2. Derivatives and Concavity of F

Volume 2012, Article ID 350183,48pages doi:10.1155/2012/350183

Research Article

Complex Hessian Equations on Some Compact K ¨ahler Manifolds

Asma Jbilou

Universit´e Internationale de Rabat (UIR), Parc Technopolis, Rocade de Rabat, Sal´e, 11 100 Sala Al Jadida, Morocco

Correspondence should be addressed to Asma Jbilou,[email protected] Received 29 March 2012; Revised 22 July 2012; Accepted 31 July 2012 Academic Editor: Aloys Krieg

Copyrightq2012 Asma Jbilou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

On a compact connected 2m-dimensional K¨ahler manifold with K¨ahler formω, given a smooth functionf :M → Rand an integer 1< k < m, we want to solve uniquely inωthe equation

ω^k∧ω^m−ke^fω^m, relying on the notion ofk-positivity forω ∈ω the extreme cases are solved:

kmbyYau in 1978, andk1 trivially. We solve by the continuity method the corresponding complex elliptickth Hessian equation, more difficult to solve than the Calabi-Yau equationkm, under the assumption that the holomorphic bisectional curvature of the manifold is nonnegative, required here only to derive an a priori eigenvalues pinching.

1. The Theorem

All manifolds considered in this paper are connected.

LetM, J, g, ωbe a compact connected K¨ahler manifold of complex dimensionm≥3.

Fix an integer 2 ≤ k ≤ m−1. Letϕ : M → Rbe a smooth function, and let us consider the1,1-formω ωi∂∂ϕand the associated 2-tensorgdefined bygX, Y ωX, JY . Consider the sesquilinear formshandhonT^1,0defined byhU, V gU, VandhU, V

gU, V. We denote byλg⁻¹gthe eigenvalues ofhwith respect to the Hermitian formh. By definition, these are the eigenvalues of the unique endomorphismAofT^1,0satisfying

hU, V hU, AV ∀U, V ∈T^1,0. 1.1

Calculations infer that the endomorphismAwrites

A:T^1,0−→T^1,0,

Uⁱ∂_i−→A^j_iUⁱ∂_jg^jg_iUⁱ∂_j.

1.2

A is a self-adjoint/Hermitian endomorphism of the Hermitian space T^1,0, h, therefore λg⁻¹g ∈ R^m. Let us consider the following cone:Γk {λ ∈ R^m/∀1 ≤ j ≤ k, σjλ > 0}, whereσ_jdenotes thejth elementary symmetric function.

Definition 1.1. ϕis said to bek-admissible if and only ifλg⁻¹g ∈Γk. In this paper, we prove the following theorem.

Theorem 1.2theσ_kequation. LetM, J, g, ωbe a compact connected K¨ahler manifold of complex dimension m ≥ 3 with nonnegative holomorphic bisectional curvature, and let f : M → Rbe a function of classC^∞satisfying

Me^fω^{m m}_k

Mω^m. There exists a unique functionϕ:M → R of classC^∞such that

1

M

ϕ ω^m0, 1.3

2ω^k∧ω^m−k e^f

^m_k

ω^m. Ek

Moreover the solutionϕisk-admissible.

This result was announced in a note in the Comptes Rendus de l’Acad´e-mie des Sciences de Paris published online in December 20091. The curvature assumption is used, inSection 6.2only, for an a priori estimate onλg⁻¹g as in2, page 408, and it should be removedas did Aubin for the casekmin3, see also4for this case. For the analogue ofEkonC^m, the Dirichlet problem is solved in5,6, and a Bedford-Taylor type theory, for weak solutions of the corresponding degenerate equations, is addressed in7. Thanks to Julien Keller, we learned of an independent work8aiming at the same result as ours, with a different gradient estimate and a similar method to estimateλg⁻¹g, but no proofs given for theC⁰and theC²estimates.

Let us notice that the functionf appearing in the second member of Ek satisfies necessarily the normalisation condition

Me^fω^{m m}_k

Mω^m. Indeed, this results from the following lemma.

Lemma 1.3. Consider

Mω^k∧ω^m−k

Mω^m. Proof. See9, page 44.

Let us writeEkdifferently.

Lemma 1.4. Considerω^k∧ω^m−k σkλg⁻¹g/ ^m_kω^m.

Proof. LetP ∈ M. It suffices to prove the equality at P in a g-normalg-adapted chart z centered atP. In such a chartg_ij0 δ_ijandg_ij0 δ_ijλ_i0, so atz0,ωidz^a∧dz^aand

ωiλa0dz^a∧dz^a. Thus

ω^k∧ω^m−k

a

iλa0dz^a∧dz^a _k

∧

b

idz^b∧dz^b _m−k

a1,..., ak∈{1,..., m}

distinct integers b1,..., b_m−k∈{1,..., m}\{a1,..., ak}

distinct integers

i^mλa10· · ·λak0

dz^a1∧dz^a1 ∧ · · · ∧

dz^ak∧dz^ak ∧

dz^b1∧dz^b1 ∧ · · · ∧

dz^bm−k∧dz^bm−k .

1.4

Now a₁, . . . , a_k, b₁, . . . , b_m−k are m distinct integers of {1, . . . , m} and 2-forms commute therefore,

ω^k∧ω^m−k

⎛

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎝

a1,..., ak∈{1,..., m}

distinct integers b1,..., bm−k∈{1,..., m}\{a1,..., ak}

distinct integers

λa10· · ·λak0

⎞

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎠

i^m

dz¹∧dz¹ ∧ · · · ∧

dz^m∧dz^m

ω^m m!

⎛

⎜⎜

⎝

a1,..., ak∈{1,..., m}

distinct integers

m−k! λ_a10· · ·λ_ak0

⎞

⎟⎟

⎠ω^m m!

ω^k∧ω^m−k m−k!

m! k!σkλ10, . . . , λm0ω^m σ_k λ

g⁻¹g ^m_k ω^m.

1.5

Consequently,Ekwrites:

σ_k λ

g⁻¹g

e^f. E_k

Let us remark that Em corresponds to the Calabi-Yau equation detg/ detg e^f, when E₁is just a linear equation in Laplacian form. Since the endomorphismAis Hermitian, the spectral theorem provides anh-orthonormal basis forT^1,0 of eigenvectorse1, . . . , em:Aei λiei,λ λ1, . . . , λm∈Γk. AtP ∈Min a chartz, we have Mat∂1,...,∂mAP Aⁱ_jz_1≤i,j≤m, thus

σkλAP σkλAⁱ_jz_{1≤i, j≤m}. In addition,A^j_i g^jg_ig^jg_i∂_iϕ δ_i^jg^j∂_iϕ, so the equation writes locally:

σk

λ

δ_i^jg^j∂_iϕ

1≤i, j≤m

e^f. E_k

Let us notice that a solution of this equation E_k

is necessarilyk-admissible 9, page 46.

Let us definef_kB σ_kλBandF_kB lnσ_kλBwhereB B^j_{i1≤i, j≤m}is a Hermitian matrix. The functionf_k is a polynomial in the variables B^j_i, specifically f_kB

|I|kB_II sum of the principal minors of orderkof the matrixB. Equivalently

E_k writes:

Fk

δ^j_i g^j∂_iϕ

1≤i, j≤m

f.

E_k

It is a nonlinear elliptic second order PDE of complex Monge-Amp`ere type. We prove the existence of ak-admissible solution by the continuity method.

2. Derivatives and Concavity of F

2.1. Calculation of the Derivatives at a Diagonal Matrix

The first derivatives of the symmetric polynomial σ_k are given by the following: for all 1 ≤ i ≤ m,∂σk/∂λ_iλ σ_{k−1, i}λ whereσ_{k−1, i}λ : σ_k−1|λi0. For 1 ≤ i /j ≤ m, let us denoteσ_{k−2, ij}λ : σ_k−2|λiλj0and σ_{k−2, ii}λ 0. The second derivatives of the polynomial σ_k are given by∂²σ_k/∂λ_i∂λ_jλ σ_{k−2, ij}λ. We calculate the derivatives of the function f_k:HmC → R, whereHmCdenotes the set of Hermitian matrices, at diagonal matrices using the formula:

fkB

1≤i1<···<ik≤m

σ∈Sk

εσBⁱ_i^σ1₁ · · ·Bⁱ_i^σk

k

1 k!

1≤i1,...,ik,j1,...,jk≤m

εⁱ_j^1···ik

1···jkB^j_i¹

1· · ·B^j_i^k

k,

2.1

where

ε_j^i1···ik

1···jk

⎧⎨

⎩

1 ifi1, . . . , ik distinct andj1, . . . , jk even permutation ofi1, . . . , ik,

−1 ifi₁, . . . , i_kdistinct andj₁, . . . , j_k odd permutation ofi₁, . . . , i_k, 0 else.

2.2

These derivatives are given by9, page 48

∂fk

∂B^j_i

diagb1, . . . , bm

0 ifi /j, σ_{k−1, i}b1, . . . , b_m ifij, ifi /j ∂²fk

∂B_j^j∂B_iⁱ

diagb1, . . . , bm

σ_k−2,ijb1, . . . , bm

∂²f_k

∂B_jⁱ∂B^j_i

diagb1, . . . , b_m

−σ_k−2,ijb1, . . . , b_m,

2.3

and all the other second derivatives offkat diagb1, . . . , bmvanish.

Consequently, the derivatives of the function F_k lnf_k : λ⁻¹Γk ⊂ HmC → R at diagonal matrices diagλ1, . . . , λ_m with λ λ1, . . . , λ_m ∈ Γk, where λ⁻¹Γk {B ∈ HmC/λB∈Γk}, are given by

∂Fk

∂B_i^j

diagλ1, . . . , λm

⎧⎨

⎩

0 ifi /j, σ_{k−1, i}λ

σ_kλ if ij, 2.4

if i /j ∂²Fk

∂B_jⁱ∂B^j_i

diagλ1, . . . , λ_m

−σ_{k−2, ij}λ σ_kλ

∂²Fk

∂B_j^j∂B_iⁱ

diagλ1, . . . , λ_m

σ_{k−2, ij}λ

σ_kλ − σ_{k−1, i}λσk−1, jλ σkλ²

∂²Fk

∂B_iⁱ∂B_iⁱ

diagλ1, . . . , λm

−σ_{k−1, i}λ² σkλ²

2.5

and all the other second derivatives ofFkat diagλ1, . . . , λmvanish.

2.2. The Invariance ofFk and of Its First and Second Differentials The functionF_k:λ⁻¹Γk → Ris invariant under unitary similitudes:

∀B∈λ⁻¹Γk, ∀U∈U_mC, F_kB F_k

tUBU . 2.6

Differentiating the previous invariance formula 2.6, we show that the first and second differentials ofF_kare also invariant under unitary similitudes:

∀B∈λ⁻¹Γk, ∀ζ∈ HmC, ∀U∈U_mC, dFk_B·ζ dFktUBU·

tUζU ,

2.7

∀B∈λ⁻¹Γk, ∀ζ∈ HmC, ∀Θ∈ HmC, ∀U∈UmC,

d²F_k

B·ζ,Θ d²F_k

tUBU·

tUζU,^tUΘU . 2.8

These invariance formulas are allowed to come down to the diagonal case, when it is useful.

2.3. Concavity ofFk

We prove in9the concavity of the functions u◦λ and more generallyu◦λB whenu ∈ Γ0R^mand is symmetric9, Theorem VII.4.2, which in particular gives the concavity of the functionsFklnσkλ9, Corollary VII.4.30and more generally lnσkλB9, Theorem VII.4.29.

In this section, let us show by an elementary calculation the concavity of the functionFk. Proposition 2.1. The functionF_k:λ⁻¹Γk → R, B→F_kB lnσ_kλBis concave (this holds for allk∈ {1, . . . , m}).

Proof. The functionFkis of classC², so its concavity is equivalent to the following inequality:

∀B∈λ⁻¹Γk, ∀ζ∈ HmC ^m

i, j, r, s1

∂²F_k

∂B_r^s∂B_i^jBζ^j_iζ^s_r ≤0. 2.9

LetB ∈ λ⁻¹Γk,ζ ∈ HmC, andU ∈U_mCsuch that ^tUBU diagλ1, . . . , λ_m. We have λ λ1, . . . , λm∈Γk. Let us denoteζ ^tUζU∈ HmC:

S: ^m

i, j, r, s1

∂²Fk

∂B^s_r∂B^j_iBζ^j_iζ^s_r

d²F_k

B·ζ, ζ so by the invariance formula2.8

d²F_k

tUBU·

tUζU,^tUζU

^m

i,j,r,s1

∂²F_k

∂B_r^s∂B_i^j

diagλ1, . . . , λ_mζ^j_iζ^s_r

^m

i /j1

−σ_k−2,ijλ σkλ ζ^j_i ζⁱ_j

ζ^j_i

^m

i /j1

σ_{k−2, ij}λ

σ_kλ −σ_{k−1, i}λσk−1, jλ σkλ²

:cij

ζⁱ_iζ^j_j^m

i1

−σ_{k−1, i}λ² σkλ²

ζⁱ_i ²

^m

i,j1

−σ_{k−2, ij}λ

σ_kλ ζ^j_i^{2 m}

i, j1

c_ijζ_iⁱζ^j_j.

2.10

But cij ∂²lnσk/∂λi∂λjλ, and ζⁱ_i ∈ R, so _m

i,j1cijζⁱ_iζ_j^j ≤ 0 by concavity of lnσk

at λ ∈ Γk 10, page 269. In addition, σ_k−2,ijλ > 0 since λ ∈ Γk 11, consequently _m

i,j1−σk−2,ijλ/σkλ|ζ^j_i|²≤0, which shows thatS≤0 and achieves the proof.

3. The Proof of Uniqueness

Letϕ0andϕ1be two smoothk-admissible solutions of E_k

such that

Mϕ0ω^m

Mϕ1ω^m 0. For allt∈0,1, let us consider the functionϕ_ttϕ₁ 1−tϕ0 ϕ₀tϕwithϕϕ₁−ϕ₀. LetP ∈ M, and let us denoteh^P_kt f_kδ^j_i g^jP∂_iϕ_tP. We haveh^P_k1−h^P_k0 0 which is equivalent to₁

0h^P_ktdt0. But

h^P_kt ^m

i, j1

_m

1

∂fk

∂B_i

δ_i^jg^jP∂_iϕtP g^jP

:α^t_ijP

∂_ijϕP.

3.1

Therefore we obtain

LϕP: ^m

i, j1

aijP∂_ijϕP 0 withaijP ₁

0

α^t_ijPdt. 3.2

We show easily that the matrixaijP_{1≤i, j≤m}is Hermitian9, page 53. Besides the function ϕis continuous on the compact manifoldMso it assumes its minimum at a pointm₀∈M, so that the complex Hessian matrix ofϕat the pointm0, namely,∂_ijϕm0_{1≤i, j≤2m}, is positive- semidefinite.

Lemma 3.1. For allt∈0,1,λg⁻¹gϕtm0∈Γk; namely, the functionsϕt_t∈0,1arek-admissible atm₀.

Proof. Let us denoteW:{t∈0,1/λg⁻¹g_ϕtm0∈Γk}. The setWis nonempty, it contains 0, and it is an open subset of0,1. Lettbe the largest number of0,1such that0, t⊂ W.

Let us suppose thatt < 1 and show that we get a contradiction. Let 1 ≤ q ≤ k, we have σ_qλg⁻¹g_ϕtm0−σ_qλg⁻¹g_ϕ0m0 h^m_q⁰t−h^m_q⁰0 _t

0h^m_q⁰sds. Let us prove that

h^m_q⁰s≥0 for alls∈0, t. Fixs∈0, t; the quantityh^m_q⁰sis intrinsic so it suffices to prove the assertion in a particular chart atm0. Now atm0in ag-unitarygϕs-adapted chart atm0

h^m_q⁰s ^m

i, j, 1

∂fq

∂B^j_i

δ^j_ig^jm0∂_iϕsm0 g^jm0∂_iϕm0

^m

i1

∂σ_q

∂λ_i

λ

g⁻¹g_ϕs m0 ∂_iiϕm0.

3.3

But λg⁻¹g_ϕsm0 ∈ Γk ⊂ Γq since s ∈ 0, t⊂ W, then ∂σq/∂λ_iλg⁻¹g_ϕsm0 > 0 for all 1 ≤ i ≤ m. Besides,∂_iiϕm0 ≥ 0 since the matrix ∂_ijϕm0_{1≤i, j≤m} is positive-semi- definite. Therefore, we infer thath^m_q⁰s≥0. Consequently, we obtain thatσ_qλg⁻¹g_ϕtm0≥ σ_qλg⁻¹g_ϕ0m0>0sinceϕ₀isk-admissible. This holds for all 1≤q≤k; we deduce then thatλg⁻¹gϕtm0∈ Γkwhich proves thatt ∈ W. This is a contradiction; we infer then that W 0,1.

We check easily that the Hermitian matrixaijm0_{1≤i, j≤m}is positive definite9, page 54and deduce then the following lemma since the mapP →a_ijP ₁

0_m

1∂fk/∂B_iδ^j_i g^jP∂_iϕ_tPg^jPdtis continuous on a neighbourhood ofm₀.

Lemma 3.2. There exists an open ballBm0 centered atm0 such that for allP ∈Bm0 the Hermitian matrixaijP_{1≤i, j≤m}is positive definite.

Consequently, the operatorLis elliptic on the open set B_m0. But the map ϕ isC^∞, assumes its minimum atm0∈Bm0, and satisfiesLϕ0; then by the Hopf maximum principle 12, we deduce thatϕP ϕm0for allP ∈Bm0. Let us denoteS:{P ∈M/ϕP ϕm0}.

This set is nonempty and it is a closed set. Let us prove thatSis an open set: letmbe a point ofS, soϕm ϕm0, then the mapϕassumes its minimum at the pointm. Therefore, by the same proof as for the pointm0, we infer that there exists an open ballBmcentered atmsuch that for allP ∈B_m ϕP ϕmso for allP ∈B_mϕP ϕm0thenB_m ⊂ S, which proves thatSis an open set. But the manifoldMis connected; thenS M, namely,ϕP ϕm0 for allP∈M. Besides

Mϕω^m0, therefore we deduce thatϕ≡0 onMnamely thatϕ1≡ϕ0

onM, which achieves the proof of uniqueness.

4. The Continuity Method

Let us consider the one parameter family ofEk,t,t∈0,1

Fk ϕ_t!

:F_k

δ^j_ig^j∂_iϕ_t

1≤i, j≤m

tfln

⎛

⎜⎝^m_k

Mω^m

Me^tfω^m

⎞

⎟⎠

At

. Ek,t

The functionϕ₀ ≡ 0 is ak-admissible solution of Ek,0:σ_kλδ^j_ig^j∂_iϕ_{01≤i, j≤m} ^mk and satisfies

Mϕ₀ω^m0. Fort1,A₁1 soEk,1corresponds to E_k

.

Let us fixl ∈N,l≥5 and 0< α <1, and let us consider the nonempty setcontaining 0:

Tl, α:

t∈0,1/Ek,thave ak-admissible solutionϕ∈C^{l, α}M such that

M

ϕω^m0

"

.

4.1

The aim is to prove that 1 ∈ Tl, α. For this we prove, using the connectedness of0,1, that Tl, α 0,1.

4.1.Tl, α Is an Open Set of0,1

This arises from the local inverse mapping theorem and from solving a linear problem. Let us consider the following sets:

Sl, α:

ϕ∈C^{l, α}M,

M

ϕω^m0

"

, Sl, α:#

ϕ∈Sl, α, k-admissible forg$ ,

4.2

whereSl, αis a vector space andSl, αis an open set ofSl, α. Using these notations, the setTl, α

writesTl, α:{t∈0,1/∃ϕ∈S_{l, α}solution ofEk,t}.

Lemma 4.1. The operatorFk : Sl,α → C^{l−2, α}M, ϕ → Fkϕ Fkδ_i^jg^j∂_iϕ_{1≤i, j≤m}, is differentiable, and its differential at a pointϕ∈S_{l, α},dF_kϕ ∈ LS_{l, α}, C^{l−2, α}Mis equal to

dFkϕ·ψ ^m

i, j1

∂F_k

∂B^j_i

δ^j_i g^j∂_iϕ g^j∂_iψ ∀ψ ∈S_l,α. 4.3

Proof. See9, page 60.

Proposition 4.2. The nonlinear operatorFkis elliptic onS_{l, α}.

Proof. Let us fix a functionϕ∈Sl, αand check that the nonlinear operatorFkis elliptic for this functionϕ. This goes back to show that the linearization atϕof the nonlinear operatorFkis elliptic. ByLemma 4.1, this linearization is the following linear operator:

dFkϕ·v ^m

i, o1

⎛

⎝^m

j1

∂F_k

∂B^j_i

δ^j_i g^{j o}∂ioϕ

1≤i, j≤m×g^{j o}

⎞

⎠∂i ov. 4.4

In order to prove that this linear operator is elliptic, it suffices to check the ellipticity in a particular chart, for example, at the center of ag-normalg_ϕ-adapted chart. At the center of such a chart,

dFkϕ·v ^m

i, o1

∂F_k

∂B_i^o

diagλ

g⁻¹g

∂_{i o}v^m

i1

σ_{k−1, i}λ g⁻¹g σkλ

g⁻¹g ∂_iiv. 4.5 But for alli∈ {1, . . . , m}we haveσ_k−1,iλg⁻¹g/σ kλg⁻¹g>0 onMsinceλg⁻¹g ∈Γk11, which proves that the linearization is elliptic and achieves the proof.

Let us denoteFkthe operator

Fk ϕ!

:f_k

δ_i^jg^j∂_iϕ

1≤i, j≤m

. 4.6

AsFk, the operatorFk:Sl, α → C^{l−2, α}Mis differentiable and elliptic onSl, αof differential dFkϕ·ψ ^m

i, j1

∂fk

∂B_i^j

δ^j_i g^j∂_iϕ g^j∂_iψ ∀ψ∈S_{l, α}. 4.7

Let us denoteaϕthe matrixδ^j_i g^j∂_iϕ_{1≤i, j≤m}and calculate this linearization in a different way, by using the expression2.1offk:

Fk ϕ! f_k

a_ϕ 1

k!

1≤i1,..., ik, j1,..., jk≤m

εⁱ_j^1···ik

1···jk

a_ϕj1

i1· · · a_ϕjk

ik. 4.8

Thus

dFkϕ·v d dt

Fk ϕtv!

|t0

d dt

⎛

⎝1 k!

1≤i1,..., ik, j1,..., jk≤m

ε_j^i1···ik

1···jk

a_ϕtvj1

i1· · · a_ϕtvjk

ik

⎞

⎠

|t0

1 k!

1≤i1,..., ik, j1,..., jk≤m

εⁱ_j^1···ik

1...jk

g^j1s∂i1sv aϕ

_j2

i2· · · aϕ

_jk

ik

1 k!

1≤i1,..., ik, j1,..., jk≤m

εⁱ_j^1···ik

1···jk

a_ϕj1

i1

g^j2s∂_i2sv · · · a_ϕjk

ik

· · · 1 k!

1≤i1,..., ik, j1,..., jk≤m

εⁱ_j^1···ik

1···jk

aϕ

_j1

i1· · · aϕ

_jk−1

ik−1

g^jks∂iksv

1 k−1!

1≤i1,..., ik, j1,..., jk≤m

εⁱ_j^1···ik

1···jk

a_ϕj1

i1· · · a_ϕjk−1

ik−1

g^jks∂_iksv

by symmetry

^m

i, j1

⎛

⎝ 1 k−1!

1≤i1,..., ik−1, j1,..., jk−1≤m

ε_j^{i1···ik−1i}

1···jk−1j

a_ϕj1

i1· · · a_ϕj_k−1

i_k−1

⎞

⎠

:Cⁱ_j^aϕ

∇^j_iv.

4.9

We infer then the following proposition.

Proposition 4.3. The linearizationdFkof the operatorFkis of divergence type:

dFkϕ∇i

Cⁱ_j a_ϕ

∇^j . 4.10

Proof. By4.9we have dFkϕ·v ^m

i, j1

Cⁱ_j a_ϕ

∇^j_iv

^m

i1

∇i

⎛

⎝^m

j1

Cⁱ_j aϕ

∇^jv

⎞

⎠−^m

j1

_m

i1

∇i

Cⁱ_j aϕ

∇^jv.

4.11

Moreover m

i1

∇i

Cⁱ_j

a_ϕ 1

k−2!

m i1

1≤i1,..., ik−1, j1,..., jk−1≤m

εⁱ_j^{1···ik−1i}

1···jk−1j

a_ϕj1

i1· · · a_ϕjk−2

ik−2∇i a_ϕjk−1

ik−1 . 4.12

But∇iaϕ^j_ik−1^k−1 ∇iδ^j_ik−1^k−1∇^j_ik−1^k−1ϕ ∇^j_ii^k−1_k−1ϕ, then m

i1

∇i

Cⁱ_j

a_ϕ 1

k−2!

m i1

1≤i1,..., ik−1, j1,..., jk−1≤m

ε_j^{i1···ik−1i}

1···jk−1j

a_ϕj1

i1· · · a_ϕj_k−2

i_k−2∇^j_ii^k−1_k−1ϕ. 4.13

Besides, the quantity∇^j_ii^k−1_k−1ϕis symmetric ini, i_k−1indeed,∇^j_ii^k−1_k−1ϕ− ∇^j_i^k−1_k−1iϕ R^j_sii^k−1

k−1∇^sϕand R^j_sii^k−1

k−1 0 since g is K¨ahler, and εⁱ_j^{1···ik−1i}

1···jk−1j is antisymmetric in i, i_k−1; it follows then that _m

i1∇iCⁱ_jaϕ 0, consequentlydFkϕ·v_m

i1∇i_m

j1Cⁱ_jaϕ∇^jv.

FromProposition 4.3, we infer easily9, page 62the following corollary.

Corollary 4.4. The mapF:S_l,α → S_{l−2, α}, ϕ→Fϕ Fkϕ−^m_kis well defined and differentiable and its differential equalsdFϕ dFkϕ∇iCⁱ_jaϕ∇^j∈ LSl, α,S_{l−2, α}.

Now, let t0 ∈ Tl, α and let ϕ0 ∈ Sl, α be a solution of the corresponding equation Ek,t0:Fϕ0 e^t0fA_t0−^m_k.

Lemma 4.5. dFϕ0 :Sl, α → S_{l−2, α}is an isomorphism.

Proof. Letψ ∈C^l−2,αMwith

Mψv_g0. Let us consider the equation

∇i

Cⁱ_j aϕ0

∇^ju ψ. 4.14

We have Cⁱ_jaϕ0∈C^l−2,αM and the matrix Cⁱ_jaϕ0_{1≤i, j≤m} ∂fk/∂B_i^jδ_i^j g^j∂_iϕ_{01≤i, j≤m}is positive definitesince Fk is elliptic atϕ₀; then by Theorem 4.7 of13, p. 104on the operators of divergence type, we deduce that there exists a unique function u∈C^{l, α}Msatisfying

Muvg0 which is solution of4.14and then solution ofdFϕ0uψ.

Thus, the linear continuous mapdF_ϕ0 :S_{l, α} → S_{l−2, α}is bijective, and its inverse is continuous by the open map theorem, which achieves the proof.

We deduce then by the local inverse mapping theorem that there exists an open set U of Sl, α containing ϕ0 and an open set V of S_{l−2, α} containing Fϕ0 such that F : U → V is a diffeomorphism. Now, let us consider a real number t ∈ 0,1 very close to t₀ and let us check that it belongs also to Tl, α: if |t−t₀| ≤ ε is sufficiently small then

e^tfA_t−^m_k−e^t0fA_t0−^m_kCl−2, αMis small enough so thate^tfA_t−^m_k ∈V, thus there

existsϕ∈U⊂S_l,αsuch thatFϕ e^tfA_t−^m_kand consequently there existsϕ∈C^{l, α}Mof vanishing integral forgwhich is solution ofEk,t. Hencet∈ Tl, α. We conclude therefore that Tl, αis an open set of0,1.

4.2.Tl,α Is a Closed Set of0,1: The Scheme of the Proof

This section is based on a priori estimates. Finding these estimates is the most difficult step of the proof. Letts_s∈Nbe a sequence of elements ofTl, αthat converges toτ ∈0,1, and let ϕts_s∈Nbe the corresponding sequence of functions:ϕtsisC^{l, α},k-admissible, has a vanishing integral, and is a solution of

Fk

δ^j_i g^j∂_iϕts

1≤i, j≤m

tsflnAts. Ek,ts

Let us prove thatτ∈ Tl, α. Here is the scheme of the proof.

1Reduction to aC^{2, β}Mestimate: ifϕts_s∈Nis bounded in aC^{2, β}Mwith 0< β <1, the inclusionC^{2, β}M ⊂C²M,Rbeing compact, we deduce that after extraction ϕts_s∈Nconverges inC²M,Rtoϕτ ∈C²M,R. We show by tending to the limit thatϕ_τis a solution ofEk,τ it is then necessarilyk-admissibleand of vanishing integral forg. We check finally by a nonlinear regularity theorem14, page 467that ϕτ ∈C^∞M,R, which allows us to deduce thatτ ∈ Tl, αsee9, pages 64–67for details.

2We show that ϕts_s∈N is bounded inC⁰M,R: first of all we prove a positivity Lemma 5.4forEk,t, inspired by the ones of15, page 843 fork m, but in a very different way, required since thek-positivity ofω_ts is weaker withk < min this case, some eigenvalues can be nonpositive, which complicates the proof, using a polarization method of7, page 1740 cf. 5.2and a G˚arding inequality 5.3; we