg in terms ofu, we have

gk¯q1=g1¯qk+χk¯q1−χ1¯qk+Wkq1¯ −W1¯qk, and hence

∇¯1∇₁g_k¯k=g_kk1¯¯ 1−2 Re(Γ^q_1k(W_kq1¯ −W_1¯qk))+O

X

q

|g_1¯qk|+λ₁

.

Returning to (3.16), and making use of (3.14), we obtain F^kkg_k¯k1¯1>−F^pq,rs∇1g_qp¯∇¯1g_s¯r−CF^kkX

q

|g1¯qk|−CFλ₁

+2 Re(F^kkΓ^q_1k(Wk¯q1−W1¯qk)).

(3.18)

To bound the term involvingW_1¯qk, we observe that

|F^kkW1¯qk|6C

Fλ1+X

p

F^kk|upk|

. (3.19)

To see (3.19), we use the fact thatλ₁>K to bound the terms involving the gradient ofu that arise fromW_1¯qk when taking the∂/∂z^k derivative ofW_1¯q=(tr_αZ)α_1¯q−(n−1)Z1¯q. For the term involvingW_kq1¯, note that

∇¯1∇₁W_k¯k=∇¯1(W_kk1¯ −Γ^q_1kW_qk¯)

=W_kk1¯¯ 1−Γ^q_1kW_k¯q1−Γ^q_1kW_q¯k¯1−(Γ^q_1k)¯1W_qk¯+Γ^p_1kΓ^q_1kW_qp¯.

In particular,

2 Re(Γ^q_1kWkq1¯ ) =W_k¯k1¯1−∇¯1∇1W_kk¯+O(K^1/2).

Using this (and that we may assumeλ1>K), we have

2F^kkRe(Γ^q_1kWk¯q1) =F^kkW_k¯k1¯1−F^kk∇¯1∇1W_k¯k+O(Fλ1). (3.20) Combining (3.18)–(3.20) gives

F^kkg_k¯k1¯1>−F^pq,rs∇1gqp¯∇¯1gs¯r+F^kkW_k¯k1¯1−F^kk∇¯1∇1W_kk¯

−C

F^kkX

q

|g1¯qk|+F^kkX

p

|upk|+Fλ1

. (3.21)

Going back to (3.13), using the square terms there to control the terms in (3.21) involving

|g1¯qk|forq6=1, we obtain

F^kk˜λ_1,k¯k>−F^pq,rs∇1gq¯p∇¯1gs¯r+F^kk(W_1¯1kk¯−∇¯1∇1W_k¯k)

−C

F^kk|g_1¯1k|+X

p

F^kk|upk|+λ1F

. (3.22)

The term F^kk∇¯1∇1W_k¯k

Using (2.5) and (2.11), we have F^kk∇¯1∇1W_kk¯= 1

n−1 X

k

X

i6=k

Feⁱⁱ∇¯1∇1W_kk¯

= 1

n−1 X

i

FeⁱⁱX

k6=i

∇¯1∇1W_k¯k=Feⁱⁱ∇¯1∇1Z_i¯i.

(3.23)

Recall from (2.12) and (2.13) thatFe¹¹= ˜f1is “large”, equivalent toF^kk=fkfor anyk>1, whileFeⁱⁱ= ˜fifori>1, is “small”, bounded byF¹¹=f1. We also recall that, as explained earlier, the crucial assumption onZ_i¯j implies that∇¯1∇1Z_1¯1 does not contain the terms u_11¯1,u11 or their complex conjugates.

Hence, using the fact that sup_i,j|u_i¯j|6Cλ1 andλ1>K, F^kk∇¯1∇₁W_k¯k6C

Fe¹¹X

k>1

(|u_k1|+|u_k¯11|)+F¹¹X

k

(|u_k1|+|u_k¯11|)+λ₁F

6C

F¹¹(|u11|+|u_1¯11|)+X

k>1

F^kk(|uk1|+|u_k¯11|)+λ1F

6C

X

p

F^kk|upk|+F^kk|u_1¯1k|+λ1F

.

(3.24)

We also have

u_1¯1k=g_1¯1k−χ_1¯1k−W_1¯1k, and so

F^kk|u_1¯1k|6F^kk|g_1¯1k|+C

X

p

F^kk|upk|+λ1F

.

From (3.24), we then obtain

F^kk∇¯1∇1W_k¯k6C X

p

F^kk|upk|+F^kk|g_1¯1k|+λ1F

. (3.25)

The term F^kkW_1¯1k¯k

Let us writeW_1¯1=W^pu_p+W^p¯u_p¯. We have

(W^pu_p)_kk¯= (W^p)_kk¯u_p+(W^p)_ku_pk¯+(W^p)¯ku_pk+W^pu_pk¯k, and so, since we may assume thatλ₁K, it follows that

F^kk(W^pu_p)_kk¯>W^pF^kku_k¯kp−C

X

p

F^kk|u_pk|+λ₁F

. (3.26)

Using (3.15), we have

|F^kku_k¯kp|=|F^kk(g_k¯kp−χ_k¯kp−W_kkp¯ )|6CF^kk|g_k¯k|+CF+|F^kkW_kkp¯ | 6CF^kk|u_kk¯|+CK^1/2F+|F^kkW_kkp¯ |.

To deal with this last term, note that, due to (2.5), as in (3.23) we have F^kkW_k¯kp=F^kk∇pW_k¯k+O(K^1/2F) =Feⁱⁱ∇pZ_i¯i+O(K^1/2F),

and using the crucial assumption on Z_i¯j, as explained earlier, we see that ∇pZ_1¯1 is independent ofu11 and u_1¯1. It follows that these Hessian terms can appear only with the “small” coefficientsFeⁱⁱ withi>1. We obtain

|F^kkW_kkp¯ |6C

Fe¹¹X

k>1

(|u_kp|+|u¯kp|)+F¹¹X

k

(|u_kp|+|ukp¯ |)+K^1/2F

6C

X

p

F^kk|u_pk|+X

p

F^kk|u_pk¯|+K^1/2F

,

and so

|F^kku_kkp¯ |6C

X

p

F^kk|upk|+X

p

F^kk|u_pk¯|+K^1/2F

. (3.27)