# Complex Hessian Equations on Some Compact Kähler Manifolds

Authors: Asma Jbilou
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## Abstract

On a compact connected2m-dimensional Kähler manifold with Kähler form ω, given a smooth function f:M→ℝ and an integer 1<k<m, we want to solve uniquely in [ω] the equation ω̃k∧ωm-k=efωm, relying on the notion of k-positivity for ω̃∈[ω] (the extreme cases are solved: k=m by (Yau in 1978), and k=1 trivially). We solve by the continuity method the corresponding complex elliptic kth Hessian equation, more difficult to solve than the Calabi-Yau equation (k=m), under the assumption that the holomorphic bisectional curvature of the manifold is nonnegative, required here only to derive an a priori eigenvalues pinching.


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## Body

## 1. The Theorem

All manifolds considered in this paper areconnected.Let(M,J,g,ω) be a compact connected Kähler manifold of complex dimension m≥3. Fix an integer 2≤k≤m-1. Let φ:M→ℝ be a smooth function, and let us consider the (1,1)-form ω~=ω+i∂∂-φ and the associated 2-tensor g~ defined by g~(X,Y)=ω~(X,JY). Consider the sesquilinear forms h and h~ on T1,0 defined by h(U,V)=g(U,V¯) and h~(U,V)=g~(U,V¯). We denote by λ(g-1g~) the eigenvalues of h~ with respect to the Hermitian form h. By definition, these are the eigenvalues of the unique endomorphism A of T1,0 satisfying
(1.1)h~(U,V)=h(U,AV)∀U,V∈T1,0.
Calculations infer that the endomorphism A writes
(1.2)A:T1,0→T1,0,Ui∂i↦AijUi∂j=gjℓ-g~iℓ-Ui∂j.A is a self-adjoint/Hermitian endomorphism of the Hermitian space (T1,0,h), therefore λ(g-1g~)∈ℝm. Let us consider the following cone: Γk={λ∈ℝm/∀1≤j≤k,σj(λ)>0}, where σj denotes the jth elementary symmetric function.Definition 1.1.
φis said to be k-admissible if and only if λ(g-1g~)∈Γk.In this paper, we prove the following theorem.Theorem 1.2 (theσk equation).
Let(M,J,g,ω) be a compact connected Kähler manifold of complex dimension m≥3 with nonnegative holomorphic bisectional curvature, and let f:M→ℝ be a function of class C∞ satisfying ∫Mefωm=(mk)∫Mωm. There exists a unique function φ:M→ℝ of class C∞ such that
(1.3)(1)∫Mφωm=0,(Ek)(2)ω~k∧ωm-k=(ef(mk))ωm.
Moreover the solutionφ is k-admissible.This result was announced in a note in the Comptes Rendus de l'Acadé-mie des Sciences de Paris published online in December 2009 [1]. The curvature assumption is used, in Section 6.2 only, for an a priori estimate on λ(g-1g~) as in [2, page 408], and it should be removed (as did Aubin for the case k=m in [3], see also [4] for this case). For the analogue of (Ek) on ℂm, the Dirichlet problem is solved in [5, 6], and a Bedford-Taylor type theory, for weak solutions of the corresponding degenerate equations, is addressed in [7]. Thanks to Julien Keller, we learned of an independent work [8] aiming at the same result as ours, with a different gradient estimate and a similar method to estimate λ(g-1g~), but no proofs given for the C0 and the C2 estimates.Let us notice that the functionf appearing in the second member of (Ek) satisfies necessarily the normalisation condition ∫M‍efωm=(mk)∫M‍ωm. Indeed, this results from the following lemma.Lemma 1.3.
Consider∫M‍ω~k∧ωm-k=∫M‍ωm.Proof.
See [9, page 44].Let us write(Ek) differently.Lemma 1.4.
Considerω~k∧ωm-k=(σk(λ(g-1g~))/(mk))ωm.Proof.
LetP∈M. It suffices to prove the equality at P in a g-normal g~-adapted chart z centered at P. In such a chart gij-(0)=δij and g~ij-(0)=δijλi(0), so at z=0, ω=idza∧dza- and ω~=iλa(0)dza∧dza-. Thus
(1.4)ω~k∧ωm-k=(∑aiλa(0)dza∧dza-)k∧(∑bidzb∧dzb-)m-k=∑(a1,…,ak)∈{1,…,m}distinct  integers(b1,…,bm-k)∈{1,…,m}∖{a1,…,ak}distinct  integersimλa1(0)⋯λak(0)(dza1∧dza-1)∧⋯∧(dzak∧dza-k)∧(dzb1∧dzb-1)∧⋯∧(dzbm-k∧dzb¯m-k).
Now a1,…,ak,b1,…,bm-k are m distinct integers of {1,…,m} and 2-forms commute therefore,
(1.5)ω~k∧ωm-k=(∑(a1,…,ak)∈{1,…,m}distinct  integers(b1,…,bm-k)∈{1,…,m}∖{a1,…,ak}distinct  integersλa1(0)⋯λak(0))im(dz1∧dz1-)∧⋯∧(dzm∧dzm-)︸=ωmm!=(∑(a1,…,ak)∈{1,…,m}distinct  integers(m-k)!λa1(0)⋯λak(0))ωmm!ω~k∧ωm-k=(m-k)!m!k!σk(λ1(0),…,λm(0))ωm=σk(λ(g-1g~))(mk)ωm.Consequently,(Ek) writes:
(Ek′)σk(λ(g-1g~))=ef.
Let us remark that Em corresponds to the Calabi-Yau equation det(g~)/det(g)=ef, when E1 is just a linear equation in Laplacian form. Since the endomorphism A is Hermitian, the spectral theorem provides an h-orthonormal basis for T1,0 of eigenvectors e1,…,em: Aei=λiei, λ=(λ1,…,λm)∈Γk. At P∈M in a chart z, we have Mat∂1,…,∂mAP=[Aji(z)]1≤i,j≤m, thus σk(λ(AP))=σk(λ([Aji(z)]1≤i,j≤m)). In addition, Aij=gjℓ-g~iℓ-=gjℓ-(giℓ-+∂iℓ-φ)=δij+gjℓ-∂iℓ-φ, so the equation writes locally:
(Ek′′)σk(λ([δij+gjℓ-∂iℓ-φ]1≤i,j≤m))=ef.
Let us notice that a solution of this equation (Ek′′) is necessarily k-admissible [9, page 46]. Let us define fk(B)=σk(λ(B)) and Fk(B)=lnσk(λ(B)) where B=[Bij]1≤i,j≤m is a Hermitian matrix. The function fk is a polynomial in the variables Bij, specifically fk(B)=∑|I|=k‍BII (sum of the principal minors of order k of the matrix B). Equivalently (Ek′′) writes:
(Ek′′′)Fk([δij+gjℓ-∂iℓ-φ]1≤i,j≤m)=f.
It is a nonlinear elliptic second order PDE of complex Monge-Ampère type. We prove the existence of a k-admissible solution by the continuity method.


## 2. Derivatives and Concavity ofFk

### 2.1. Calculation of the Derivatives at a Diagonal Matrix

The first derivatives of the symmetric polynomialσk are given by the following: for all 1≤i≤m, (∂σk/∂λi)(λ)=σk-1,i(λ) where σk-1,i(λ):=σk-1|λi=0. For 1≤i≠j≤m, let us denote σk-2,ij(λ):=σk-2|λi=λj=0 and σk-2,ii(λ)=0. The second derivatives of the polynomial σk are given by (∂2σk/∂λi∂λj)(λ)=σk-2,ij(λ). We calculate the derivatives of the function fk:ℋm(ℂ)→ℝ, where ℋm(ℂ) denotes the set of Hermitian matrices, at diagonal matrices using the formula:
(2.1)fk(B)=∑1≤i1<⋯<ik≤m∑σ∈Skε(σ)Bi1iσ(1)⋯Bikiσ(k)=1k!∑1≤i1,…,ik,j1,…,jk≤mεj1⋯jki1⋯ikBi1j1⋯Bikjk,
where
(2.2)εj1⋯jki1⋯ik={1ifi1,…,ikdistinctandj1,…,jkevenpermutationofi1,…,ik,-1ifi1,…,ikdistinctandj1,…,jkoddpermutationofi1,…,ik,0else.
These derivatives are given by [9, page 48]
(2.3)∂fk∂Bij(diag(b1,…,bm))={0ifi≠j,σk-1,i(b1,…,bm)ifi=j,ifi≠j∂2fk∂Bjj∂Bii(diag(b1,…,bm))=σk-2,ij(b1,…,bm)∂2fk∂Bji∂Bij(diag(b1,…,bm))=-σk-2,ij(b1,…,bm),
and all the other second derivatives of fk at diag(b1,…,bm) vanish.Consequently, the derivatives of the functionFk=lnfk:λ-1(Γk)⊂ℋm(ℂ)→ℝ at diagonal matrices diag(λ1,…,λm) with λ=(λ1,…,λm)∈Γk, where λ-1(Γk)={B∈ℋm(ℂ)/λ(B)∈Γk}, are given by
(2.4)∂Fk∂Bij(diag(λ1,…,λm))={0ifi≠j,σk-1,i(λ)σk(λ)ifi=j,(2.5)ifi≠j∂2Fk∂Bji∂Bij(diag(λ1,…,λm))=-σk-2,ij(λ)σk(λ)∂2Fk∂Bjj∂Bii(diag(λ1,…,λm))=σk-2,ij(λ)σk(λ)-σk-1,i(λ)σk-1,j(λ)(σk(λ))2∂2Fk∂Bii∂Bii(diag(λ1,…,λm))=-(σk-1,i(λ))2(σk(λ))2
and all the other second derivatives of Fk at diag(λ1,…,λm) vanish.


### 2.2. The Invariance ofFk and of Its First and Second Differentials

The functionFk:λ-1(Γk)→ℝ is invariant under unitary similitudes:
(2.6)∀B∈λ-1(Γk),∀U∈Um(ℂ),Fk(B)=Fk(U¯tBU).
Differentiating the previous invariance formula (2.6), we show that the first and second differentials of Fk are also invariant under unitary similitudes:
(2.7)∀B∈λ-1(Γk),∀ζ∈ℋm(ℂ),∀U∈Um(ℂ),(dFk)B·ζ=(dFk)U¯tBU·(U¯tζU),(2.8)∀B∈λ-1(Γk),∀ζ∈ℋm(ℂ),∀Θ∈ℋm(ℂ),∀U∈Um(ℂ),(d2Fk)B·(ζ,Θ)=(d2Fk)U¯tBU·(U¯tζU,U¯tΘU).
These invariance formulas are allowed to come down to the diagonal case, when it is useful.


### 2.3. Concavity ofFk

We prove in [9] the concavity of the functions u∘λ and more generally u∘λB when u∈Γ0(ℝm) and is symmetric [9, Theorem  VII.4.2], which in particular gives the concavity of the functions Fk=lnσkλ [9, Corollary VII.4.30] and more generally lnσkλB [9, Theorem VII.4.29]. In this section, let us show by an elementary calculation the concavity of the function Fk.Proposition 2.1.
The functionFk:λ-1(Γk)→ℝ,B↦Fk(B)=lnσk(λ(B)) is concave (this holds for all k∈{1,…,m}).Proof.
The functionFk is of class C2, so its concavity is equivalent to the following inequality:
(2.9)∀B∈λ-1(Γk),∀ζ∈ℋm(ℂ)∑i,j,r,s=1m‍∂2Fk∂Brs∂Bij(B)ζijζrs≤0.
Let B∈λ-1(Γk), ζ∈ℋm(ℂ), and U∈Um(ℂ) such that U¯tBU=diag(λ1,…,λm). We have λ=(λ1,…,λm)∈Γk. Let us denote ζ~=U¯tζU∈ℋm(ℂ):
(2.10)S:=∑i,j,r,s=1m‍∂2Fk∂Brs∂Bij(B)ζijζrs=(d2Fk)B·(ζ,ζ)sobytheinvarianceformula(2.8)=(d2Fk)U¯tBU·(U¯tζU,U¯tζU)=∑i,j,r,s=1m‍∂2Fk∂Brs∂Bij(diag(λ1,…,λm))ζ~ijζ~rs=∑i≠j=1m‍-σk-2,ij(λ)σk(λ)ζ~ijζ~ji︸=ζ~ij¯+∑i≠j=1m‍(σk-2,ij(λ)σk(λ)-σk-1,i(λ)σk-1,j(λ)(σk(λ))2)︸=:cijζ~iiζ~jj+∑i=1m‍-(σk-1,i(λ))2(σk(λ))2(ζ~ii)2=∑i,j=1m‍-σk-2,ij(λ)σk(λ)|ζ~ij|2+∑i,j=1m‍cijζ~iiζ~jj.
But cij=(∂2(lnσk)/∂λi∂λj)(λ), and ζ~ii∈ℝ, so ∑i,j=1m‍cijζ~iiζ~jj≤0 by concavity of lnσk at λ∈Γk [10, page 269]. In addition, σk-2,ij(λ)>0 since λ∈Γk [11], consequently ∑i,j=1m‍-(σk-2,ij(λ)/σk(λ))|ζ~ij|2≤0, which shows that S≤0 and achieves the proof.


## 2.1. Calculation of the Derivatives at a Diagonal Matrix

The first derivatives of the symmetric polynomialσk are given by the following: for all 1≤i≤m, (∂σk/∂λi)(λ)=σk-1,i(λ) where σk-1,i(λ):=σk-1|λi=0. For 1≤i≠j≤m, let us denote σk-2,ij(λ):=σk-2|λi=λj=0 and σk-2,ii(λ)=0. The second derivatives of the polynomial σk are given by (∂2σk/∂λi∂λj)(λ)=σk-2,ij(λ). We calculate the derivatives of the function fk:ℋm(ℂ)→ℝ, where ℋm(ℂ) denotes the set of Hermitian matrices, at diagonal matrices using the formula:
(2.1)fk(B)=∑1≤i1<⋯<ik≤m∑σ∈Skε(σ)Bi1iσ(1)⋯Bikiσ(k)=1k!∑1≤i1,…,ik,j1,…,jk≤mεj1⋯jki1⋯ikBi1j1⋯Bikjk,
where
(2.2)εj1⋯jki1⋯ik={1ifi1,…,ikdistinctandj1,…,jkevenpermutationofi1,…,ik,-1ifi1,…,ikdistinctandj1,…,jkoddpermutationofi1,…,ik,0else.
These derivatives are given by [9, page 48]
(2.3)∂fk∂Bij(diag(b1,…,bm))={0ifi≠j,σk-1,i(b1,…,bm)ifi=j,ifi≠j∂2fk∂Bjj∂Bii(diag(b1,…,bm))=σk-2,ij(b1,…,bm)∂2fk∂Bji∂Bij(diag(b1,…,bm))=-σk-2,ij(b1,…,bm),
and all the other second derivatives of fk at diag(b1,…,bm) vanish.Consequently, the derivatives of the functionFk=lnfk:λ-1(Γk)⊂ℋm(ℂ)→ℝ at diagonal matrices diag(λ1,…,λm) with λ=(λ1,…,λm)∈Γk, where λ-1(Γk)={B∈ℋm(ℂ)/λ(B)∈Γk}, are given by
(2.4)∂Fk∂Bij(diag(λ1,…,λm))={0ifi≠j,σk-1,i(λ)σk(λ)ifi=j,(2.5)ifi≠j∂2Fk∂Bji∂Bij(diag(λ1,…,λm))=-σk-2,ij(λ)σk(λ)∂2Fk∂Bjj∂Bii(diag(λ1,…,λm))=σk-2,ij(λ)σk(λ)-σk-1,i(λ)σk-1,j(λ)(σk(λ))2∂2Fk∂Bii∂Bii(diag(λ1,…,λm))=-(σk-1,i(λ))2(σk(λ))2
and all the other second derivatives of Fk at diag(λ1,…,λm) vanish.


## 2.2. The Invariance ofFk and of Its First and Second Differentials

The functionFk:λ-1(Γk)→ℝ is invariant under unitary similitudes:
(2.6)∀B∈λ-1(Γk),∀U∈Um(ℂ),Fk(B)=Fk(U¯tBU).
Differentiating the previous invariance formula (2.6), we show that the first and second differentials of Fk are also invariant under unitary similitudes:
(2.7)∀B∈λ-1(Γk),∀ζ∈ℋm(ℂ),∀U∈Um(ℂ),(dFk)B·ζ=(dFk)U¯tBU·(U¯tζU),(2.8)∀B∈λ-1(Γk),∀ζ∈ℋm(ℂ),∀Θ∈ℋm(ℂ),∀U∈Um(ℂ),(d2Fk)B·(ζ,Θ)=(d2Fk)U¯tBU·(U¯tζU,U¯tΘU).
These invariance formulas are allowed to come down to the diagonal case, when it is useful.


## 2.3. Concavity ofFk

We prove in [9] the concavity of the functions u∘λ and more generally u∘λB when u∈Γ0(ℝm) and is symmetric [9, Theorem  VII.4.2], which in particular gives the concavity of the functions Fk=lnσkλ [9, Corollary VII.4.30] and more generally lnσkλB [9, Theorem VII.4.29]. In this section, let us show by an elementary calculation the concavity of the function Fk.Proposition 2.1.
The functionFk:λ-1(Γk)→ℝ,B↦Fk(B)=lnσk(λ(B)) is concave (this holds for all k∈{1,…,m}).Proof.
The functionFk is of class C2, so its concavity is equivalent to the following inequality:
(2.9)∀B∈λ-1(Γk),∀ζ∈ℋm(ℂ)∑i,j,r,s=1m‍∂2Fk∂Brs∂Bij(B)ζijζrs≤0.
Let B∈λ-1(Γk), ζ∈ℋm(ℂ), and U∈Um(ℂ) such that U¯tBU=diag(λ1,…,λm). We have λ=(λ1,…,λm)∈Γk. Let us denote ζ~=U¯tζU∈ℋm(ℂ):
(2.10)S:=∑i,j,r,s=1m‍∂2Fk∂Brs∂Bij(B)ζijζrs=(d2Fk)B·(ζ,ζ)sobytheinvarianceformula(2.8)=(d2Fk)U¯tBU·(U¯tζU,U¯tζU)=∑i,j,r,s=1m‍∂2Fk∂Brs∂Bij(diag(λ1,…,λm))ζ~ijζ~rs=∑i≠j=1m‍-σk-2,ij(λ)σk(λ)ζ~ijζ~ji︸=ζ~ij¯+∑i≠j=1m‍(σk-2,ij(λ)σk(λ)-σk-1,i(λ)σk-1,j(λ)(σk(λ))2)︸=:cijζ~iiζ~jj+∑i=1m‍-(σk-1,i(λ))2(σk(λ))2(ζ~ii)2=∑i,j=1m‍-σk-2,ij(λ)σk(λ)|ζ~ij|2+∑i,j=1m‍cijζ~iiζ~jj.
But cij=(∂2(lnσk)/∂λi∂λj)(λ), and ζ~ii∈ℝ, so ∑i,j=1m‍cijζ~iiζ~jj≤0 by concavity of lnσk at λ∈Γk [10, page 269]. In addition, σk-2,ij(λ)>0 since λ∈Γk [11], consequently ∑i,j=1m‍-(σk-2,ij(λ)/σk(λ))|ζ~ij|2≤0, which shows that S≤0 and achieves the proof.


## 3. The Proof of Uniqueness

Letφ0 and φ1 be two smooth k-admissible solutions of (Ek′′′) such that ∫Mφ0ωm=∫Mφ1ωm=0. For all t∈[0,1], let us consider the function φt=tφ1+(1-t)φ0=φ0+tφ with φ=φ1-φ0. Let P∈M, and let us denote hkP(t)=fk([δij+gjℓ-(P)∂iℓ-φt(P)]). We have hkP(1)-hkP(0)=0 which is equivalent to ∫01hkP′(t)dt=0. But
(3.1)hkP′(t)=∑i,j=1m‍(∑ℓ=1m‍∂fk∂Biℓ([δij+gjℓ-(P)∂iℓ-φt(P)])gℓj-(P))︸=:αijt(P)∂ij-φ(P).
Therefore we obtain
(3.2)ℒφ(P):=∑i,j=1m‍aij(P)∂ij-φ(P)=0withaij(P)=∫01αijt(P)dt.
We show easily that the matrix [aij(P)]1≤i,j≤m is Hermitian [9, page 53]. Besides the function φ is continuous on the compact manifold M so it assumes its minimum at a point m0∈M, so that the complex Hessian matrix of φ at the point m0, namely, [∂ij-φ(m0)]1≤i,j≤2m, is positive-semidefinite.Lemma 3.1.
For allt∈[0,1], λ(g-1g~φt)(m0)∈Γk; namely, the functions (φt)t∈[0,1] are k-admissible at m0.Proof.
Let us denote𝒲:={t∈[0,1]/λ(g-1g~φt)(m0)∈Γk}. The set 𝒲 is nonempty, it contains 0, and it is an open subset of [0,1]. Let t be the largest number of [0,1] such that [0,t[⊂𝒲. Let us suppose that t<1 and show that we get a contradiction. Let 1≤q≤k, we have σq(λ(g-1g~φt)(m0))-σq(λ(g-1g~φ0)(m0))=hqm0(t)-hqm0(0)=∫0thqm0′(s)ds. Let us prove that hqm0′(s)≥0 for all s∈[0,t[. Fix s∈[0,t[; the quantity hqm0′(s) is intrinsic so it suffices to prove the assertion in a particular chart at m0. Now at m0 in a g-unitary g~φs-adapted chart at m0(3.3)hqm0′(s)=∑i,j,ℓ=1m‍∂fq∂Bij([δij+gjℓ-(m0)∂iℓ-φs(m0)])gjℓ-(m0)∂iℓ-φ(m0)=∑i=1m‍∂σq∂λi(λ(g-1g~φs)(m0))∂ii-φ(m0).
But λ(g-1g~φs)(m0)∈Γk⊂Γq since s∈[0,t[⊂𝒲, then (∂σq/∂λi)(λ(g-1g~φs)(m0))>0 for all 1≤i≤m. Besides, ∂ii-φ(m0)≥0 since the matrix [∂ij-φ(m0)]1≤i,j≤m is positive-semidefinite. Therefore, we infer that hqm0′(s)≥0. Consequently, we obtain that σq(λ(g-1g~φt)(m0))≥σq(λ(g-1g~φ0)(m0))>0 (since φ0 is k-admissible). This holds for all 1≤q≤k; we deduce then that λ(g-1g~φt)(m0)∈Γk which proves that t∈𝒲. This is a contradiction; we infer then that 𝒲=[0,1].We check easily that the Hermitian matrix[aij(m0)]1≤i,j≤m is positive definite [9, page 54] and deduce then the following lemma since the map P↦aij(P)=∫01‍(∑ℓ=1m(∂fk/∂Biℓ)([δij+gjℓ-(P)∂iℓ-φt(P)])gℓj-(P))dt is continuous on a neighbourhood of m0.Lemma 3.2.
There exists an open ballBm0 centered at m0 such that for all P∈Bm0 the Hermitian matrix [aij(P)]1≤i,j≤m is positive definite.Consequently, the operatorℒ is elliptic on the open set Bm0. But the map φ is C∞, assumes its minimum at m0∈Bm0, and satisfies ℒφ=0; then by the Hopf maximum principle [12], we deduce that φ(P)=φ(m0) for all P∈Bm0. Let us denote 𝒮:={P∈M/φ(P)=φ(m0)}. This set is nonempty and it is a closed set. Let us prove that 𝒮 is an open set: let m be a point of 𝒮, so φ(m)=φ(m0), then the map φ assumes its minimum at the point m. Therefore, by the same proof as for the point m0, we infer that there exists an open ball Bm centered at m such that for all P∈Bmφ(P)=φ(m) so for all P∈Bmφ(P)=φ(m0) then Bm⊂𝒮, which proves that 𝒮 is an open set. But the manifold M is connected; then 𝒮=M, namely, φ(P)=φ(m0) for all P∈M. Besides ∫M‍φωm=0, therefore we deduce that φ≡0 on M namely that φ1≡φ0 on M, which achieves the proof of uniqueness.


## 4. The Continuity Method

Let us consider the one parameter family of(Ek,t), t∈[0,1](Ek,t)ℱk[φt]:=Fk([δij+gjℓ-∂iℓ-φt]1≤i,j≤m)=tf+ln((mk)∫M‍ωm∫M‍etfωm︸)At.
The function φ0≡0 is a k-admissible solution of (Ek,0): σk(λ([δij+gjℓ-∂iℓ-φ0]1≤i,j≤m))=(mk) and satisfies ∫Mφ0ωm=0. For t=1, A1=1 so (Ek,1) corresponds to (Ek′′′).Let us fixl∈ℕ, l≥5 and 0<α<1, and let us consider the nonempty set (containing 0):
(4.1)𝒯l,α≔{∫Mφωm=0t∈[0,1]/(Ek,t)have  ak-admissible  solutionφ∈Cl,α(M)such  that∫Mφωm=0}.
The aim is to prove that 1∈𝒯l,α. For this we prove, using the connectedness of [0,1], that 𝒯l,α=[0,1].


### 4.1.𝒯l,α Is an Open Set of [0,1]

This arises from the local inverse mapping theorem and from solving a linear problem. Let us consider the following sets:(4.2)S~l,α:={φ∈Cl,α(M),∫Mφωm=0},Sl,α:={φ∈S~l,α,k-admissible  forg},
where S~l,α is a vector space and Sl,α is an open set of S~l,α. Using these notations, the set 𝒯l,α writes 𝒯l,α:={t∈[0,1]/∃φ∈Sl,α solution of (Ek,t)}.Lemma 4.1.
The operatorℱk:Sl,α→Cl-2,α(M), φ↦ℱk[φ]=Fk([δij+gjℓ-∂iℓ-φ]1≤i,j≤m), is differentiable, and its differential at a point φ∈Sl,α, dℱkφ∈ℒ(S~l,α,Cl-2,α(M)) is equal to
(4.3)dℱkφ·ψ=∑i,j=1m‍∂Fk∂Bij([δij+gjℓ-∂iℓ-φ])gjℓ-∂iℓ-ψ∀ψ∈S~l,α.Proof.
See [9, page 60].Proposition 4.2.
The nonlinear operatorℱk is elliptic on Sl,α.Proof.
Let us fix a functionφ∈Sl,α and check that the nonlinear operator ℱk is elliptic for this function φ. This goes back to show that the linearization at φ of the nonlinear operator ℱk is elliptic. By Lemma 4.1, this linearization is the following linear operator:
(4.4)dℱkφ·v=∑i,o=1m‍(∑j=1m‍∂Fk∂Bij[δij+gjo-∂io-φ]1≤i,j≤m×gjo-)∂io-v.
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 a g-normal g~φ-adapted chart. At the center of such a chart,
(4.5)dℱkφ·v=∑i,o=1m‍(∂Fk∂Bio(diagλ(g-1g~)))∂io-v=∑i=1m‍σk-1,iλ(g-1g~)σkλ(g-1g~)∂ii-v.
But for all i∈{1,…,m} we have σk-1,iλ(g-1g~)/σkλ(g-1g~)>0 on M since λ(g-1g~)∈Γk [11], which proves that the linearization is elliptic and achieves the proof.Let us denote𝔉k the operator
(4.6)𝔉k[φ]:=fk([δij+gjℓ-∂iℓ-φ]1≤i,j≤m).
As ℱk, the operator 𝔉k:Sl,α→Cl-2,α(M) is differentiable and elliptic on Sl,α of differential
(4.7)d𝔉kφ·ψ=∑i,j=1m‍∂fk∂Bij([δij+gjℓ-∂iℓ-φ])gjℓ-∂iℓ-ψ∀ψ∈S~l,α.
Let us denote aφ the matrix [δij+gjℓ-∂iℓ-φ]1≤i,j≤m and calculate this linearization in a different way, by using the expression (2.1) of fk:
(4.8)𝔉k[φ]=fk(aφ)=1k!∑1≤i1,…,ik,j1,…,jk≤m‍εj1⋯jki1⋯ik(aφ)i1j1⋯(aφ)ikjk.
Thus
(4.9)d𝔉kφ·v=ddt(𝔉k[φ+tv])|t=0=ddt(1k!∑1≤i1,…,ik,j1,…,jk≤m‍εj1⋯jki1⋯ik(aφ+tv)i1j1⋯(aφ+tv)ikjk)|t=0=1k!∑1≤i1,…,ik,j1,…,jk≤m‍εj1...jki1⋯ik(gj1s-∂i1s-v)(aφ)i2j2⋯(aφ)ikjk+1k!∑1≤i1,…,ik,j1,…,jk≤m‍εj1⋯jki1⋯ik(aφ)i1j1(gj2s-∂i2s-v)⋯(aφ)ikjk+⋯+1k!∑1≤i1,…,ik,j1,…,jk≤m‍εj1⋯jki1⋯ik(aφ)i1j1⋯(aφ)ik-1jk-1(gjks-∂iks-v)=1(k-1)!∑1≤i1,…,ik,j1,…,jk≤m‍εj1⋯jki1⋯ik(aφ)i1j1⋯(aφ)ik-1jk-1(gjks-∂iks-v)by  symmetry=∑i,j=1m‍(1(k-1)!∑1≤i1,…,ik-1,j1,…,jk-1≤m‍εj1⋯jk-1ji1⋯ik-1i(aφ)i1j1⋯(aφ)ik-1jk-1)︸=:𝒞ji(aφ)∇ijv.
We infer then the following proposition.Proposition 4.3.
The linearizationd𝔉k of the operator 𝔉k is of divergence type:
(4.10)d𝔉kφ=∇i(𝒞ji(aφ)∇j).Proof.
By (4.9) we have
(4.11)d𝔉kφ·v=∑i,j=1m‍𝒞ji(aφ)∇ijv=∑i=1m‍∇i(∑j=1m‍𝒞ji(aφ)∇jv)-∑j=1m‍(∑i=1m‍∇i(𝒞ji(aφ)))∇jv.
Moreover
(4.12)∑i=1m‍∇i(𝒞ji(aφ))=1(k-2)!∑i=1m‍∑1≤i1,…,ik-1,j1,…,jk-1≤m‍εj1⋯jk-1ji1⋯ik-1i(aφ)i1j1⋯(aφ)ik-2jk-2∇i((aφ)ik-1jk-1).
But ∇i((aφ)ik-1jk-1)=∇i(δik-1jk-1+∇ik-1jk-1φ)=∇iik-1jk-1φ, then
(4.13)∑i=1m∇i(𝒞ji(aφ))=1(k-2)!∑i=1m‍∑1≤i1,…,ik-1,j1,…,jk-1≤m‍εj1⋯jk-1ji1⋯ik-1i(aφ)i1j1⋯(aφ)ik-2jk-2∇iik-1jk-1φ.
Besides, the quantity ∇iik-1jk-1φ is symmetric in i,ik-1 (indeed, ∇iik-1jk-1φ-∇ik-1ijk-1φ=Rsiik-1jk-1∇sφ and Rsiik-1jk-1=0 since g is Kähler), and εj1⋯jk-1ji1⋯ik-1i is antisymmetric in i,ik-1; it follows then that ∑i=1m‍∇i(𝒞ji(aφ))=0, consequently d𝔉kφ·v=∑i=1m‍∇i(∑j=1m‍𝒞ji(aφ)∇jv).From Proposition4.3, we infer easily [9, page 62] the following corollary.Corollary 4.4.
The mapF:Sl,α→S~l-2,α,φ↦F(φ)=𝔉k[φ]-(mk) is well defined and differentiable and its differential equals dFφ=d𝔉kφ=∇i(𝒞ji(aφ)∇j)∈ℒ(S~l,α,S~l-2,α).Now, lett0∈𝒯l,α and let φ0∈Sl,α be a solution of the corresponding equation (Ek,t0): F(φ0)=et0fAt0-(mk).Lemma 4.5.
d
F
φ
0
:
S
~
l
,
α
→
S
~
l
-
2
,
α is an isomorphism.Proof.
Letψ∈Cl-2,α(M) with ∫M‍ψvg=0. Let us consider the equation
(4.14)∇i(𝒞ji(aφ0)∇ju)=ψ.
We have 𝒞ji(aφ0)∈Cl-2,α(M) and the matrix [𝒞ji(aφ0)]1≤i,j≤m=[(∂fk/∂Bij)([δij+gjℓ-∂iℓ-φ0])]1≤i,j≤m is positive definite (since 𝔉k is elliptic at φ0); then by Theorem 4.7 of [13, p. 104] on the operators of divergence type, we deduce that there exists a unique function u∈Cl,α(M) satisfying ∫M‍uvg=0 which is solution of (4.14) and then solution of dFφ0u=ψ. Thus, the linear continuous map dFφ0: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 setU of Sl,α containing φ0 and an open set V of S~l-2,α containing F(φ0) 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 ∥(etfAt-(mk))-(et0fAt0-(mk))∥Cl-2,α(M) is small enough so that etfAt-(mk)∈V, thus there exists φ∈U⊂Sl,α such that F(φ)=etfAt-(mk) and consequently there exists φ∈Cl,α(M) of vanishing integral for g which is solution of (Ek,t). Hence t∈𝒯l,α. We conclude therefore that 𝒯l,α is an open set of [0,1].


### 4.2.𝒯l,α Is a Closed Set of [0,1]: The Scheme of the Proof

This section is based on a priori estimates. Finding these estimates is the most difficult step of the proof. Let(ts)s∈ℕ be a sequence of elements of 𝒯l,α that converges to τ∈[0,1], and let (φts)s∈ℕ be the corresponding sequence of functions: φts is Cl,α, k-admissible, has a vanishing integral, and is a solution of
(Ek,ts)Fk([δij+gjℓ-∂iℓ-φts]1≤i,j≤m)=tsf+ln(Ats).
Let us prove that τ∈𝒯l,α. Here is the scheme of the proof.(1)
Reduction to aC2,β(M) estimate: if (φts)s∈ℕ is bounded in a C2,β(M) with 0<β<1, the inclusion C2,β(M)⊂C2(M,ℝ) being compact, we deduce that after extraction (φts)s∈ℕ converges in C2(M,ℝ) to φτ∈C2(M,ℝ). We show by tending to the limit that φτ is a solution of (Ek,τ) (it is then necessarily k-admissible) and of vanishing integral for g. We check finally by a nonlinear regularity theorem [14, page 467] that φτ∈C∞(M,ℝ), which allows us to deduce that τ∈𝒯l,α (see [9, pages 64–67] for details).(2)
We show that(φts)s∈ℕ is bounded in C0(M,ℝ): first of all we prove a positivity Lemma 5.4 for (Ek,t), inspired by the ones of [15, page 843] (for k=m), but in a very different way, required since the k-positivity of ω~ts is weaker with k<m (in this case, some eigenvalues can be nonpositive, which complicates the proof), using a polarization method of [7, page 1740] (cf. 5.2) and a Gårding inequality 5.3; we infer then from this lemma a fundamental inequality 5.5 as Proposition 7.18 of [13, page 262]. We conclude the proof using the Moser's iteration technique exactly as for the equation of Calabi-Yau. We deal with this C0 estimate in Section 5.(3)
We establish the key point of the proof, namely, aC2 a priori estimate (Section 6).(4)
With the uniform ellipticity at hand (consequence of the previous step), we obtain the neededC2,β(M) estimate by the Evans-Trudinger theory (Section 7).


## 4.1.𝒯l,α Is an Open Set of [0,1]

This arises from the local inverse mapping theorem and from solving a linear problem. Let us consider the following sets:(4.2)S~l,α:={φ∈Cl,α(M),∫Mφωm=0},Sl,α:={φ∈S~l,α,k-admissible  forg},
where S~l,α is a vector space and Sl,α is an open set of S~l,α. Using these notations, the set 𝒯l,α writes 𝒯l,α:={t∈[0,1]/∃φ∈Sl,α solution of (Ek,t)}.Lemma 4.1.
The operatorℱk:Sl,α→Cl-2,α(M), φ↦ℱk[φ]=Fk([δij+gjℓ-∂iℓ-φ]1≤i,j≤m), is differentiable, and its differential at a point φ∈Sl,α, dℱkφ∈ℒ(S~l,α,Cl-2,α(M)) is equal to
(4.3)dℱkφ·ψ=∑i,j=1m‍∂Fk∂Bij([δij+gjℓ-∂iℓ-φ])gjℓ-∂iℓ-ψ∀ψ∈S~l,α.Proof.
See [9, page 60].Proposition 4.2.
The nonlinear operatorℱk is elliptic on Sl,α.Proof.
Let us fix a functionφ∈Sl,α and check that the nonlinear operator ℱk is elliptic for this function φ. This goes back to show that the linearization at φ of the nonlinear operator ℱk is elliptic. By Lemma 4.1, this linearization is the following linear operator:
(4.4)dℱkφ·v=∑i,o=1m‍(∑j=1m‍∂Fk∂Bij[δij+gjo-∂io-φ]1≤i,j≤m×gjo-)∂io-v.
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 a g-normal g~φ-adapted chart. At the center of such a chart,
(4.5)dℱkφ·v=∑i,o=1m‍(∂Fk∂Bio(diagλ(g-1g~)))∂io-v=∑i=1m‍σk-1,iλ(g-1g~)σkλ(g-1g~)∂ii-v.
But for all i∈{1,…,m} we have σk-1,iλ(g-1g~)/σkλ(g-1g~)>0 on M since λ(g-1g~)∈Γk [11], which proves that the linearization is elliptic and achieves the proof.Let us denote𝔉k the operator
(4.6)𝔉k[φ]:=fk([δij+gjℓ-∂iℓ-φ]1≤i,j≤m).
As ℱk, the operator 𝔉k:Sl,α→Cl-2,α(M) is differentiable and elliptic on Sl,α of differential
(4.7)d𝔉kφ·ψ=∑i,j=1m‍∂fk∂Bij([δij+gjℓ-∂iℓ-φ])gjℓ-∂iℓ-ψ∀ψ∈S~l,α.
Let us denote aφ the matrix [δij+gjℓ-∂iℓ-φ]1≤i,j≤m and calculate this linearization in a different way, by using the expression (2.1) of fk:
(4.8)𝔉k[φ]=fk(aφ)=1k!∑1≤i1,…,ik,j1,…,jk≤m‍εj1⋯jki1⋯ik(aφ)i1j1⋯(aφ)ikjk.
Thus
(4.9)d𝔉kφ·v=ddt(𝔉k[φ+tv])|t=0=ddt(1k!∑1≤i1,…,ik,j1,…,jk≤m‍εj1⋯jki1⋯ik(aφ+tv)i1j1⋯(aφ+tv)ikjk)|t=0=1k!∑1≤i1,…,ik,j1,…,jk≤m‍εj1...jki1⋯ik(gj1s-∂i1s-v)(aφ)i2j2⋯(aφ)ikjk+1k!∑1≤i1,…,ik,j1,…,jk≤m‍εj1⋯jki1⋯ik(aφ)i1j1(gj2s-∂i2s-v)⋯(aφ)ikjk+⋯+1k!∑1≤i1,…,ik,j1,…,jk≤m‍εj1⋯jki1⋯ik(aφ)i1j1⋯(aφ)ik-1jk-1(gjks-∂iks-v)=1(k-1)!∑1≤i1,…,ik,j1,…,jk≤m‍εj1⋯jki1⋯ik(aφ)i1j1⋯(aφ)ik-1jk-1(gjks-∂iks-v)by  symmetry=∑i,j=1m‍(1(k-1)!∑1≤i1,…,ik-1,j1,…,jk-1≤m‍εj1⋯jk-1ji1⋯ik-1i(aφ)i1j1⋯(aφ)ik-1jk-1)︸=:𝒞ji(aφ)∇ijv.
We infer then the following proposition.Proposition 4.3.
The linearizationd𝔉k of the operator 𝔉k is of divergence type:
(4.10)d𝔉kφ=∇i(𝒞ji(aφ)∇j).Proof.
By (4.9) we have
(4.11)d𝔉kφ·v=∑i,j=1m‍𝒞ji(aφ)∇ijv=∑i=1m‍∇i(∑j=1m‍𝒞ji(aφ)∇jv)-∑j=1m‍(∑i=1m‍∇i(𝒞ji(aφ)))∇jv.
Moreover
(4.12)∑i=1m‍∇i(𝒞ji(aφ))=1(k-2)!∑i=1m‍∑1≤i1,…,ik-1,j1,…,jk-1≤m‍εj1⋯jk-1ji1⋯ik-1i(aφ)i1j1⋯(aφ)ik-2jk-2∇i((aφ)ik-1jk-1).
But ∇i((aφ)ik-1jk-1)=∇i(δik-1jk-1+∇ik-1jk-1φ)=∇iik-1jk-1φ, then
(4.13)∑i=1m∇i(𝒞ji(aφ))=1(k-2)!∑i=1m‍∑1≤i1,…,ik-1,j1,…,jk-1≤m‍εj1⋯jk-1ji1⋯ik-1i(aφ)i1j1⋯(aφ)ik-2jk-2∇iik-1jk-1φ.
Besides, the quantity ∇iik-1jk-1φ is symmetric in i,ik-1 (indeed, ∇iik-1jk-1φ-∇ik-1ijk-1φ=Rsiik-1jk-1∇sφ and Rsiik-1jk-1=0 since g is Kähler), and εj1⋯jk-1ji1⋯ik-1i is antisymmetric in i,ik-1; it follows then that ∑i=1m‍∇i(𝒞ji(aφ))=0, consequently d𝔉kφ·v=∑i=1m‍∇i(∑j=1m‍𝒞ji(aφ)∇jv).From Proposition4.3, we infer easily [9, page 62] the following corollary.Corollary 4.4.
The mapF:Sl,α→S~l-2,α,φ↦F(φ)=𝔉k[φ]-(mk) is well defined and differentiable and its differential equals dFφ=d𝔉kφ=∇i(𝒞ji(aφ)∇j)∈ℒ(S~l,α,S~l-2,α).Now, lett0∈𝒯l,α and let φ0∈Sl,α be a solution of the corresponding equation (Ek,t0): F(φ0)=et0fAt0-(mk).Lemma 4.5.
d
F
φ
0
:
S
~
l
,
α
→
S
~
l
-
2
,
α is an isomorphism.Proof.
Letψ∈Cl-2,α(M) with ∫M‍ψvg=0. Let us consider the equation
(4.14)∇i(𝒞ji(aφ0)∇ju)=ψ.
We have 𝒞ji(aφ0)∈Cl-2,α(M) and the matrix [𝒞ji(aφ0)]1≤i,j≤m=[(∂fk/∂Bij)([δij+gjℓ-∂iℓ-φ0])]1≤i,j≤m is positive definite (since 𝔉k is elliptic at φ0); then by Theorem 4.7 of [13, p. 104] on the operators of divergence type, we deduce that there exists a unique function u∈Cl,α(M) satisfying ∫M‍uvg=0 which is solution of (4.14) and then solution of dFφ0u=ψ. Thus, the linear continuous map dFφ0: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 setU of Sl,α containing φ0 and an open set V of S~l-2,α containing F(φ0) 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 ∥(etfAt-(mk))-(et0fAt0-(mk))∥Cl-2,α(M) is small enough so that etfAt-(mk)∈V, thus there exists φ∈U⊂Sl,α such that F(φ)=etfAt-(mk) and consequently there exists φ∈Cl,α(M) of vanishing integral for g which is solution of (Ek,t). Hence t∈𝒯l,α. We conclude therefore that 𝒯l,α is an open set of [0,1].


## 4.2.𝒯l,α Is a Closed Set of [0,1]: The Scheme of the Proof

This section is based on a priori estimates. Finding these estimates is the most difficult step of the proof. Let(ts)s∈ℕ be a sequence of elements of 𝒯l,α that converges to τ∈[0,1], and let (φts)s∈ℕ be the corresponding sequence of functions: φts is Cl,α, k-admissible, has a vanishing integral, and is a solution of
(Ek,ts)Fk([δij+gjℓ-∂iℓ-φts]1≤i,j≤m)=tsf+ln(Ats).
Let us prove that τ∈𝒯l,α. Here is the scheme of the proof.(1)
Reduction to aC2,β(M) estimate: if (φts)s∈ℕ is bounded in a C2,β(M) with 0<β<1, the inclusion C2,β(M)⊂C2(M,ℝ) being compact, we deduce that after extraction (φts)s∈ℕ converges in C2(M,ℝ) to φτ∈C2(M,ℝ). We show by tending to the limit that φτ is a solution of (Ek,τ) (it is then necessarily k-admissible) and of vanishing integral for g. We check finally by a nonlinear regularity theorem [14, page 467] that φτ∈C∞(M,ℝ), which allows us to deduce that τ∈𝒯l,α (see [9, pages 64–67] for details).(2)
We show that(φts)s∈ℕ is bounded in C0(M,ℝ): first of all we prove a positivity Lemma 5.4 for (Ek,t), inspired by the ones of [15, page 843] (for k=m), but in a very different way, required since the k-positivity of ω~ts is weaker with k<m (in this case, some eigenvalues can be nonpositive, which complicates the proof), using a polarization method of [7, page 1740] (cf. 5.2) and a Gårding inequality 5.3; we infer then from this lemma a fundamental inequality 5.5 as Proposition 7.18 of [13, page 262]. We conclude the proof using the Moser's iteration technique exactly as for the equation of Calabi-Yau. We deal with this C0 estimate in Section 5.(3)
We establish the key point of the proof, namely, aC2 a priori estimate (Section 6).(4)
With the uniform ellipticity at hand (consequence of the previous step), we obtain the neededC2,β(M) estimate by the Evans-Trudinger theory (Section 7).


## 5. TheC0 A Priori Estimate

### 5.1. The Positivity Lemma

Our first three lemmas are based on the ideas of [7, Section 2].Lemma 5.1.
Letπ be a real (1-1)-form, it then writes π=ipab-dza∧dzb-, with pab-=p(∂a,∂b-) where p is the symmetric tensor p(U,V)=π(U,JV); hence
(5.1)∀ℓ≤mπℓ∧ωm-ℓ=ℓ!(m-ℓ)!m!σℓ(λ[g-1p])ωm.Proof.
The same proof as Lemma1.4.We consider for1≤ℓ≤m the map fℓ=σℓ∘λ:ℋm→ℝ where ℋm denotes the ℝ-vector space of Hermitian square matrices of size m. fℓ is a real polynomial of degree ℓ and in m2 real variables. Moreover, it is I hyperbolic (cf. [16] for the proof) and it satisfies fℓ(I)=σℓ(1,…,1)=(mℓ)>0. Let f~ℓ be the totally polarized form of fℓ. This polarized form f~ℓ:ℋm×⋯×ℋm︸ℓtimes→ℝ is characterized by the following properties: (i)
f~ℓ is ℓ-linear.(ii)
f~ℓ is symmetric.(iii)
For allB∈ℋm, f~ℓ(B,…,B)=fℓ(B). Using these notations, we infer from Lemma 5.1 that at the center of a g-unitary chart (this guarantees that the matrix g-1p is Hermitian), we have
(5.2)πℓ∧ωm-ℓ=ℓ!(m-ℓ)!m!fℓ(g-1p)ωm.
By transition to the polarized form in this equality we obtain the following lemma.Lemma 5.2.
Let1≤ℓ≤m and π1,…,πℓ be real (1-1)-forms. These forms write πα=i(pα)ab-dza∧dzb-, with (pα)ab-=pα(∂a,∂b-) where pα is the symmetric tensor pα(U,V)=πα(U,JV). Then, at the center of a g-unitary chart we have
(5.3)π1∧⋯∧πℓ∧ωm-ℓ=ℓ!(m-ℓ)!m!f~ℓ(g-1p1,…,g-1pℓ)ωm.Proof.
See [9, page 71].Theorem 5 of Gårding [16] applies to fℓ with 2≤ℓ≤m.Lemma 5.3 (the Gårding inequality forfℓ).
Let2≤ℓ≤m, for all y1,…,yℓ∈Γ(fℓ,I),
(5.4)f~ℓ(y1,…,yℓ)≥fℓ(y1)1/ℓ⋯fℓ(yℓ)1/ℓ.Let us recall thatΓ(fℓ,I) is the connected component of {y∈ℋm/fℓ(y)>0} containing I. The same proof as [17, pages 129, 130] implies that
(5.5)Γ(fℓ,I)={y∈ℋm/∀1≤i≤ℓfi(y)>0}={y∈ℋm/λ(y)∈Γℓ}=λ-1(Γℓ).Note that the Gårding inequality (Lemma5.3) holds for Γ~(fℓ,I)={y∈ℋm/∀1≤i≤ℓfi(y)≥0}.Let us now apply the previous lemmas in order to prove the following positivity lemma inspired by the ones of [15, page 843] (for k=m); let us emphasize that the proof is very different since the k-positivity is weaker.Lemma 5.4 (positivity lemma).
Letα be a real 1-form on M and j∈{1,…,k-1}, then the function f:M→ℝ defined by fωm=Jtα∧α∧ωm-1-j∧ω~j is nonnegative.Proof.
Let1≤j≤k-1, then 2≤ℓ=j+1≤k. Let α be a real 1-form, it then writes α=αadza+αa¯dza-. Let π1=Jtα∧α, hence π1(∂a,∂b-)=α(J∂a)α(∂b-)-α(J∂b-)α(∂a)=iαaαb¯-(-i)αb¯αa=2iαaαb¯. Similarly, we prove that π1(∂a,∂b)=π1(∂a-,∂b-)=0, consequently π1=i2αaαb¯︸=:pab-dza∧dzb-. Besides, set π2=⋯=πj+1=ω~=ig~ab-dza∧dzb-. Now, let x∈M and ϕ be a g-unitary chart centered at x. Using Lemma 5.2, we infer that at x in the chart ϕ:
(5.6)Jtα∧α∧ω~j∧ωm-(j+1)=π1∧⋯∧πj+1∧ωm-(j+1)=(m-j-1)!(j+1)!m!f~j+1(g-1p,g-1g~,…,g-1g~)ωm.
But at x, g-1g~=g~∈Γ(fj+1,I) and g-1p=p∈Γ~(fj+1,I). Indeed, λ(g-1g~)∈Γk since φ is k-admissible and Γk⊂Γj+1. Moreover, the Hermitian matrix [2αaαb¯]1≤a,b≤m is positive-semidefinite since for all ξ∈ℂm, we have ∑a,b=1m‍2αaαb¯ξaξb¯=2|∑a=1m‍αaξa|2≥0; we then deduce that for all 1≤i≤j+1, we have at x, fi(g-1p)=σi(λ(g-1p))≥0. Finally, we infer by the Gårding inequality that at x in the chart ϕ we have
(5.7)f~j+1(g-1p,g-1g~,…,g-1g~)≥fj+1(g-1p)1/(j+1)fj+1(g-1g~)j/(j+1)≥0
which proves the positivity lemma.


### 5.2. The Fundamental Inequality

TheC0 a priori estimate is based on the following crucial proposition which is a generalization of the Proposition 7.18 of [13, page 262].Proposition 5.5.
Leth(t) be an increasing function of class C1 defined on ℝ, and let φ be a C2  k-admissible function defined on M, then the following inequality is satisfied:
(5.8)∫M[(mk)-fk(g-1g~)]h(φ)ωm≥12m(mk)∫M‍h′(φ)|∇φ|g2ωm.Proof.
We have the equality∫M‍[(mk)-fk(g-1g~)]h(φ)ωm=(mk)∫M‍h(φ)(ωm-ω~k∧ωm-k). Besides, since Λ2M is commutative ωm-ω~k∧ωm-k=(ω-ω~)∧(ωm-1+ωm-2∧ω~+⋯+ωm-k∧ω~k-1)︸=:Ω, namely, ωm-ω~k∧ωm-k=-(1/2)ddcφ∧Ω, then ∫M[(mk)-fk(g-1g~)]h(φ)ωm=-(1/2)(mk)∫Mddcφ∧(h(φ)Ω). But d(dcφ∧h(φ)Ω)=ddcφ∧h(φ)Ω+(-1)1dcφ∧d(h(φ)Ω), and d(h(φ)Ω)=h′(φ)dφ∧Ω+(-1)0h(φ)dΩ︸=0sinceωandω-areclosed so ddcφ∧h(φ)Ω=dcφ∧h′(φ)dφ∧Ω+d(something). In addition by Stokes' theorem, ∫Md(something)=0; therefore,
(5.9)∫M[(mk)-fk(g-1g~)]h(φ)ωm=-12(mk)∫Mh′(φ)dcφ∧dφ∧Ω=12(mk)(∫M‍h′(φ)(-dcφ)∧dφ∧ωm-1︸T1+∑j=1k-1‍∫M‍h′(φ)(-dcφ)∧dφ∧ωm-1-j∧ω~j︸)T2.
Let us prove that T2≥0 (using the positivity lemma) and that T1=(1/m)∫M‍h′(φ)|∇φ|g2ωm. Let us apply the positivity lemma to dφ: the function f:M→ℝ defined by fωm=Jtdφ∧dφ∧ωm-1-j∧ω~j is nonnegative for all 1≤j≤k-1. But Jtdφ=-dcφ and h is an increasing function; then T2≥0. Let us now calculate T1. Fix x∈M, and let us work in a g-unitary chart centered at x and satisfying dφ/|dφ|g=(dzm+dzm-)/2 at x. We have then ω=idza∧dza- at x and Jtdφ∧dφ=i|dφ|g2dzm∧dzm-; therefore,
(5.10)Jtdφ∧dφ∧ωm-1=∑a1,…,am-1∈{1,…,m-1}2by2≠‍im|dφ|g2(dzm∧dzm-)∧(dza1∧dza-1)∧⋯∧(dzam-1∧dza-m-1)=(∑a1,…,am-1∈{1,…,m-1}2by2≠‍1)|dφ|g2im(dz1∧dz1-)∧⋯∧(dzm∧dzm-)=(m-1)!|dφ|g2ωmm!=1m|∇φ|g2ωm.
Thus T1=(1/m)∫M‍h′(φ)|∇φ|g2ωm, consequently ∫M‍[(mk)-fk(g-1g~)]h(φ)ωm≥(1/2)(mk)T1=(1/2m)(mk)∫M‍h′(φ)|∇φ|g2ωm, which achieves the proof of the proposition.


### 5.3. The Moser Iteration Technique

We conclude the proof using the Moser's iteration technique exactly as for the equation of Calabi-Yau. Let us apply the proposition toφts in order to obtain a crucial inequality (the inequality (IN1) from which we will infer the a priori estimate of ∥φts∥C0. Let p≥2 be a real number. The function φts is C2 admissible. Let us consider the function h(u):=u|u|p-2 : ℝ→ℝ. This function is of class C1 and h′(u)=|u|p-2+u(p-2)u|u|p-4=(p-1)|u|p-2≥0, so h is increasing. Therefore we infer by the previous proposition that
(5.11)p-12m(mk)∫M‍|φts|p-2|∇φts|2vg≤∫M‍[(mk)-fk(g-1g~)]φts|φts|p-2vg.
Besides, |∇|φts|p/2|2=2gab-∂a|φts|p/2∂b-|φts|p/2=2gab-((p/2)φts|φts|p/2-2)2∂aφts∂b-φts=(p2/4)|φts|p-2|∇φts|2, so the previous inequality writes:
(IN1)∫M‍|∇|φts|p/2|2vg≤mp22(p-1)(mk)∫M‍[(mk)-fk(g-1g~)]φts|φts|p-2vg.
Let us infer from the inequality (IN1) another inequality (the inequality (IN4) that is required for the proof. It follows from the continuous Sobolev embedding H12(M)⊂L2m/(m-1)(M) that
(IN2)∥|φts|p∥m/(m-1)=∥|φts|p/2∥2m/(m-1)2≤Cste(∫M‍|∇|φts|p/2|2+∫M‍|φts|(p/2)·2),
where Cste is independent of p. Besides, fk(g-1g~) is uniformly bounded; indeed,
(IN3)|fk(g-1g~)|=etsf(mk)Vol(M)∫M‍etsfvg≤(mk)e2ts∥f∥∞≤(mk)e2∥f∥∞.
Using the inequalities (IN1), (IN2), (IN3), and p2/2(p-1)≤p we obtain
(IN4)∥|φts|p∥m/(m-1)≤Cste′×p(∫M‍|φts|p-1+∫M‍|φts|p)(p≥2),
where Cste′ is independent of p. Suppose that Cste′≥1.Using the Green's formula and the inequalities of Sobolev-Poincaré(IN2) and of Hölder, we prove following [13] these Lq estimates.Lemma 5.6.
There exists a constantμ such that for all 1≤q≤2m/(m-1),
(5.12)∥φts∥q≤μ.Proof.
M is a compact Riemannian manifold and φts∈C2, so by the Green's formula φts(x)=(1/Vol(M))∫Mφtsdv︸=0+∫MG(x,y)▵φts(y)dv(y), where G(x,y)≥0 and ∫M‍G(x,y)dv(y) is independent of x. Here ▵φts denotes the real Laplacian. Then, we infer that ∥φts∥1≤C∥▵φts∥1. But ∥▵φts∥1=∫M‍▵φts++▵φts- and ∫M‍▵φts=∫M‍▵φts+-▵φts-=0; then ∥▵φts∥1=2∫M‍▵φts+. Besides ▵φts<2m since φts is k-admissible: indeed, at x in a g-normal g~-adapted chart, namely, a chart satisfying gab-=δab, g~ab-=δabλa and ∂νgab-=0 for all 1≤a, b≤m, ν∈{1,…,m,1-,…,m-}, we have λ(g-1g~)=(λ1,…,λm) so λ=(λ1…,λm)∈Γk since φts is k-admissible; consequently ▵φts=-2gab-∂ab-φts=-2∑a‍∂aa-φts=2∑a‍(1-λa)=2m-2σ1(λ), but σ1(λ)>0 since λ∈Γk which proves that ▵φts<2m. Therefore ▵φts+<2m and ∥▵φts∥1≤4mVol(M). We infer then that ∥φts∥1≤4mCVol(M). Now let us take p=2 in the inequality (IN2): ∥φts∥2m/(m-1)2≤Cste(∫M‍|∇|φts||2+∫M‍|φts|2). Besides, φts∈H12(M) and has a vanishing integral; then by the Sobolev-Poincaré inequality we infer ∥φts∥2≤𝒜∥∇φts∥2. But |∇|φts||=|∇φts| almost everywhere; therefore ∥φts∥2m/(m-1)≤Cste∥∇|φts|∥2. Using the inequality (IN1) with p=2 and the fact that fk(g-1g~) is uniformly bounded, we obtain that ∥∇|φts|∥22≤Cste∥φts∥1≤Cste′. Consequently, we infer that ∥φts∥2m/(m-1)≤Cste.
Let1≤q≤2m/(m-1)=:2δ. By the Hölder inequality we have ∥φts∥qq=∫M|φts|q·1≤(∫M|φts|q·(2δ/q))q/2δVol(M)1-q/2δ. Therefore ∥φts∥q≤Vol(M)(1/q)-(1/2δ)∥φts∥2δ. But
(5.13)Vol(M)1/q-1/2δ=e(1/q-1/2δ)ln(Vol(M))≤{1ifVol(M)≤1,Vol(M)1-1/2δifVol(M)≥1
and ∥φts∥2δ≤Cste, thus ∥φts∥q≤μ:=Cste×Max(1,Vol(M)1-1/2δ).Suppose without limitation of generality thatμ≥1. Now, we deduce from the previous lemma and the inequality (IN4), by induction, these more general Lp estimates using the same method as [13].Lemma 5.7.
There exists a constantC0 such that for all p≥2,
(5.14)∥φts∥p≤C0(δm-1Cp)-m/p,
with δ=m/(m-1) and C=Cste′(1+Max(1,Vol(M)1/2))≥1 where Cste′ is the constant of the inequality (IN4).Proof.
We prove this lemma by induction: first we check that the inequality is satisfied for2≤p≤2δ=2m/(m-1); afterwards we show that if the inequality is true for p, then it is satisfied for δp too. Denote C0=μδm(m-1)Cmem/e. For 2≤p≤2δ we have ∥φts∥p≤μ, so it suffices to check that μ≤C0(δm-1Cp)-m/p. This inequality is equivalent to δm(m-1)Cmem/e(δm-1Cp)-m/p≥1; then (δm(m-1)Cm)em/e≥(δm(m-1)Cm)1/ppm/p. But if x≥1, then x≥x1/p (since p≥1), and δm(m-1)Cm≥1 (since C≥1,m≥1 and δ≥1); therefore δm(m-1)Cm≥(δm(m-1)Cm)1/p. Besides, pm/p=em(lnp/p)≤em/e, which proves the inequality for 2≤p≤2δ. Now let us fix p≥2. Suppose that ∥φts∥p≤C0(δm-1Cp)-m/p and prove that ∥φts∥δp≤C0(δm-1Cδp)-m/δp. The inequality (IN4) proved previously writes:
(IN4′)∥|φts|p∥δ≤Cste′×p(∫M|φts|p-1+∫M‍|φts|p‍)(p≥2),
where Cste′ is independent of p, namely, ∥φts∥δpp≤Cste′×p(∥φts∥p-1p-1+∥φts∥pp). But since 1≤p-1≤p, we have by the Hölder inequality that ∥φts∥p-1≤Vol(M)1/(p-1)-1/p∥φts∥p; therefore ∥φts∥δpp≤Cste′×p(Vol(M)1/p∥φts∥pp-1+∥φts∥pp).(i)
If∥φts∥p≤1, then ∥φts∥δpp≤C×p; therefore ∥φts∥δp≤(Cp)1/p. Let us check that (Cp)1/p≤C0(δm-1Cδp)-m/δp. This inequality is equivalent to p(1/p)(1+m/δ)≤μδm(m-1)(1-1/p)em/e×Cm-m/δp-1/p, but 1+m/δ=m so it is equivalent to pm/p≤μδm(m-1)(1-1/p)em/e×Cm(1-1/p). Besides pm/p≤em/e and μδm(m-1)(1-1/p)≥1, then it suffices to have Cm(1-1/p)≥1, and this is satisfied since C≥1.(ii)
If∥φts∥p≥1, we infer that ∥φts∥δpp≤C×p∥φts∥pp, therefore ∥φts∥δp≤C1/p×p1/p∥φts∥p≤(Cp)1/pC0(δm-1Cp)-m/p by the induction hypothesis. But (1-m)/p=-m/δp; then we obtain the required inequality ∥φts∥δp≤C0δ-m2/δp(Cp)-m/δp=C0(δm-1Cδp)-m/δp.By tending to the limitp→+∞ in the inequality of the previous lemma, we obtain the needed C0 a priori estimate.Corollary 5.8.
Consider(5.15)∥φts∥C0≤C0.


## 5.1. The Positivity Lemma

Our first three lemmas are based on the ideas of [7, Section 2].Lemma 5.1.
Letπ be a real (1-1)-form, it then writes π=ipab-dza∧dzb-, with pab-=p(∂a,∂b-) where p is the symmetric tensor p(U,V)=π(U,JV); hence
(5.1)∀ℓ≤mπℓ∧ωm-ℓ=ℓ!(m-ℓ)!m!σℓ(λ[g-1p])ωm.Proof.
The same proof as Lemma1.4.We consider for1≤ℓ≤m the map fℓ=σℓ∘λ:ℋm→ℝ where ℋm denotes the ℝ-vector space of Hermitian square matrices of size m. fℓ is a real polynomial of degree ℓ and in m2 real variables. Moreover, it is I hyperbolic (cf. [16] for the proof) and it satisfies fℓ(I)=σℓ(1,…,1)=(mℓ)>0. Let f~ℓ be the totally polarized form of fℓ. This polarized form f~ℓ:ℋm×⋯×ℋm︸ℓtimes→ℝ is characterized by the following properties: (i)
f~ℓ is ℓ-linear.(ii)
f~ℓ is symmetric.(iii)
For allB∈ℋm, f~ℓ(B,…,B)=fℓ(B). Using these notations, we infer from Lemma 5.1 that at the center of a g-unitary chart (this guarantees that the matrix g-1p is Hermitian), we have
(5.2)πℓ∧ωm-ℓ=ℓ!(m-ℓ)!m!fℓ(g-1p)ωm.
By transition to the polarized form in this equality we obtain the following lemma.Lemma 5.2.
Let1≤ℓ≤m and π1,…,πℓ be real (1-1)-forms. These forms write πα=i(pα)ab-dza∧dzb-, with (pα)ab-=pα(∂a,∂b-) where pα is the symmetric tensor pα(U,V)=πα(U,JV). Then, at the center of a g-unitary chart we have
(5.3)π1∧⋯∧πℓ∧ωm-ℓ=ℓ!(m-ℓ)!m!f~ℓ(g-1p1,…,g-1pℓ)ωm.Proof.
See [9, page 71].Theorem 5 of Gårding [16] applies to fℓ with 2≤ℓ≤m.Lemma 5.3 (the Gårding inequality forfℓ).
Let2≤ℓ≤m, for all y1,…,yℓ∈Γ(fℓ,I),
(5.4)f~ℓ(y1,…,yℓ)≥fℓ(y1)1/ℓ⋯fℓ(yℓ)1/ℓ.Let us recall thatΓ(fℓ,I) is the connected component of {y∈ℋm/fℓ(y)>0} containing I. The same proof as [17, pages 129, 130] implies that
(5.5)Γ(fℓ,I)={y∈ℋm/∀1≤i≤ℓfi(y)>0}={y∈ℋm/λ(y)∈Γℓ}=λ-1(Γℓ).Note that the Gårding inequality (Lemma5.3) holds for Γ~(fℓ,I)={y∈ℋm/∀1≤i≤ℓfi(y)≥0}.Let us now apply the previous lemmas in order to prove the following positivity lemma inspired by the ones of [15, page 843] (for k=m); let us emphasize that the proof is very different since the k-positivity is weaker.Lemma 5.4 (positivity lemma).
Letα be a real 1-form on M and j∈{1,…,k-1}, then the function f:M→ℝ defined by fωm=Jtα∧α∧ωm-1-j∧ω~j is nonnegative.Proof.
Let1≤j≤k-1, then 2≤ℓ=j+1≤k. Let α be a real 1-form, it then writes α=αadza+αa¯dza-. Let π1=Jtα∧α, hence π1(∂a,∂b-)=α(J∂a)α(∂b-)-α(J∂b-)α(∂a)=iαaαb¯-(-i)αb¯αa=2iαaαb¯. Similarly, we prove that π1(∂a,∂b)=π1(∂a-,∂b-)=0, consequently π1=i2αaαb¯︸=:pab-dza∧dzb-. Besides, set π2=⋯=πj+1=ω~=ig~ab-dza∧dzb-. Now, let x∈M and ϕ be a g-unitary chart centered at x. Using Lemma 5.2, we infer that at x in the chart ϕ:
(5.6)Jtα∧α∧ω~j∧ωm-(j+1)=π1∧⋯∧πj+1∧ωm-(j+1)=(m-j-1)!(j+1)!m!f~j+1(g-1p,g-1g~,…,g-1g~)ωm.
But at x, g-1g~=g~∈Γ(fj+1,I) and g-1p=p∈Γ~(fj+1,I). Indeed, λ(g-1g~)∈Γk since φ is k-admissible and Γk⊂Γj+1. Moreover, the Hermitian matrix [2αaαb¯]1≤a,b≤m is positive-semidefinite since for all ξ∈ℂm, we have ∑a,b=1m‍2αaαb¯ξaξb¯=2|∑a=1m‍αaξa|2≥0; we then deduce that for all 1≤i≤j+1, we have at x, fi(g-1p)=σi(λ(g-1p))≥0. Finally, we infer by the Gårding inequality that at x in the chart ϕ we have
(5.7)f~j+1(g-1p,g-1g~,…,g-1g~)≥fj+1(g-1p)1/(j+1)fj+1(g-1g~)j/(j+1)≥0
which proves the positivity lemma.


## 5.2. The Fundamental Inequality

TheC0 a priori estimate is based on the following crucial proposition which is a generalization of the Proposition 7.18 of [13, page 262].Proposition 5.5.
Leth(t) be an increasing function of class C1 defined on ℝ, and let φ be a C2  k-admissible function defined on M, then the following inequality is satisfied:
(5.8)∫M[(mk)-fk(g-1g~)]h(φ)ωm≥12m(mk)∫M‍h′(φ)|∇φ|g2ωm.Proof.
We have the equality∫M‍[(mk)-fk(g-1g~)]h(φ)ωm=(mk)∫M‍h(φ)(ωm-ω~k∧ωm-k). Besides, since Λ2M is commutative ωm-ω~k∧ωm-k=(ω-ω~)∧(ωm-1+ωm-2∧ω~+⋯+ωm-k∧ω~k-1)︸=:Ω, namely, ωm-ω~k∧ωm-k=-(1/2)ddcφ∧Ω, then ∫M[(mk)-fk(g-1g~)]h(φ)ωm=-(1/2)(mk)∫Mddcφ∧(h(φ)Ω). But d(dcφ∧h(φ)Ω)=ddcφ∧h(φ)Ω+(-1)1dcφ∧d(h(φ)Ω), and d(h(φ)Ω)=h′(φ)dφ∧Ω+(-1)0h(φ)dΩ︸=0sinceωandω-areclosed so ddcφ∧h(φ)Ω=dcφ∧h′(φ)dφ∧Ω+d(something). In addition by Stokes' theorem, ∫Md(something)=0; therefore,
(5.9)∫M[(mk)-fk(g-1g~)]h(φ)ωm=-12(mk)∫Mh′(φ)dcφ∧dφ∧Ω=12(mk)(∫M‍h′(φ)(-dcφ)∧dφ∧ωm-1︸T1+∑j=1k-1‍∫M‍h′(φ)(-dcφ)∧dφ∧ωm-1-j∧ω~j︸)T2.
Let us prove that T2≥0 (using the positivity lemma) and that T1=(1/m)∫M‍h′(φ)|∇φ|g2ωm. Let us apply the positivity lemma to dφ: the function f:M→ℝ defined by fωm=Jtdφ∧dφ∧ωm-1-j∧ω~j is nonnegative for all 1≤j≤k-1. But Jtdφ=-dcφ and h is an increasing function; then T2≥0. Let us now calculate T1. Fix x∈M, and let us work in a g-unitary chart centered at x and satisfying dφ/|dφ|g=(dzm+dzm-)/2 at x. We have then ω=idza∧dza- at x and Jtdφ∧dφ=i|dφ|g2dzm∧dzm-; therefore,
(5.10)Jtdφ∧dφ∧ωm-1=∑a1,…,am-1∈{1,…,m-1}2by2≠‍im|dφ|g2(dzm∧dzm-)∧(dza1∧dza-1)∧⋯∧(dzam-1∧dza-m-1)=(∑a1,…,am-1∈{1,…,m-1}2by2≠‍1)|dφ|g2im(dz1∧dz1-)∧⋯∧(dzm∧dzm-)=(m-1)!|dφ|g2ωmm!=1m|∇φ|g2ωm.
Thus T1=(1/m)∫M‍h′(φ)|∇φ|g2ωm, consequently ∫M‍[(mk)-fk(g-1g~)]h(φ)ωm≥(1/2)(mk)T1=(1/2m)(mk)∫M‍h′(φ)|∇φ|g2ωm, which achieves the proof of the proposition.


## 5.3. The Moser Iteration Technique

We conclude the proof using the Moser's iteration technique exactly as for the equation of Calabi-Yau. Let us apply the proposition toφts in order to obtain a crucial inequality (the inequality (IN1) from which we will infer the a priori estimate of ∥φts∥C0. Let p≥2 be a real number. The function φts is C2 admissible. Let us consider the function h(u):=u|u|p-2 : ℝ→ℝ. This function is of class C1 and h′(u)=|u|p-2+u(p-2)u|u|p-4=(p-1)|u|p-2≥0, so h is increasing. Therefore we infer by the previous proposition that
(5.11)p-12m(mk)∫M‍|φts|p-2|∇φts|2vg≤∫M‍[(mk)-fk(g-1g~)]φts|φts|p-2vg.
Besides, |∇|φts|p/2|2=2gab-∂a|φts|p/2∂b-|φts|p/2=2gab-((p/2)φts|φts|p/2-2)2∂aφts∂b-φts=(p2/4)|φts|p-2|∇φts|2, so the previous inequality writes:
(IN1)∫M‍|∇|φts|p/2|2vg≤mp22(p-1)(mk)∫M‍[(mk)-fk(g-1g~)]φts|φts|p-2vg.
Let us infer from the inequality (IN1) another inequality (the inequality (IN4) that is required for the proof. It follows from the continuous Sobolev embedding H12(M)⊂L2m/(m-1)(M) that
(IN2)∥|φts|p∥m/(m-1)=∥|φts|p/2∥2m/(m-1)2≤Cste(∫M‍|∇|φts|p/2|2+∫M‍|φts|(p/2)·2),
where Cste is independent of p. Besides, fk(g-1g~) is uniformly bounded; indeed,
(IN3)|fk(g-1g~)|=etsf(mk)Vol(M)∫M‍etsfvg≤(mk)e2ts∥f∥∞≤(mk)e2∥f∥∞.
Using the inequalities (IN1), (IN2), (IN3), and p2/2(p-1)≤p we obtain
(IN4)∥|φts|p∥m/(m-1)≤Cste′×p(∫M‍|φts|p-1+∫M‍|φts|p)(p≥2),
where Cste′ is independent of p. Suppose that Cste′≥1.Using the Green's formula and the inequalities of Sobolev-Poincaré(IN2) and of Hölder, we prove following [13] these Lq estimates.Lemma 5.6.
There exists a constantμ such that for all 1≤q≤2m/(m-1),
(5.12)∥φts∥q≤μ.Proof.
M is a compact Riemannian manifold and φts∈C2, so by the Green's formula φts(x)=(1/Vol(M))∫Mφtsdv︸=0+∫MG(x,y)▵φts(y)dv(y), where G(x,y)≥0 and ∫M‍G(x,y)dv(y) is independent of x. Here ▵φts denotes the real Laplacian. Then, we infer that ∥φts∥1≤C∥▵φts∥1. But ∥▵φts∥1=∫M‍▵φts++▵φts- and ∫M‍▵φts=∫M‍▵φts+-▵φts-=0; then ∥▵φts∥1=2∫M‍▵φts+. Besides ▵φts<2m since φts is k-admissible: indeed, at x in a g-normal g~-adapted chart, namely, a chart satisfying gab-=δab, g~ab-=δabλa and ∂νgab-=0 for all 1≤a, b≤m, ν∈{1,…,m,1-,…,m-}, we have λ(g-1g~)=(λ1,…,λm) so λ=(λ1…,λm)∈Γk since φts is k-admissible; consequently ▵φts=-2gab-∂ab-φts=-2∑a‍∂aa-φts=2∑a‍(1-λa)=2m-2σ1(λ), but σ1(λ)>0 since λ∈Γk which proves that ▵φts<2m. Therefore ▵φts+<2m and ∥▵φts∥1≤4mVol(M). We infer then that ∥φts∥1≤4mCVol(M). Now let us take p=2 in the inequality (IN2): ∥φts∥2m/(m-1)2≤Cste(∫M‍|∇|φts||2+∫M‍|φts|2). Besides, φts∈H12(M) and has a vanishing integral; then by the Sobolev-Poincaré inequality we infer ∥φts∥2≤𝒜∥∇φts∥2. But |∇|φts||=|∇φts| almost everywhere; therefore ∥φts∥2m/(m-1)≤Cste∥∇|φts|∥2. Using the inequality (IN1) with p=2 and the fact that fk(g-1g~) is uniformly bounded, we obtain that ∥∇|φts|∥22≤Cste∥φts∥1≤Cste′. Consequently, we infer that ∥φts∥2m/(m-1)≤Cste.
Let1≤q≤2m/(m-1)=:2δ. By the Hölder inequality we have ∥φts∥qq=∫M|φts|q·1≤(∫M|φts|q·(2δ/q))q/2δVol(M)1-q/2δ. Therefore ∥φts∥q≤Vol(M)(1/q)-(1/2δ)∥φts∥2δ. But
(5.13)Vol(M)1/q-1/2δ=e(1/q-1/2δ)ln(Vol(M))≤{1ifVol(M)≤1,Vol(M)1-1/2δifVol(M)≥1
and ∥φts∥2δ≤Cste, thus ∥φts∥q≤μ:=Cste×Max(1,Vol(M)1-1/2δ).Suppose without limitation of generality thatμ≥1. Now, we deduce from the previous lemma and the inequality (IN4), by induction, these more general Lp estimates using the same method as [13].Lemma 5.7.
There exists a constantC0 such that for all p≥2,
(5.14)∥φts∥p≤C0(δm-1Cp)-m/p,
with δ=m/(m-1) and C=Cste′(1+Max(1,Vol(M)1/2))≥1 where Cste′ is the constant of the inequality (IN4).Proof.
We prove this lemma by induction: first we check that the inequality is satisfied for2≤p≤2δ=2m/(m-1); afterwards we show that if the inequality is true for p, then it is satisfied for δp too. Denote C0=μδm(m-1)Cmem/e. For 2≤p≤2δ we have ∥φts∥p≤μ, so it suffices to check that μ≤C0(δm-1Cp)-m/p. This inequality is equivalent to δm(m-1)Cmem/e(δm-1Cp)-m/p≥1; then (δm(m-1)Cm)em/e≥(δm(m-1)Cm)1/ppm/p. But if x≥1, then x≥x1/p (since p≥1), and δm(m-1)Cm≥1 (since C≥1,m≥1 and δ≥1); therefore δm(m-1)Cm≥(δm(m-1)Cm)1/p. Besides, pm/p=em(lnp/p)≤em/e, which proves the inequality for 2≤p≤2δ. Now let us fix p≥2. Suppose that ∥φts∥p≤C0(δm-1Cp)-m/p and prove that ∥φts∥δp≤C0(δm-1Cδp)-m/δp. The inequality (IN4) proved previously writes:
(IN4′)∥|φts|p∥δ≤Cste′×p(∫M|φts|p-1+∫M‍|φts|p‍)(p≥2),
where Cste′ is independent of p, namely, ∥φts∥δpp≤Cste′×p(∥φts∥p-1p-1+∥φts∥pp). But since 1≤p-1≤p, we have by the Hölder inequality that ∥φts∥p-1≤Vol(M)1/(p-1)-1/p∥φts∥p; therefore ∥φts∥δpp≤Cste′×p(Vol(M)1/p∥φts∥pp-1+∥φts∥pp).(i)
If∥φts∥p≤1, then ∥φts∥δpp≤C×p; therefore ∥φts∥δp≤(Cp)1/p. Let us check that (Cp)1/p≤C0(δm-1Cδp)-m/δp. This inequality is equivalent to p(1/p)(1+m/δ)≤μδm(m-1)(1-1/p)em/e×Cm-m/δp-1/p, but 1+m/δ=m so it is equivalent to pm/p≤μδm(m-1)(1-1/p)em/e×Cm(1-1/p). Besides pm/p≤em/e and μδm(m-1)(1-1/p)≥1, then it suffices to have Cm(1-1/p)≥1, and this is satisfied since C≥1.(ii)
If∥φts∥p≥1, we infer that ∥φts∥δpp≤C×p∥φts∥pp, therefore ∥φts∥δp≤C1/p×p1/p∥φts∥p≤(Cp)1/pC0(δm-1Cp)-m/p by the induction hypothesis. But (1-m)/p=-m/δp; then we obtain the required inequality ∥φts∥δp≤C0δ-m2/δp(Cp)-m/δp=C0(δm-1Cδp)-m/δp.By tending to the limitp→+∞ in the inequality of the previous lemma, we obtain the needed C0 a priori estimate.Corollary 5.8.
Consider(5.15)∥φts∥C0≤C0.


## 6. TheC2 A Priori Estimate

### 6.1. Strategy for aC2 Estimate

First, we will look for a uniform upper bound on the eigenvaluesλ([δij+gjℓ-∂iℓ-φt]1≤i,j≤m. Secondly, we will infer from it the uniform ellipticity of the continuity equation (Ek,t) and a uniform gradient bound. Thirdly, with the uniform ellipticity at hand, we will derive a one-sided estimate on pure second derivatives and finally get the needed C2 bound.


### 6.2. Eigenvalues Upper Bound

#### 6.2.1. The Functional

Lett∈𝒯l,α, and let φt : M→ℝ be a Cl,α  k-admissible solution of (Ek,t) satisfying ∫M‍φtωm=0. Consider the following functional:
(6.1)B:UT1,0→ℝ(P,ξ)↦B(P,ξ)=h~P(ξ,ξ)-φt(P),
where UT1,0 is the unit sphere bundle associated to (T1,0,h) and g~ is related to g by: ω~=ω+i∂∂-φt. B is continuous on the compact set UT1,0, so it assumes its maximum at a point (P0,ξ0)∈UT1,0. In addition, for fixed P∈M, ξ∈UTP1,0↦h~P(ξ,ξ) is continuous on the compact subset UTP1,0 (the fiber); therefore it attains its maximum at a unit vector ξP∈UTP1,0, and by the min-max principle we can choose ξP as the direction of the largest eigenvalue of AP, λmax(AP). Specifically, we have the following.Lemma 6.1 (min-max principle).
Consider(6.2)h~P(ξP,ξP)=maxξ∈TP1,0,hP(ξ,ξ)=1h~P(ξ,ξ)=λmax(AP).For fixedP, we have maxhP(ξ,ξ)=1B(P,ξ)=B(P,ξP)=λmax(AP)-φt(P); therefore max(P,ξ)∈UT1,0B(P,ξ)=maxP∈MB(P,ξP)=B(P0,ξ0)≤B(P0,ξP0); hence,
(6.3)max(P,ξ)∈UT1,0B(P,ξ)=B(P0,ξP0)=λmax(AP0)-φt(P0).
At the point P0, consider e1P0,…,emP0 an hP0-orthonormal basis of (TP01,0,hP0) made of eigenvectors of AP0 that satisfies the following properties: (1)
hP0-orthonormal: [hij(P0)]1≤i,j≤m=Im.(2)
h~P0-diagonal: [h~ij(P0)]1≤i,j≤m=MatAP0=diag(λ1,…,λm), λ∈Γk.(3)
λmax(AP0) is achieved in the direction e1P0=ξP0: AP0(ξP0)=λmax(AP0)ξP0=λ1ξP0 and λ1≥⋯≥λm. In other words, it is a basis satisfying(1)
[gij-(P0)]1≤i,j≤m=Im,(2)
[g~ij~(P0)]1≤i,j≤m=MatAP0=diag(λ1,…,λm), λ∈Γk,(3)
λmax(AP0)=λ1≥⋯≥λm. Let us consider a holomorphic normal chart (U0,ψ0) centered at P0 such that ψ0(P0)=0 and ∂i|P0=eiP0 for all i∈{1⋯m}.


#### 6.2.2. Auxiliary Local Functional

From now on, we work in the chart(U0,ψ0) constructed at P0. The map P↦g11-(P) is continuous on U0 and is equal to 1 at P0, so there exists an open subset U1⊂U0 such that g11-(P)>0 for all P∈U1. Let B1 be the functional
(6.4)B1:U1→ℝP↦B1(P)=g~11-(P)g11-(P)-φt(P).
We claim that B1 assumes a local maximum at P0. Indeed, we have at each P∈U1: g~11-(P)/g11-(P)=g~P(∂1,∂1-)/gP(∂1,∂1-)=h~P(∂1,∂1)/hP(∂1,∂1)=h~P(∂1/|∂1|hP,∂1/|∂1|hP)≤λmax(AP) (see Lemma 6.1); thus B1(P)≤λmax(AP)-φt(P)≤λmax(AP0)-φt(P0)=B1(P0).


#### 6.2.3. Differentiating the Equation

For short, we drop henceforth the subscriptt of φt. Let us differentiate (Ek,t) at P, in a chart z:
(6.5)t∂1-f=dFk[δij+gjℓ-(P)∂iℓ-φ(P)]1≤i,j≤m·[∂1-(gjℓ-∂iℓ-φ)]1≤i,j≤m=∑i,j=1m‍∂Fk∂Bij[δij+gjℓ-∂iℓ-φ](∂1-gjℓ-∂iℓ-φ+gjℓ-∂1-iℓ-φ).
Differentiating once again, we find
(6.6)t∂11-f=∑i,j,r,s=1m‍∂2Fk∂Brs∂Bij[δij+gjℓ-∂iℓ-φ](∂1gso-∂ro-φ+gso-∂1ro-φ)×(∂1-gjℓ-∂iℓ-φ+gjℓ-∂1-iℓ-φ)+∑i,j=1m‍∂Fk∂Bij[δij+gjℓ-∂iℓ-φ]×(∂11-gjℓ-∂iℓ-φ+∂1-gjℓ-∂1iℓ-φ+∂1gjℓ-∂1-iℓ-φ+gjℓ-∂11-iℓ-φ).
Using the above chart (U1,ψ0) at the point P0, normality yields gjℓ-=δjℓ, ∂αgiℓ-=0 and ∂αgiℓ-=0. Furthermore [δij+gjℓ-∂iℓ-φ]=[δij+∂ij-φ]=[g~ij-]=diag(λ1,…,λm). In this chart, we can simplify the previous expression; we get then at P0,
(6.7)t∂11-f=∑i,j,r,s=1m‍∂2Fk∂Brs∂Bij(diag(λ1,…,λm))∂1rs-φ∂1-ij-φ+∑i,j=1m‍∂Fk∂Bij(diag(λ1,…,λm))(∂11-gji-∂ii-φ+∂11-ij-φ).
Besides, ∂11-gji-=∂1-(-gjs-goi-∂1gos-), so still by normality, we obtain at P0 that ∂11-gji-=-gjs-goi-∂11-gos-=-∂11-gij--R11-ij-. Therefore we get
(6.8)t∂11-f=∑i,j=1m‍∂Fk∂Bij(diag(λ1,…,λm))(∂11-ij-φ-R11-ij-∂ii-φ)+∑i,j,r,s=1m‍∂2Fk∂Brs∂Bij(diag(λ1,…,λm))∂1rs-φ∂1-ij-φ.


#### 6.2.4. Using Concavity

Now, using the concavity oflnσk [10], we prove for Proposition 2.1 that the second sum of (6.8) is negative [9, page 84]. This is not a direct consequence of the concavity of the function Fk since the matrix [∂1ij-φ]1≤i,j≤m is not Hermitian.Lemma 6.2.
Consider(6.9)S:=∑i,j,r,s=1m‍∂2Fk∂Brs∂Bij(diag(λ1,…,λm))∂1rs-φ∂1-ij-φ≤0.Hence, from (6.8) combined with Lemma 6.2 we infer
(6.10)t∂11-f≤∑i=1m‍σk-1,i(λ)σk(λ)(∂11-ii-φ-R11-ii-∂ii-φ).


#### 6.2.5. Differentiation of the FunctionalB1

Let us differentiate twice the functionalB1:
(6.11)B1(P)=g~11-(P)g11-(P)-φ(P),∂i-B1=∂i-g~11-g11--g~11-∂i-g11-(g11-)2-∂i-φ,∂ii-B1=∂ii-g~11-g11--∂ig11-∂i-g~11-+∂ig~11-∂i-g11-+g~11-∂ii-g11-(g11-)2+2g~11-∂ig11-∂i-g11-(g11-)3-∂ii-φ.
Therefore at P0, in the above chart (U1,ψ0) we find ∂ii-B1=∂ii-(g11-+∂11-φ)-λ1∂ii-g11--∂ii-φ=R11-ii-+∂11-ii-φ-λ1R11-ii--∂ii-φ. Let us define the operator:
(6.12)L:=∑i,j=1m‍∂Fk∂Bij([δij+gjℓ-∂iℓ-φ]1≤i,j≤m)∇ij.
Thus, we have at P0(6.13)L(B1)=∑i=1m‍σk-1,i(λ)σk(λ)(∂11-ii-φ+(1-λ1)R11-ii--∂ii-φ).
Combining (6.13) with (6.10), we get rid of the fourth derivatives:
(6.14)t∂11-f-L(B1)≤∑i=1m‍σk-1,i(λ)σk(λ)R11-ii-(λ1-1-λi+1)+∑i=1m‍σk-1,i(λ)σk(λ)(λi-1).
Since B1 assumes its maximum at P0, we have at P0 that L(B1)≤0. So we are left with the following inequality at P0:
(6.15)0≥∑i=2m‍σk-1,i(λ)σk(λ)(-R11-ii-)(λ1-λi)-∑i=1m‍σk-1,i(λ)σk(λ)λi+∑i=1m‍σk-1,i(λ)σk(λ)+t∂11-f.Curvature Assumption
Henceforth, we will suppose that the holomorphic bisectional curvature is nonnegative at anyP∈M. Thus in a holomorphic normal chart centered at P we have Raa-bb-(P)≤0 for all 1≤a, b≤m. This holds in particular at P0 in the previous chart ψ0. This assumption will be used only to derive an a priori eigenvalues pinching and is not required in the other sections.Back to the inequality (6.15), we have σk(λ)>0 and σk-1,i(λ)>0 since λ∈Γk, and under our curvature assumption (-R11-ii-)≥0 for all i≥2. Besides, λi≤λ1 for all i; therefore ∑i=2m‍(σk-1,i(λ)/σk(λ))(-R11-ii-)(λ1-λi)≥0. So we can get rid of the curvature terms in (6.15) and infer from it the inequality
(6.16)0≥-∑i=1mσk-1,i(λ)σk(λ)λi+∑i=1mσk-1,i(λ)σk(λ)+t∂11-f.


#### 6.2.6. Aλ1’s Upper Bound

Here, we require elementary identities satisfied by theσℓ's [11], namely:
(6.17)∀1≤ℓ≤mσℓ(λ)=σℓ,i(λ)+λiσℓ-1,i(λ),∀1≤ℓ≤m∑i=1mσℓ-1,i(λ)λi=ℓσℓ(λ),so in particular∑i=1mσk-1,i(λ)σk(λ)λi=k,∀1≤ℓ≤m∑i=1mσℓ,i(λ)=(m-ℓ)σℓ(λ),so in particular∑i=1mσk-1,i(λ)σk(λ)=(m-k+1)σk-1(λ)σk(λ).
Consequently, (6.16) writes:
(6.18)qk:=(m-k+1)kσk-1(λ)σk(λ)≤1-tk∂11-f.
So qk≤1+(1/k)|∂11-f|. But at P0, |∇2f|g2=2gac-gdb-(∇ab-f∇c-df+∇adf∇c-b-f)=2∑a,b=1m‍(|∂ab-f|2+|∂abf|2), then |∂11-f|≤|∇2f|g, and consequently qk≤1+(1/k)∥f∥C2(M)=:C1. In other words, there exists a constant C1 independent of t∈[0,1] such that
(6.19)qk≤C1.
To proceed further, we recall the followingLemma 6.3 (Newton inequalities).
For allℓ≥2, λ∈ℝm:
(6.20)σℓ(λ)σℓ-2(λ)≤(ℓ-1)(m-ℓ+1)ℓ(m-ℓ+2)[σℓ-1(λ)]2.Let us use Newton inequalities to relateqk to σ1. Since for 2≤ℓ≤k and λ∈Γk we have σℓ(λ)>0,σℓ-1(λ)>0 and σℓ-2(λ)>0 (σ0(λ)=1 by convention), Newton inequalities imply then that ((m-ℓ+2)/(ℓ-1))(σℓ-2(λ)/σℓ-1(λ))≤((m-ℓ+1)/ℓ)(σℓ-1(λ)/σℓ(λ)), or else qℓ-1≤qℓ, consequently qk≥qk-1≥⋯≥q2=(m-1)σ1(λ)/2σ2(λ). By induction, we get σ1(λ)≤(ℓ!(m-ℓ)!/(m-1)!)σℓ(λ)(qℓ)ℓ-1 for all 2≤ℓ≤k. In particular
(6.21)σ1(λ)≤k!(m-k)!(m-1)!σk(λ)(qk)k-1.
But σk(λ)≤e2∥f∥∞(mk); combining this with (6.19) and (6.21) we obtain at P0 that
(6.22)σ1(λ)≤me2∥f∥∞(C1)k-1=:C2.
Hence we may state the following.Theorem 6.4.
There exists a constantC2>0 depending only on m,k,∥f∥∞ and ∥f∥C2 such that for all 1≤i≤mλi(P0)≤C2.Combining this result with theC0 a priori estimate ∥φt∥C0≤C0 immediately yields the following.Theorem 6.5.
There exists a constantC2′>0 depending only on m,k,∥f∥C2 and C0 such that for all P∈M, for all 1≤i≤m, λi(P)≤C2+2C0=:C2′.


#### 6.2.7. Uniform Pinching of the Eigenvalues

We infer automatically the following pinchings of the eigenvalues.Proposition 6.6.
For all1≤i≤m, -(m-1)C2≤λi(P0)≤C2.Proposition 6.7.
For allP∈M, for all 1≤i≤m,-(m-1)C2′≤λi(P)≤C2′.


### 6.3. Uniform Ellipticity of the Continuity Equation

To prove the next proposition on uniform ellipticity, we require some inequalities satisfied by theσℓ’s.Lemma 6.8 (Maclaurin inequalities).
For all1≤ℓ≤s  for all λ∈Γs¯,(σs(λ)/(ms))1/s≤(σℓ(λ)/(mℓ))1/ℓ.Proposition 6.9 (uniform ellipticity).
There exist constantsE>0 and F>0 depending only on m,k,∥f∥∞ and C2 such that: E≤σk-1,1(λ)≤⋯≤σk-1,m(λ)≤F where λ=λ(P0).Proof.
We have∂σk/∂λ1=σk-1,1(λ)≤⋯≤σk-1,m(λ)≤(m-1k-1)(C2)k-1=:F where, indeed, the constant F so defined depends only on m,k, and C2. Let us look for a uniform lower bound on σk-1,1(λ), using the identity σk(λ)=λ1σk-1,1(λ)+σk,1(λ). We distinguish two cases.
Case  1. (σk,1(λ)≤0). When so, we have σk(λ)≤λ1σk-1,1(λ); therefore σk-1,1(λ)≥σk(λ)/λ1. But σk(λ)≥e-2∥f∥∞(mk) and 0<λ1≤C2; hence σk-1,1(λ)≥e-2∥f∥∞(mk)/C2.
Case  2. (σk,1(λ)>0). For 1≤j≤k-1,σj(λ2,…,λm)=σj,1(λ)>0 since j+1≤k and λ∈Γk. Besides σk(λ2,…,λm)=σk,1(λ)>0 by hypothesis, therefore (λ2,…,λm)∈Γk,1={β∈ℝm-1/∀1≤j≤k,σj(β)>0}. From the latter, we infer by Maclaurin inequalities (σk(λ2,…,λm)/(m-1k))1/k≤(σk-1(λ2,…,λm)/(m-1k-1))1/(k-1) or else (σk,1(λ)/(m-1k))1/k≤(σk-1,1(λ)/(m-1k-1))1/(k-1); thus we have σk,1(λ)≤(m-1k)(σk-1,1(λ)/(m-1k-1))1+1/(k-1), consequently
(6.23)σk(λ)=λ1σk-1,1(λ)+σk,1(λ)≤λ1σk-1,1(λ)+(m-1k)(σk-1,1(λ)(m-1k-1))1+1/(k-1)≤σk-1,1(λ)[λ1+(m-1k)(m-1k-1)(σk-1,1(λ)(m-1k-1))1/(k-1)].
Here, let us distinguish two subcases of Case 2.(i)
Ifσk-1,1(λ)>(m-1k-1), then we have the uniform lower bound that we look for.(ii)
Elseσk-1,1(λ)≤(m-1k-1), thus (σk-1,1(λ)/(m-1k-1))1/(k-1)≤1, therefore σk(λ)≤σk-1,1(λ)[λ1+(m-1k)/(m-1k-1)]=σk-1,1(λ)(λ1+m/k-1); then we get σk-1,1(λ)≥σk(λ)/(λ1+m/k-1)≥e-2∥f∥∞(mk)/(C2+m/k-1). Consequently σk-1,1(λ)≥min(e-2∥f∥∞(mk)/C2,(m-1k-1),e-2∥f∥∞(mk)/(C2+m/k-1)) or finally σk-1,1(λ)≥min((m-1k-1),e-2∥f∥∞(mk)/(C2+m/k-1))=:E, where the constant E so defined depends only on m,k,∥f∥∞ and C2.Similarly we prove the following.Proposition 6.10 (uniform ellipticity).
There exists constantsE0>0 and F0>0 depending only on m,k,∥f∥∞ and C2′ such that for all P∈M, for all 1≤i≤m, E0≤σk-1,i(λ(P))≤F0.


### 6.4. Gradient Uniform Estimate

The manifoldM is Riemannian compact and φt∈C2(M), so by the Green's formula
(6.24)φt(P)=1Vol(M)∫Mφt(Q)dvg(Q)+∫MG(P,Q)▵φt(Q)dvg(Q),
where G(P,Q) is the Green's function of the Laplacian ▵. By differentiating locally under the integral sign we obtain ∂uiφt(P)=∫M‍▵φt(Q)(∂ui)PG(P,Q)dvg(Q). We infer then that at P in a holomorphic normal chart, we have
(6.25)|(∇φt)P|≤2m∫M‍|▵φt(Q)||∇PG(P,Q)|dvg(Q).
Now, using the uniform pinching of the eigenvalues, we prove easily the following estimate of the Laplacian.Lemma 6.11.
There exists a constantC3>0 depending on m and C2′ such that ∥▵φt∥∞,M≤C3.Combining Lemma6.11 with (6.25), we deduce that |(∇φt)P|≤2mC3∫M|∇PG(P,Q)|dvg(Q). Besides, classically [13, page 109], there exists constants 𝒞 and 𝒞' such that
(6.26)|∇PG(P,Q)|≤𝒞dg(P,Q)2m-1,∫M‍1dg(P,Q)2m-1dvg(Q)≤𝒞′.
We thus obtain the following result.Proposition 6.12.
There exists a constantC5>0 depending on m,C2′, and (M,g) such that for all P∈M|(∇φt)P|≤C5.Specifically, we can chooseC5=2mC3𝒞𝒞'.


### 6.5. Second Derivatives Estimate

Our equation is of type:(E)F(P,[∂uiujφ]1≤i,j≤2m)=v,P∈M.


#### 6.5.1. The Functional

Consider the following functional:(6.27)Φ:UT→ℝ(P,ξ)↦(∇2φt)P(ξ,ξ)+12|(∇φt)P|g2,
where UT is the real unit sphere bundle associated to (TM,g). Φ is continuous on the compact set UT, so it assumes its maximum at a point (P1,ξ1)∈UT.


#### 6.5.2. Reduction to Finding a One-Sided Estimate for(∇2φt)P1(ξ1,ξ1)

If we find a uniform upper bound for(∇2φt)P1(ξ1,ξ1), since |∇φt|∞≤C5, we readily deduce that there exists a constant C6>0 such that
(6.28)(∇2φt)P(ξ,ξ)≤C6∀(P,ξ)∈UT.
Fix P∈M. Let (UP,ψP) be a holomorphic g-normal g~-adapted chart centered at P, namely, [gij-(P)]1≤i,j≤m=Im, ∂ℓgij-(P)=0 and [g~ij-(P)]1≤i,j≤m=[diag(λ1(P),…,λm(P))]. Since |∂xj|g=2, we obtain ∂xjxjφt(P)=2(∇2φt)P(∂xj/2,∂xj/2)≤2C6 and similarly ∂yjyjφt(P)=2(∇2φt)P(∂yj/2,∂yj/2)≤2C6 for all 1≤j≤m. Besides, we have ∂xjxjφt(P)+∂yjyjφt(P)=4∂jj-φt(P)=4(λj(P)-1)≥-4[(m-1)C2′+1]; therefore we obtain
(6.29)∂xjxjφt(P)≥-4[(m-1)C2′+1]-2C6=:-C7,∂yjyjφt(P)≥-C7,∀1≤j≤m.
Let us now bound second derivatives of mixed type ∂urusφt(P). Let 1≤r≠s≤2m. Since |∂xr±∂xs|g=2, we have (∇2φt)P((∂xr±∂xs)/2,(∂xr±∂xs)/2)=(1/4)∂xrxrφt(P)+(1/4)∂xsxsφt(P)±(1/2)∂xrxsφt(P)≤C6, which yields ±∂xrxsφt(P)≤2C6-(1/2)∂xrxrφt(P)-(1/2)∂xsxsφt(P), hence as well |∂xrxsφt(P)|≤2C6+C7. Similarly we prove that at P, in the above chart ψP, we have |∂yrysφt(P)|≤2C6+C7 for all 1≤r≠s≤m and |∂xrysφt(P)|≤2C6+C7 for all 1≤r,s≤m. Consequently |∂uiujφt(P)|≤2C6+C7 for all 1≤i, j≤2m. Therefore we deduce that
(6.30)|(∇2φt)(P)|g2=14∑1≤i,j≤2m‍(∂uiujφt(P))2≤m2(2C6+C7)2.Theorem 6.13 (second derivatives uniform estimate).
There exists a constantC8>0 depending only on m,C2′, and C6 such that for all P∈M, |(∇2φt)P|g≤C8.This allows to deduce the needed uniformC2 estimate:
(6.31)∥φ∥C2(M,ℝ)≤C0+C5+C8.


#### 6.5.3. Chart Choice

For fixedP∈M, ξ∈UTP↦(∇2φt)P(ξ,ξ) is continuous on the compact subset UTP (the fiber); therefore it assumes its maximum at a unit vector ξP∈UTP. Besides, (∇2φt)P is a symmetric bilinear form on TPM, so by the min-max principle we have (∇2φt)P(ξP,ξP)=maxξ∈TPM,g(ξ,ξ)=1(∇2φt)P(ξ,ξ)=βmax(P), where βmax(P) denotes the largest eigenvalue of (∇2φt)P with respect to gP; furthermore we can choose ξP as the direction of the largest eigenvalue βmax(P). For fixed P, we now have maxξ∈TPM,gP(ξ,ξ)=1Φ(P,ξ)=Φ(P,ξP)=(∇2φt)P(ξP,ξP)+(1/2)|(∇φt)P|g2=βmax(P)+(1/2)|(∇φt)P|g2, consequently max(P,ξ)∈UTΦ(P,ξ)=maxP∈MΦ(P,ξP)=Φ(P1,ξ1)≤Φ(P1,ξP1), hence max(P,ξ)∈UTΦ(P,ξ)=Φ(P1,ξP1)=βmax(P1)+(1/2)|(∇φt)P1|g2.At the pointP1, consider ε1P1,…,ε2mP1 a (real) basis of (TP1M,gP1) that satisfies the following properties: (i)
[gij(P1)]1≤i,j≤2m=I2m,(ii)
[(∇2φt)ij(P0)]1≤i,j≤2m=diag(β1,…,β2m),(iii)
β1=βmax(P1)≥β2≥⋯≥β2m. Let (U1′,ψ1) be a C∞g-normal real chart at P1 obtained from a holomorphic chart z1,…,zm by setting (u1,…,u2m)=(x1,…,xm,y1,…,ym) where zj=xj+iyj (namely, [gij(P1)]1≤i,j≤2m=I2m and ∂uℓgij=0 for all 1≤i,j,ℓ≤2m) satisfying ψ1(P1)=0 and ∂ui|P1=εiP1, so that ∂u1|P1 is the direction of the largest eigenvalue βmax(P1).


#### 6.5.4. Auxiliary Local Functional

From now on, we work in thereal chart(U1′,ψ1) so constructed at P1.LetU2⊂U1′ be an open subset such that g11(P)>0 for all P∈U2, and let Φ1 be the functional
(6.32)Φ1:U2→ℝP↦Φ1(P)=(∇2φt)11(P)g11(P)+12|(∇φt)P|g2.
We claim that Φ1 assumes its maximum at P1. Indeed, (∇2φt)11(P)/g11(P)=(∇2φ)P(∂u1,∂u1)/gP(∂u1,∂u1)=(∇2φ)P(∂u1/|∂u1|g,∂u1/|∂u1|g)≤βmax(P), so Φ1(P)≤βmax(P)+(1/2)|(∇φt)P|g2≤βmax(P1)+(1/2)|(∇φt)P1|g2=Φ1(P1) proving our claim.Let us now differentiate twice in thereal direction∂u1 the equation
(E*)F(P,[∂uiujφ]1≤i,j≤2m)=v.
At the point P, in a chart u, we obtain
(6.33)∂u1v=∂F∂u1[φ]+∑i,j=12m‍∂F∂rij[φ]∂u1uiujφ.
Differentiating once again
(6.34)∂u1u1v=∂2F∂u1∂u1[φ]+∑i,j=12m‍∂2F∂rij∂u1[φ]∂u1uiujφ+∑i,j=12m‍[∂2F∂u1∂rij[φ]+∑e,s=12m‍∂2F∂res∂rij[φ]∂u1ueusφ]∂u1uiujφ+∑i,j=12m‍∂F∂rij[φ]∂u1u1uiujφ.
But at the point P1, for our function F(P,r)=Fk[δij+(1/4)gjℓ-(P)(riℓ+r(i+m)(ℓ+m)+iri(ℓ+m)-ir(i+m)ℓ)]1≤i,j≤m, we have (∂2F/∂rij∂u1)[φ]=0 since ∂u1gsq-(P1)=0. Hence, we infer that
(6.35)∂u1u1v=∂2F∂u1∂u1[φ]+∑i,j,e,s=12m‍∂2F∂res∂rij[φ]∂u1ueusφ∂u1uiujφ+∑i,j=12m‍∂F∂rij[φ]∂u1u1uiujφ.


#### 6.5.5. Using Concavity

The functionF is concave with respect to the variable r. Indeed
(6.36)F(P,r)=Fk[δij+14gjℓ-(P)(riℓ+r(i+m)(ℓ+m)+iri(ℓ+m)-ir(i+m)ℓ)]1≤i,j≤m=Fk(g-1(P)r~),wherer~=[gij-(P)+14(rij+r(i+m)(j+m)+iri(j+m)-ir(i+m)j)]1≤i,j≤m=Fk(g-1/2(P)r~g-1/2(P)︸)∈ℋm(ℂ)=Fk(ρP(r)),whereρP(r)≔[δij+14∑ℓ,s=1m‍(g-1/2(P))iℓ(g-1/2(P))sj(rℓs+r(ℓ+m)(s+m)+irℓ(s+m)-ir(ℓ+m)s)]1≤i,j≤m
but for a fixed point P the function r∈S2m(ℝ)↦ρP(r)∈ℋm(ℂ) is affine (where S2m(ℝ) denotes the set of symmetric matrices of size 2m); we deduce then that the composition F(P,·)=Fk∘ρP is concave on the set {r∈S2m(ℝ)/ρP(r)∈λ-1(Γk)}=ρP-1(λ-1(Γk)), which proves our claim. Hence, since the matrix [∂u1uiujφ]1≤i,j≤m is symmetric, we obtain that
(6.37)∑i,j,e,s=12m‍∂2F∂res∂rij[φ]∂u1ueusφ∂u1uiujφ≤0.
Consequently
(6.38)∂u1u1v-∂u1u1F[φ]≤∑i,j=12m‍∂F∂rij[φ]∂u1u1uiujφ.
Let us now calculate the quantity ∂u1u1F[φ] (at P1). Since ∂u1gsq-(P1)=0, we have
(6.39)∂2F∂u1∂u1(P1,D2φ(P1))=∑s=1m‍σk-1,s(λ(P1))σk(λ(P1))×∂u1u1gss-(P1)∂ss-φ(P1).
But at P1, ∂u1u1gss-=-gso-gqs-∂u1u1gqo- and [gij-]1≤i,j≤m=2Im, then ∂u1u1gss-=-4∂u1u1gss- so that ∂u1u1gss-=-∂u1u1gss-∂u1u1g(s+m)(s+m). Moreover Γujusur=(1/2)(∂ujgos+∂usgoj-∂uogjs)gor, thus ∂uiΓujusur=(1/2)(∂uiujgrs+∂uiusgrj-∂uiurgjs). Similarly, we have at P1 : ∂uiΓujurus=(1/2)(∂uiujgrs+∂uiurgsj-∂uiusgjr). Consequently, we deduce that ∂uiujgrs=∂uiΓujusur+∂uiΓujurus. Hence, we have at P1: ∂u1u1gss-=-2∂u1Γu1usus-2∂u1Γu1us+mus+m. Besides, ∂ss-φ=(1/4)(∂ususφ+∂us+mus+mφ), which infers that at P1(6.40)∂u1u1F[φ]=∑s=1m‍σk-1,s(λ(P1))σk(λ(P1))×(-2∂u1Γu1usus-2∂u1Γu1us+mus+m)×14(∂ususφ+∂us+mus+mφ).
Consequently, the inequality (6.38) becomes
(6.41)∂u1u1v≤∑i,j=12m‍∂F∂rij[φ]∂u1u1uiujφ-12∑s=1m‍σk-1,s(λ(P1))σk(λ(P1))×(∂u1Γu1usus+∂u1Γu1us+mus+m)×(∂ususφ+∂us+mus+mφ).


#### 6.5.6. Differentiation of the FunctionalΦ1

We differentiate twice the functionalΦ1:
(6.42)Φ1(P)=(∇2φ)11(P)g11(P)+12|(∇φ)P|g2,∂ujΦ1(P)=∂uj(∇2φ)11g11(P)-(∇2φ)11∂ujg11(P)g11(P)2+12∂uj|(∇φ)P|g2,∂uiujΦ1(P)=∂uiuj(∇2φ)11g11(P)-∂uj(∇2φ)11∂uig11(P)g11(P)2-∂ui(∇2φ)11∂ujg11(P)+(∇2φ)11(P)∂uiujg11(P)g11(P)2-(∇2φ)11(P)∂ujg11(P)∂ui(1g11(P)2)+12∂uiuj|(∇φ)P|g2.
Hence, at P1 in the chart ψ1, we obtain
(6.43)∂uiujΦ1=∂uiuj(∇2φ)11-(∇2φ)11(P1)∂uiujg11+12∂uiuj|(∇φ)P|g2(P1).
Let us now calculate the different terms of this formula (at P1 in the chart ψ1):
(6.44)∂uiuj(∇2φ)11=∂uiuj(∂u1u1φ-Γu1u1us∂usφ)=∂uiuju1u1φ-∂uiujΓu1u1us∂usφ-∂ujΓu1u1us∂uiusφ-∂uiΓu1u1us∂ujusφ.
Besides, we have Γuju1u1=(1/2)(∂ujgs1+∂u1gsj-∂usgj1)gs1; therefore we deduce that ∂uiΓuju1u1=(1/2)(∂uiujgs1+∂uiu1gsj-∂uiusgj1)gs1+0=(1/2)∂uiujg11; namely, ∂uiujg11=2∂uiΓuju1u1. Moreover, we have at P1(6.45)∂uiuj|(∇φ)P|g2=∂uiuj(∑r,s=12m‍grs∂urφ∂usφ)=∑r,s=12m‍∂uiujgrs∂urφ∂usφ+grs∂uiujurφ∂usφ+grs∂ujurφ∂uiusφ+grs∂uiurφ∂ujusφ+grs∂urφ∂uiujusφ=∑r,s=12m‍∂uiujgrs∂urφ∂usφ+2∑s=12m‍∂uiujusφ∂usφ+2∑s=12m‍∂uiusφ∂ujusφ.
But at P1, ∂uiujgrs=-∂uiujgrs, in addition at this point ∂uiujgrs=∂uiΓujusur+∂uiΓujurus; therefore we obtain at P1 in the chart ψ1(6.46)∂uiuj|(∇φ)P|g2=-2∑r,s=12m∂uiΓujusur∂urφ∂usφ+2∑s=12m∂uiujusφ∂usφ+2∑s=12m∂uiusφ∂ujusφ.
Henceforth, and in order to lighten the notations, we use ∂i instead of ∂ui and Γijs instead of Γuiujus, so we have
(6.47)∂ijΦ1=∂ij11φ-∂ijΓ11s∂sφ-∂jΓ11s∂isφ-∂iΓ11s∂jsφ-2∂iΓj11(∇2φ)11(P1)-∑r,s=12m‍∂iΓjsr∂rφ∂sφ+∑s=12m‍∂ijsφ∂sφ+∑s=12m‍∂isφ∂jsφ.
Let us now consider the second order linear operator:
(6.48)L~=∑i,j=12m∂F∂rij[φ]∂ij.
Since the functional Φ1 assumes its maximum at the point P1, we have L~(Φ1)≤0 at P1 in the chart ψ1. Besides, combining the inequalities (6.41) and (6.47), we obtain
(6.49)2L~Φ1︸≤0-∂11v≥∑i,j=12m∂F∂rij[φ][∑r,s=12m∂iΓjsr∂rφ∂sφ∂ij11φ-∂ijΓ11s∂sφ-∂jΓ11i∂iiφ-∂iΓ11j∂jjφ-2∂iΓj11(∇2φ)11(P1)-∑r,s=12m∂iΓjsr∂rφ∂sφ+∑s=12m∂ijsφ∂sφ+δij(∂iiφ)2-∂11ijφ]+12∑s=1mσk-1,s(λ(P1))σk(λ(P1))(∂1Γ1ss+∂1Γ1(s+m)s+m)(∂ssφ+∂(s+m)(s+m)φ).
The fourth derivatives are simplified. Moreover, we have at P1 : ∂sv=(∂F/∂u1)[φ]+∑i,j=12m‍(∂F/∂rij)[φ]∂sijφ with (∂F/∂u1)(P1,D2φ(P1))=0, consequently:
(6.50)0≥∂11v+∑s=12m∂sv∂sφ+∑i,j=12m∂F∂rij[φ][∑r,s=12m‍-2∂iΓj11(∇2φ)11(P1)-∂iΓ11j∂jjφ-∂jΓ11i∂iiφ-∑s=12m∂ijΓ11s∂sφ-∑r,s=12m‍∂iΓjsr∂rφ∂sφ+δij(∂iiφ)2]+12∑s=1mσk-1,s(λ(P1))σk(λ(P1))(∂1Γ1ss+∂1Γ1(s+m)s+m)(∂ssφ+∂(s+m)(s+m)φ).
Let us now express the quantities ∂iΓj11,∂iΓ11j,∂jΓ11i,∂iΓjsr and ∂ijΓ11s using the components of the Riemann curvature (at the point P1 in the normal chart ψ1):
(6.51)∂iΓj11=13(Rj11i+Rji11︸=0)=13Rj11i,∂iΓ11j=13(R1j1i+R1i1j)=23R1j1i,∂jΓ11i=23R1i1j,∂iΓjsr=13(Rjrsi+Rjisr),∂1Γ1ss=13(R1ss1+R11ss︸=0)=13R1ss1,∂1Γ1(s+m)s+m=13R1(s+m)(s+m)1,∂ijΓ11s=14(∇iR1j1s+∇iR1s1j+∇jR1s1i+∇jR1i1s)-112(∇sR1i1j+∇sR1j1i)=12(∇iR1s1j+∇jR1s1i)-16∇sR1i1j.
We then obtain
(6.52)0≥∂11v+∑s=12m∂sv∂sφ+∑i,j=12m∂F∂rij[φ][∑i,j=12m∂F∂rij[φ]-23Rj11i(∇2φ)11(P1)-23R1j1i∂jjφ-23R1i1j∂iiφ-∑s=12m(12∇iR1s1j+12∇jR1s1i-16∇sR1i1j)∂sφ-∑r,s=12m13(Rjrsi+Rjisr)∂rφ∂sφ+δij(∂iiφ)2]+12∑s=1mσk-1,s(λ(P1))σk(λ(P1))13(R1ss1+R1(s+m)(s+m)1)(∂ssφ+∂(s+m)(s+m)φ).


#### 6.5.7. The Uniform Upper Bound ofβ1=(∇2φ)P1(ξ1,ξ1)

By the uniform estimate of the gradient we have|∂jφt|≤C5 for all 1≤j≤2m. Moreover, at P1 in the chart ψ1: [(∇2φ)ij(P1)]1≤i,j≤2m=[∂ijφ(P1)]1≤i,j≤2m=diag(β1,…,β2m). Consequently
(6.53)0≥∂11v+∑s=12m∂sv∂sφ+∑i,j=12m∂F∂rij[φ][δij(βi)2-23Rj11iβ1-23R1j1iβj-23R1i1jβi-13∑r,s=12m‍(Rjrsi+Rjisr)∂rφ∂sφ-12∑s=12m(∇iR1s1j+∇jR1s1i-13∇sR1i1j)∂sφ]+16∑i=1mσk-1,i(λ(P1))σk(λ(P1))×(R1ii1+R1(i+m)(i+m)1)(βi+βi+m).
But for F[φ]=Fk([δij+gjℓ-∂iℓ-φ]1≤i,j≤m) since ∂ss-φ=(1/4)(∂usus+∂us+mus+m), we obtain at P1 in the chart ψ1 that
(6.54)∂F∂rij[φ]=∑s=1m∂Fk∂Bss(diag(λ(P1)))14∂(rss+r(s+m)(s+m))∂rij.
Then
(6.55)∀1≤i≠j≤2m∂F∂rij[φ]=0,∀1≤i≤m∂F∂rii[φ]=∂F∂r(i+m)(i+m)[φ]=14∂Fk∂Bii(diag(λ(P1)))=14σk-1,i(λ(P1))σk(λ(P1))︸>0sinceλ(P1)∈Γk.
Hence
(6.56)0≥∂11v+∑s=12m∂sv∂sφ+∑i-12m∂F∂rii[φ]×[(βi)2+23R1i1i(β1-2βi)+13∑r,s=12m‍Riris∂rφ∂sφ-∑s=12m‍(∇iR1s1i-16∇sR1i1i)∂sφ]+16∑i=1mσk-1,i(λ(P1))σk(λ(P1))(R1ii1+R1(i+m)(i+m)1)(βi+βi+m).
But at P1 in the chart ψ1, ∥R∥g2=gaigbjgcrgdsRabcdRijrs=∑a,b,c,d=12m‍(Rabcd)2; then |Rabcd|≤∥R∥g for all a,b,c,d∈{1,…,2m}, consequently
(6.57)|13∑r,s=12mRiris∂rφ∂sφ|≤13∑r,s=12m∥R∥g(C5)2=13(2m)2∥R∥g(C5)2=43m2(C5)2∥R∥g.
Besides, at P1 in the chart ψ1, we have ∥∇R∥g2=gelgaigbjgcrgds∇eRabcd∇lRijrs=
∑
e
,
a
,
b
,
c
,
d
=
1
2
m
‍
(
∇
e
R
a
b
c
d
)
2, so |∇eRabcd|≤∥∇R∥g for all e,a,b,c,d∈{1,…,2m}, therefore
(6.58)|-∑s=12m(∇iR1s1i-16∇sR1i1i)∂sφ|≤∑s=12m76∥∇R∥gC5=2m76∥∇R∥gC5=73mC5∥∇R∥g.
Hence at P1 in the chart ψ1, we obtain
(6.59)-t∂11f-t∑s=12m∂sf∂sφ≥∑i=12m∂F∂rii[φ][(βi)2+23R1i1i(β1-2βi)]+16∑i=1mσk-1,i(λ(P1))σk(λ(P1))×(R1ii1+R1(i+m)(i+m)1)(βi+βi+m)+(∑i=12m∂F∂rii[φ])×[-43m2(C5)2∥R∥g-73mC5∥∇R∥g].
But |∂11f(P1)|≤∥f∥C2(M), |∂sf(P1)|≤∥f∥C2(M) and |∂sφ|≤C5 for all s then
(6.60)-t∂11f-t∑s=12m∂sf∂sφ≤∥f∥C2(M)(1+2mC5).
Besides
(6.61)∑i=12m‍∂F∂rii[φ]=∑i=1m‍∂F∂rii[φ]+∂F∂r(i+m)(i+m)[φ]=12∑i=1m‍σk-1,i(λ(P1))σk(λ(P1)).
Consequently, we obtain
(6.62)∥f∥C2(M)(1+2mC5)≥∂F∂r11[φ](β1)2+23∑i=12m∂F∂rii[φ]R1i1i(β1-2βi)+16∑i=1mσk-1,i(λ(P1))σk(λ(P1))×(R1ii1+R1(i+m)(i+m)1)(βi+βi+m)+12(∑i=1mσk-1,i(λ(P1))σk(λ(P1)))×[-43m2(C5)2∥R∥g-73mC5∥∇R∥g].
Let us now estimate |βi| for 1≤i≤m using β1. We follow the same method as for the proof of Theorem 6.13. For all (P,ξ)∈UT, we have the inequality (∇2φt)P(ξ,ξ)≤β1+(1/2)(C5)2; then at P in a holomorphic g-normal g~-adapted chart ψP, namely, a chart such that [gij-(P)]1≤i,j≤m=Im, ∂ℓgij-(P)=0 and [g~ij-(P)]1≤i,j≤m=diag(λ1(P),…,λm(P)), we deduce that for all j∈{1,…,m}(6.63)∂xjxjφt(P)=2(∇2φt)P(∂xj2,∂xj2)≤2β1+(C5)2,∂yjyjφt(P)≤2β1+(C5)2.
Since λj(P)≥-(m-1)C2′, we infer the following inequalities:
(6.64)∀j∈{1,…,m}∂xjxjφt(P)≥-4[(m-1)C2′+1]-2β1-(C5)2,∂yjyjφt(P)≥-4[(m-1)C2′+1]-2β1-(C5)2.
Consequently
(6.65)∀1≤i,j≤2m|∂uiujφt(P)|≤4β1+2(C5)2+4[(m-1)C2′+1]︸=:C9,
in the chart ψP.Hence we infer that(6.66)|(∇2φt)P|g2=14∑i,j=12m(∂uiujφt(P))2≤m2[4β1+2(C5)2+C9]2∀P.
But at P1 in the chart ψ1, |(∇2φt)P1|g2=∑i=12m‍(∂uiuiφt(P1))2=∑i=12m‍(βi)2; consequently we obtain
(6.67)∀1≤i≤2m|βi|≤m(4β1+2(C5)2+C9).
Thus
(6.68)|(R1i1i)(β1-2βi)|≤|R1i1i|(|β1|+2|βi|)≤3m∥R∥g(4β1+2(C5)2+C9).
Besides
(6.69)|(R1ii1+R1(i+m)(i+m)1)(βi+βi+m)|≤(|R1ii1|+|R1(i+m)(i+m)1|)(|βi|+|βi+m|)≤4m∥R∥g(4β1+2(C5)2+C9).
Hence
(6.70)∥f∥C2(M)(1+2mC5)≥14σk-1,1(λ(P1))σk(λ(P1))(β1)2+(∑i=1mσk-1,i(λ(P1))σk(λ(P1)))(-m)∥R∥g(4β1+2(C5)2+C9)+(∑i=1mσk-1,i(λ(P1))σk(λ(P1)))(-23m)∥R∥g(4β1+2(C5)2+C9)+(∑i=1mσk-1,i(λ(P1))σk(λ(P1)))[-76mC5∥∇R∥g-23m2(C5)2∥R∥g].
Then
(6.71)∥f∥C2(M)(1+2mC5)≥14σk-1,1(λ(P1))σk(λ(P1))(β1)2+(∑i=1mσk-1,i(λ(P1))σk(λ(P1)))(-53m)∥R∥g(4β1+2(C5)2+C9)+(∑i=1mσk-1,i(λ(P1))σk(λ(P1)))[-76mC5∥∇R∥g-23m2(C5)2∥R∥g].
But using the uniform ellipticity and the inequalities e-2∥f∥∞(mk)≤σk(λ(P))≤e2∥f∥∞(mk), we obtain
(6.72)∑i=1m‍σk-1,i(λ(P1))σk(λ(P1))≤me2∥f∥∞F0(mk),(6.73)σk-1,1(λ(P1))σk(λ(P1))≥e-2∥f∥∞E0(mk).
Then at P1 in the chart ψ1, we have
(6.74)0≥14e-2∥f∥∞E0(mk)(β1)2+me2∥f∥∞F0(mk)(-53m)∥R∥g(4β1+2(C5)2+C9)-me2∥f∥∞F0(mk)[76mC5∥∇R∥g+23m2(C5)2∥R∥g]-∥f∥C2(M)(1+2mC5).
The previous inequality means that some polynomial of second order in the variable β1 is negative:
(6.75)0≥14e-2∥f∥∞E0(mk)(β1)2+me2∥f∥∞F0(mk)(-203m)∥R∥gβ1-me2∥f∥∞F0(mk)[76mC5∥∇R∥g+23m2(C5)2∥R∥g+53m∥R∥g(2(C5)2+C9)]-∥f∥C2(M)(1+2mC5).
Set
(6.76)I:=803m2e4∥f∥∞F0E0∥R∥g>0,J≔4m2e4∥f∥∞F0E0[76C5∥∇R∥g+23m(C5)2∥R∥g+53(2(C5)2+C9)∥R∥g]+4(mk)e2∥f∥∞E0∥f∥C2(M)(1+2mC5)>0.
The previous inequality writes then:
(6.77)(β1)2-Iβ1-J≤0.
The discriminant of this polynomial of second order is equal to ▵=I2+4J>0, which gives an upper bound for β1.


## 6.1. Strategy for aC2 Estimate

First, we will look for a uniform upper bound on the eigenvaluesλ([δij+gjℓ-∂iℓ-φt]1≤i,j≤m. Secondly, we will infer from it the uniform ellipticity of the continuity equation (Ek,t) and a uniform gradient bound. Thirdly, with the uniform ellipticity at hand, we will derive a one-sided estimate on pure second derivatives and finally get the needed C2 bound.


## 6.2. Eigenvalues Upper Bound

### 6.2.1. The Functional

Lett∈𝒯l,α, and let φt : M→ℝ be a Cl,α  k-admissible solution of (Ek,t) satisfying ∫M‍φtωm=0. Consider the following functional:
(6.1)B:UT1,0→ℝ(P,ξ)↦B(P,ξ)=h~P(ξ,ξ)-φt(P),
where UT1,0 is the unit sphere bundle associated to (T1,0,h) and g~ is related to g by: ω~=ω+i∂∂-φt. B is continuous on the compact set UT1,0, so it assumes its maximum at a point (P0,ξ0)∈UT1,0. In addition, for fixed P∈M, ξ∈UTP1,0↦h~P(ξ,ξ) is continuous on the compact subset UTP1,0 (the fiber); therefore it attains its maximum at a unit vector ξP∈UTP1,0, and by the min-max principle we can choose ξP as the direction of the largest eigenvalue of AP, λmax(AP). Specifically, we have the following.Lemma 6.1 (min-max principle).
Consider(6.2)h~P(ξP,ξP)=maxξ∈TP1,0,hP(ξ,ξ)=1h~P(ξ,ξ)=λmax(AP).For fixedP, we have maxhP(ξ,ξ)=1B(P,ξ)=B(P,ξP)=λmax(AP)-φt(P); therefore max(P,ξ)∈UT1,0B(P,ξ)=maxP∈MB(P,ξP)=B(P0,ξ0)≤B(P0,ξP0); hence,
(6.3)max(P,ξ)∈UT1,0B(P,ξ)=B(P0,ξP0)=λmax(AP0)-φt(P0).
At the point P0, consider e1P0,…,emP0 an hP0-orthonormal basis of (TP01,0,hP0) made of eigenvectors of AP0 that satisfies the following properties: (1)
hP0-orthonormal: [hij(P0)]1≤i,j≤m=Im.(2)
h~P0-diagonal: [h~ij(P0)]1≤i,j≤m=MatAP0=diag(λ1,…,λm), λ∈Γk.(3)
λmax(AP0) is achieved in the direction e1P0=ξP0: AP0(ξP0)=λmax(AP0)ξP0=λ1ξP0 and λ1≥⋯≥λm. In other words, it is a basis satisfying(1)
[gij-(P0)]1≤i,j≤m=Im,(2)
[g~ij~(P0)]1≤i,j≤m=MatAP0=diag(λ1,…,λm), λ∈Γk,(3)
λmax(AP0)=λ1≥⋯≥λm. Let us consider a holomorphic normal chart (U0,ψ0) centered at P0 such that ψ0(P0)=0 and ∂i|P0=eiP0 for all i∈{1⋯m}.


### 6.2.2. Auxiliary Local Functional

From now on, we work in the chart(U0,ψ0) constructed at P0. The map P↦g11-(P) is continuous on U0 and is equal to 1 at P0, so there exists an open subset U1⊂U0 such that g11-(P)>0 for all P∈U1. Let B1 be the functional
(6.4)B1:U1→ℝP↦B1(P)=g~11-(P)g11-(P)-φt(P).
We claim that B1 assumes a local maximum at P0. Indeed, we have at each P∈U1: g~11-(P)/g11-(P)=g~P(∂1,∂1-)/gP(∂1,∂1-)=h~P(∂1,∂1)/hP(∂1,∂1)=h~P(∂1/|∂1|hP,∂1/|∂1|hP)≤λmax(AP) (see Lemma 6.1); thus B1(P)≤λmax(AP)-φt(P)≤λmax(AP0)-φt(P0)=B1(P0).


### 6.2.3. Differentiating the Equation

For short, we drop henceforth the subscriptt of φt. Let us differentiate (Ek,t) at P, in a chart z:
(6.5)t∂1-f=dFk[δij+gjℓ-(P)∂iℓ-φ(P)]1≤i,j≤m·[∂1-(gjℓ-∂iℓ-φ)]1≤i,j≤m=∑i,j=1m‍∂Fk∂Bij[δij+gjℓ-∂iℓ-φ](∂1-gjℓ-∂iℓ-φ+gjℓ-∂1-iℓ-φ).
Differentiating once again, we find
(6.6)t∂11-f=∑i,j,r,s=1m‍∂2Fk∂Brs∂Bij[δij+gjℓ-∂iℓ-φ](∂1gso-∂ro-φ+gso-∂1ro-φ)×(∂1-gjℓ-∂iℓ-φ+gjℓ-∂1-iℓ-φ)+∑i,j=1m‍∂Fk∂Bij[δij+gjℓ-∂iℓ-φ]×(∂11-gjℓ-∂iℓ-φ+∂1-gjℓ-∂1iℓ-φ+∂1gjℓ-∂1-iℓ-φ+gjℓ-∂11-iℓ-φ).
Using the above chart (U1,ψ0) at the point P0, normality yields gjℓ-=δjℓ, ∂αgiℓ-=0 and ∂αgiℓ-=0. Furthermore [δij+gjℓ-∂iℓ-φ]=[δij+∂ij-φ]=[g~ij-]=diag(λ1,…,λm). In this chart, we can simplify the previous expression; we get then at P0,
(6.7)t∂11-f=∑i,j,r,s=1m‍∂2Fk∂Brs∂Bij(diag(λ1,…,λm))∂1rs-φ∂1-ij-φ+∑i,j=1m‍∂Fk∂Bij(diag(λ1,…,λm))(∂11-gji-∂ii-φ+∂11-ij-φ).
Besides, ∂11-gji-=∂1-(-gjs-goi-∂1gos-), so still by normality, we obtain at P0 that ∂11-gji-=-gjs-goi-∂11-gos-=-∂11-gij--R11-ij-. Therefore we get
(6.8)t∂11-f=∑i,j=1m‍∂Fk∂Bij(diag(λ1,…,λm))(∂11-ij-φ-R11-ij-∂ii-φ)+∑i,j,r,s=1m‍∂2Fk∂Brs∂Bij(diag(λ1,…,λm))∂1rs-φ∂1-ij-φ.


### 6.2.4. Using Concavity

Now, using the concavity oflnσk [10], we prove for Proposition 2.1 that the second sum of (6.8) is negative [9, page 84]. This is not a direct consequence of the concavity of the function Fk since the matrix [∂1ij-φ]1≤i,j≤m is not Hermitian.Lemma 6.2.
Consider(6.9)S:=∑i,j,r,s=1m‍∂2Fk∂Brs∂Bij(diag(λ1,…,λm))∂1rs-φ∂1-ij-φ≤0.Hence, from (6.8) combined with Lemma 6.2 we infer
(6.10)t∂11-f≤∑i=1m‍σk-1,i(λ)σk(λ)(∂11-ii-φ-R11-ii-∂ii-φ).


### 6.2.5. Differentiation of the FunctionalB1

Let us differentiate twice the functionalB1:
(6.11)B1(P)=g~11-(P)g11-(P)-φ(P),∂i-B1=∂i-g~11-g11--g~11-∂i-g11-(g11-)2-∂i-φ,∂ii-B1=∂ii-g~11-g11--∂ig11-∂i-g~11-+∂ig~11-∂i-g11-+g~11-∂ii-g11-(g11-)2+2g~11-∂ig11-∂i-g11-(g11-)3-∂ii-φ.
Therefore at P0, in the above chart (U1,ψ0) we find ∂ii-B1=∂ii-(g11-+∂11-φ)-λ1∂ii-g11--∂ii-φ=R11-ii-+∂11-ii-φ-λ1R11-ii--∂ii-φ. Let us define the operator:
(6.12)L:=∑i,j=1m‍∂Fk∂Bij([δij+gjℓ-∂iℓ-φ]1≤i,j≤m)∇ij.
Thus, we have at P0(6.13)L(B1)=∑i=1m‍σk-1,i(λ)σk(λ)(∂11-ii-φ+(1-λ1)R11-ii--∂ii-φ).
Combining (6.13) with (6.10), we get rid of the fourth derivatives:
(6.14)t∂11-f-L(B1)≤∑i=1m‍σk-1,i(λ)σk(λ)R11-ii-(λ1-1-λi+1)+∑i=1m‍σk-1,i(λ)σk(λ)(λi-1).
Since B1 assumes its maximum at P0, we have at P0 that L(B1)≤0. So we are left with the following inequality at P0:
(6.15)0≥∑i=2m‍σk-1,i(λ)σk(λ)(-R11-ii-)(λ1-λi)-∑i=1m‍σk-1,i(λ)σk(λ)λi+∑i=1m‍σk-1,i(λ)σk(λ)+t∂11-f.Curvature Assumption
Henceforth, we will suppose that the holomorphic bisectional curvature is nonnegative at anyP∈M. Thus in a holomorphic normal chart centered at P we have Raa-bb-(P)≤0 for all 1≤a, b≤m. This holds in particular at P0 in the previous chart ψ0. This assumption will be used only to derive an a priori eigenvalues pinching and is not required in the other sections.Back to the inequality (6.15), we have σk(λ)>0 and σk-1,i(λ)>0 since λ∈Γk, and under our curvature assumption (-R11-ii-)≥0 for all i≥2. Besides, λi≤λ1 for all i; therefore ∑i=2m‍(σk-1,i(λ)/σk(λ))(-R11-ii-)(λ1-λi)≥0. So we can get rid of the curvature terms in (6.15) and infer from it the inequality
(6.16)0≥-∑i=1mσk-1,i(λ)σk(λ)λi+∑i=1mσk-1,i(λ)σk(λ)+t∂11-f.


### 6.2.6. Aλ1’s Upper Bound

Here, we require elementary identities satisfied by theσℓ's [11], namely:
(6.17)∀1≤ℓ≤mσℓ(λ)=σℓ,i(λ)+λiσℓ-1,i(λ),∀1≤ℓ≤m∑i=1mσℓ-1,i(λ)λi=ℓσℓ(λ),so in particular∑i=1mσk-1,i(λ)σk(λ)λi=k,∀1≤ℓ≤m∑i=1mσℓ,i(λ)=(m-ℓ)σℓ(λ),so in particular∑i=1mσk-1,i(λ)σk(λ)=(m-k+1)σk-1(λ)σk(λ).
Consequently, (6.16) writes:
(6.18)qk:=(m-k+1)kσk-1(λ)σk(λ)≤1-tk∂11-f.
So qk≤1+(1/k)|∂11-f|. But at P0, |∇2f|g2=2gac-gdb-(∇ab-f∇c-df+∇adf∇c-b-f)=2∑a,b=1m‍(|∂ab-f|2+|∂abf|2), then |∂11-f|≤|∇2f|g, and consequently qk≤1+(1/k)∥f∥C2(M)=:C1. In other words, there exists a constant C1 independent of t∈[0,1] such that
(6.19)qk≤C1.
To proceed further, we recall the followingLemma 6.3 (Newton inequalities).
For allℓ≥2, λ∈ℝm:
(6.20)σℓ(λ)σℓ-2(λ)≤(ℓ-1)(m-ℓ+1)ℓ(m-ℓ+2)[σℓ-1(λ)]2.Let us use Newton inequalities to relateqk to σ1. Since for 2≤ℓ≤k and λ∈Γk we have σℓ(λ)>0,σℓ-1(λ)>0 and σℓ-2(λ)>0 (σ0(λ)=1 by convention), Newton inequalities imply then that ((m-ℓ+2)/(ℓ-1))(σℓ-2(λ)/σℓ-1(λ))≤((m-ℓ+1)/ℓ)(σℓ-1(λ)/σℓ(λ)), or else qℓ-1≤qℓ, consequently qk≥qk-1≥⋯≥q2=(m-1)σ1(λ)/2σ2(λ). By induction, we get σ1(λ)≤(ℓ!(m-ℓ)!/(m-1)!)σℓ(λ)(qℓ)ℓ-1 for all 2≤ℓ≤k. In particular
(6.21)σ1(λ)≤k!(m-k)!(m-1)!σk(λ)(qk)k-1.
But σk(λ)≤e2∥f∥∞(mk); combining this with (6.19) and (6.21) we obtain at P0 that
(6.22)σ1(λ)≤me2∥f∥∞(C1)k-1=:C2.
Hence we may state the following.Theorem 6.4.
There exists a constantC2>0 depending only on m,k,∥f∥∞ and ∥f∥C2 such that for all 1≤i≤mλi(P0)≤C2.Combining this result with theC0 a priori estimate ∥φt∥C0≤C0 immediately yields the following.Theorem 6.5.
There exists a constantC2′>0 depending only on m,k,∥f∥C2 and C0 such that for all P∈M, for all 1≤i≤m, λi(P)≤C2+2C0=:C2′.


### 6.2.7. Uniform Pinching of the Eigenvalues

We infer automatically the following pinchings of the eigenvalues.Proposition 6.6.
For all1≤i≤m, -(m-1)C2≤λi(P0)≤C2.Proposition 6.7.
For allP∈M, for all 1≤i≤m,-(m-1)C2′≤λi(P)≤C2′.


## 6.2.1. The Functional

Lett∈𝒯l,α, and let φt : M→ℝ be a Cl,α  k-admissible solution of (Ek,t) satisfying ∫M‍φtωm=0. Consider the following functional:
(6.1)B:UT1,0→ℝ(P,ξ)↦B(P,ξ)=h~P(ξ,ξ)-φt(P),
where UT1,0 is the unit sphere bundle associated to (T1,0,h) and g~ is related to g by: ω~=ω+i∂∂-φt. B is continuous on the compact set UT1,0, so it assumes its maximum at a point (P0,ξ0)∈UT1,0. In addition, for fixed P∈M, ξ∈UTP1,0↦h~P(ξ,ξ) is continuous on the compact subset UTP1,0 (the fiber); therefore it attains its maximum at a unit vector ξP∈UTP1,0, and by the min-max principle we can choose ξP as the direction of the largest eigenvalue of AP, λmax(AP). Specifically, we have the following.Lemma 6.1 (min-max principle).
Consider(6.2)h~P(ξP,ξP)=maxξ∈TP1,0,hP(ξ,ξ)=1h~P(ξ,ξ)=λmax(AP).For fixedP, we have maxhP(ξ,ξ)=1B(P,ξ)=B(P,ξP)=λmax(AP)-φt(P); therefore max(P,ξ)∈UT1,0B(P,ξ)=maxP∈MB(P,ξP)=B(P0,ξ0)≤B(P0,ξP0); hence,
(6.3)max(P,ξ)∈UT1,0B(P,ξ)=B(P0,ξP0)=λmax(AP0)-φt(P0).
At the point P0, consider e1P0,…,emP0 an hP0-orthonormal basis of (TP01,0,hP0) made of eigenvectors of AP0 that satisfies the following properties: (1)
hP0-orthonormal: [hij(P0)]1≤i,j≤m=Im.(2)
h~P0-diagonal: [h~ij(P0)]1≤i,j≤m=MatAP0=diag(λ1,…,λm), λ∈Γk.(3)
λmax(AP0) is achieved in the direction e1P0=ξP0: AP0(ξP0)=λmax(AP0)ξP0=λ1ξP0 and λ1≥⋯≥λm. In other words, it is a basis satisfying(1)
[gij-(P0)]1≤i,j≤m=Im,(2)
[g~ij~(P0)]1≤i,j≤m=MatAP0=diag(λ1,…,λm), λ∈Γk,(3)
λmax(AP0)=λ1≥⋯≥λm. Let us consider a holomorphic normal chart (U0,ψ0) centered at P0 such that ψ0(P0)=0 and ∂i|P0=eiP0 for all i∈{1⋯m}.


## 6.2.2. Auxiliary Local Functional

From now on, we work in the chart(U0,ψ0) constructed at P0. The map P↦g11-(P) is continuous on U0 and is equal to 1 at P0, so there exists an open subset U1⊂U0 such that g11-(P)>0 for all P∈U1. Let B1 be the functional
(6.4)B1:U1→ℝP↦B1(P)=g~11-(P)g11-(P)-φt(P).
We claim that B1 assumes a local maximum at P0. Indeed, we have at each P∈U1: g~11-(P)/g11-(P)=g~P(∂1,∂1-)/gP(∂1,∂1-)=h~P(∂1,∂1)/hP(∂1,∂1)=h~P(∂1/|∂1|hP,∂1/|∂1|hP)≤λmax(AP) (see Lemma 6.1); thus B1(P)≤λmax(AP)-φt(P)≤λmax(AP0)-φt(P0)=B1(P0).


## 6.2.3. Differentiating the Equation

For short, we drop henceforth the subscriptt of φt. Let us differentiate (Ek,t) at P, in a chart z:
(6.5)t∂1-f=dFk[δij+gjℓ-(P)∂iℓ-φ(P)]1≤i,j≤m·[∂1-(gjℓ-∂iℓ-φ)]1≤i,j≤m=∑i,j=1m‍∂Fk∂Bij[δij+gjℓ-∂iℓ-φ](∂1-gjℓ-∂iℓ-φ+gjℓ-∂1-iℓ-φ).
Differentiating once again, we find
(6.6)t∂11-f=∑i,j,r,s=1m‍∂2Fk∂Brs∂Bij[δij+gjℓ-∂iℓ-φ](∂1gso-∂ro-φ+gso-∂1ro-φ)×(∂1-gjℓ-∂iℓ-φ+gjℓ-∂1-iℓ-φ)+∑i,j=1m‍∂Fk∂Bij[δij+gjℓ-∂iℓ-φ]×(∂11-gjℓ-∂iℓ-φ+∂1-gjℓ-∂1iℓ-φ+∂1gjℓ-∂1-iℓ-φ+gjℓ-∂11-iℓ-φ).
Using the above chart (U1,ψ0) at the point P0, normality yields gjℓ-=δjℓ, ∂αgiℓ-=0 and ∂αgiℓ-=0. Furthermore [δij+gjℓ-∂iℓ-φ]=[δij+∂ij-φ]=[g~ij-]=diag(λ1,…,λm). In this chart, we can simplify the previous expression; we get then at P0,
(6.7)t∂11-f=∑i,j,r,s=1m‍∂2Fk∂Brs∂Bij(diag(λ1,…,λm))∂1rs-φ∂1-ij-φ+∑i,j=1m‍∂Fk∂Bij(diag(λ1,…,λm))(∂11-gji-∂ii-φ+∂11-ij-φ).
Besides, ∂11-gji-=∂1-(-gjs-goi-∂1gos-), so still by normality, we obtain at P0 that ∂11-gji-=-gjs-goi-∂11-gos-=-∂11-gij--R11-ij-. Therefore we get
(6.8)t∂11-f=∑i,j=1m‍∂Fk∂Bij(diag(λ1,…,λm))(∂11-ij-φ-R11-ij-∂ii-φ)+∑i,j,r,s=1m‍∂2Fk∂Brs∂Bij(diag(λ1,…,λm))∂1rs-φ∂1-ij-φ.


## 6.2.4. Using Concavity

Now, using the concavity oflnσk [10], we prove for Proposition 2.1 that the second sum of (6.8) is negative [9, page 84]. This is not a direct consequence of the concavity of the function Fk since the matrix [∂1ij-φ]1≤i,j≤m is not Hermitian.Lemma 6.2.
Consider(6.9)S:=∑i,j,r,s=1m‍∂2Fk∂Brs∂Bij(diag(λ1,…,λm))∂1rs-φ∂1-ij-φ≤0.Hence, from (6.8) combined with Lemma 6.2 we infer
(6.10)t∂11-f≤∑i=1m‍σk-1,i(λ)σk(λ)(∂11-ii-φ-R11-ii-∂ii-φ).


## 6.2.5. Differentiation of the FunctionalB1

Let us differentiate twice the functionalB1:
(6.11)B1(P)=g~11-(P)g11-(P)-φ(P),∂i-B1=∂i-g~11-g11--g~11-∂i-g11-(g11-)2-∂i-φ,∂ii-B1=∂ii-g~11-g11--∂ig11-∂i-g~11-+∂ig~11-∂i-g11-+g~11-∂ii-g11-(g11-)2+2g~11-∂ig11-∂i-g11-(g11-)3-∂ii-φ.
Therefore at P0, in the above chart (U1,ψ0) we find ∂ii-B1=∂ii-(g11-+∂11-φ)-λ1∂ii-g11--∂ii-φ=R11-ii-+∂11-ii-φ-λ1R11-ii--∂ii-φ. Let us define the operator:
(6.12)L:=∑i,j=1m‍∂Fk∂Bij([δij+gjℓ-∂iℓ-φ]1≤i,j≤m)∇ij.
Thus, we have at P0(6.13)L(B1)=∑i=1m‍σk-1,i(λ)σk(λ)(∂11-ii-φ+(1-λ1)R11-ii--∂ii-φ).
Combining (6.13) with (6.10), we get rid of the fourth derivatives:
(6.14)t∂11-f-L(B1)≤∑i=1m‍σk-1,i(λ)σk(λ)R11-ii-(λ1-1-λi+1)+∑i=1m‍σk-1,i(λ)σk(λ)(λi-1).
Since B1 assumes its maximum at P0, we have at P0 that L(B1)≤0. So we are left with the following inequality at P0:
(6.15)0≥∑i=2m‍σk-1,i(λ)σk(λ)(-R11-ii-)(λ1-λi)-∑i=1m‍σk-1,i(λ)σk(λ)λi+∑i=1m‍σk-1,i(λ)σk(λ)+t∂11-f.Curvature Assumption
Henceforth, we will suppose that the holomorphic bisectional curvature is nonnegative at anyP∈M. Thus in a holomorphic normal chart centered at P we have Raa-bb-(P)≤0 for all 1≤a, b≤m. This holds in particular at P0 in the previous chart ψ0. This assumption will be used only to derive an a priori eigenvalues pinching and is not required in the other sections.Back to the inequality (6.15), we have σk(λ)>0 and σk-1,i(λ)>0 since λ∈Γk, and under our curvature assumption (-R11-ii-)≥0 for all i≥2. Besides, λi≤λ1 for all i; therefore ∑i=2m‍(σk-1,i(λ)/σk(λ))(-R11-ii-)(λ1-λi)≥0. So we can get rid of the curvature terms in (6.15) and infer from it the inequality
(6.16)0≥-∑i=1mσk-1,i(λ)σk(λ)λi+∑i=1mσk-1,i(λ)σk(λ)+t∂11-f.


## 6.2.6. Aλ1’s Upper Bound

Here, we require elementary identities satisfied by theσℓ's [11], namely:
(6.17)∀1≤ℓ≤mσℓ(λ)=σℓ,i(λ)+λiσℓ-1,i(λ),∀1≤ℓ≤m∑i=1mσℓ-1,i(λ)λi=ℓσℓ(λ),so in particular∑i=1mσk-1,i(λ)σk(λ)λi=k,∀1≤ℓ≤m∑i=1mσℓ,i(λ)=(m-ℓ)σℓ(λ),so in particular∑i=1mσk-1,i(λ)σk(λ)=(m-k+1)σk-1(λ)σk(λ).
Consequently, (6.16) writes:
(6.18)qk:=(m-k+1)kσk-1(λ)σk(λ)≤1-tk∂11-f.
So qk≤1+(1/k)|∂11-f|. But at P0, |∇2f|g2=2gac-gdb-(∇ab-f∇c-df+∇adf∇c-b-f)=2∑a,b=1m‍(|∂ab-f|2+|∂abf|2), then |∂11-f|≤|∇2f|g, and consequently qk≤1+(1/k)∥f∥C2(M)=:C1. In other words, there exists a constant C1 independent of t∈[0,1] such that
(6.19)qk≤C1.
To proceed further, we recall the followingLemma 6.3 (Newton inequalities).
For allℓ≥2, λ∈ℝm:
(6.20)σℓ(λ)σℓ-2(λ)≤(ℓ-1)(m-ℓ+1)ℓ(m-ℓ+2)[σℓ-1(λ)]2.Let us use Newton inequalities to relateqk to σ1. Since for 2≤ℓ≤k and λ∈Γk we have σℓ(λ)>0,σℓ-1(λ)>0 and σℓ-2(λ)>0 (σ0(λ)=1 by convention), Newton inequalities imply then that ((m-ℓ+2)/(ℓ-1))(σℓ-2(λ)/σℓ-1(λ))≤((m-ℓ+1)/ℓ)(σℓ-1(λ)/σℓ(λ)), or else qℓ-1≤qℓ, consequently qk≥qk-1≥⋯≥q2=(m-1)σ1(λ)/2σ2(λ). By induction, we get σ1(λ)≤(ℓ!(m-ℓ)!/(m-1)!)σℓ(λ)(qℓ)ℓ-1 for all 2≤ℓ≤k. In particular
(6.21)σ1(λ)≤k!(m-k)!(m-1)!σk(λ)(qk)k-1.
But σk(λ)≤e2∥f∥∞(mk); combining this with (6.19) and (6.21) we obtain at P0 that
(6.22)σ1(λ)≤me2∥f∥∞(C1)k-1=:C2.
Hence we may state the following.Theorem 6.4.
There exists a constantC2>0 depending only on m,k,∥f∥∞ and ∥f∥C2 such that for all 1≤i≤mλi(P0)≤C2.Combining this result with theC0 a priori estimate ∥φt∥C0≤C0 immediately yields the following.Theorem 6.5.
There exists a constantC2′>0 depending only on m,k,∥f∥C2 and C0 such that for all P∈M, for all 1≤i≤m, λi(P)≤C2+2C0=:C2′.


## 6.2.7. Uniform Pinching of the Eigenvalues

We infer automatically the following pinchings of the eigenvalues.Proposition 6.6.
For all1≤i≤m, -(m-1)C2≤λi(P0)≤C2.Proposition 6.7.
For allP∈M, for all 1≤i≤m,-(m-1)C2′≤λi(P)≤C2′.


## 6.3. Uniform Ellipticity of the Continuity Equation

To prove the next proposition on uniform ellipticity, we require some inequalities satisfied by theσℓ’s.Lemma 6.8 (Maclaurin inequalities).
For all1≤ℓ≤s  for all λ∈Γs¯,(σs(λ)/(ms))1/s≤(σℓ(λ)/(mℓ))1/ℓ.Proposition 6.9 (uniform ellipticity).
There exist constantsE>0 and F>0 depending only on m,k,∥f∥∞ and C2 such that: E≤σk-1,1(λ)≤⋯≤σk-1,m(λ)≤F where λ=λ(P0).Proof.
We have∂σk/∂λ1=σk-1,1(λ)≤⋯≤σk-1,m(λ)≤(m-1k-1)(C2)k-1=:F where, indeed, the constant F so defined depends only on m,k, and C2. Let us look for a uniform lower bound on σk-1,1(λ), using the identity σk(λ)=λ1σk-1,1(λ)+σk,1(λ). We distinguish two cases.
Case  1. (σk,1(λ)≤0). When so, we have σk(λ)≤λ1σk-1,1(λ); therefore σk-1,1(λ)≥σk(λ)/λ1. But σk(λ)≥e-2∥f∥∞(mk) and 0<λ1≤C2; hence σk-1,1(λ)≥e-2∥f∥∞(mk)/C2.
Case  2. (σk,1(λ)>0). For 1≤j≤k-1,σj(λ2,…,λm)=σj,1(λ)>0 since j+1≤k and λ∈Γk. Besides σk(λ2,…,λm)=σk,1(λ)>0 by hypothesis, therefore (λ2,…,λm)∈Γk,1={β∈ℝm-1/∀1≤j≤k,σj(β)>0}. From the latter, we infer by Maclaurin inequalities (σk(λ2,…,λm)/(m-1k))1/k≤(σk-1(λ2,…,λm)/(m-1k-1))1/(k-1) or else (σk,1(λ)/(m-1k))1/k≤(σk-1,1(λ)/(m-1k-1))1/(k-1); thus we have σk,1(λ)≤(m-1k)(σk-1,1(λ)/(m-1k-1))1+1/(k-1), consequently
(6.23)σk(λ)=λ1σk-1,1(λ)+σk,1(λ)≤λ1σk-1,1(λ)+(m-1k)(σk-1,1(λ)(m-1k-1))1+1/(k-1)≤σk-1,1(λ)[λ1+(m-1k)(m-1k-1)(σk-1,1(λ)(m-1k-1))1/(k-1)].
Here, let us distinguish two subcases of Case 2.(i)
Ifσk-1,1(λ)>(m-1k-1), then we have the uniform lower bound that we look for.(ii)
Elseσk-1,1(λ)≤(m-1k-1), thus (σk-1,1(λ)/(m-1k-1))1/(k-1)≤1, therefore σk(λ)≤σk-1,1(λ)[λ1+(m-1k)/(m-1k-1)]=σk-1,1(λ)(λ1+m/k-1); then we get σk-1,1(λ)≥σk(λ)/(λ1+m/k-1)≥e-2∥f∥∞(mk)/(C2+m/k-1). Consequently σk-1,1(λ)≥min(e-2∥f∥∞(mk)/C2,(m-1k-1),e-2∥f∥∞(mk)/(C2+m/k-1)) or finally σk-1,1(λ)≥min((m-1k-1),e-2∥f∥∞(mk)/(C2+m/k-1))=:E, where the constant E so defined depends only on m,k,∥f∥∞ and C2.Similarly we prove the following.Proposition 6.10 (uniform ellipticity).
There exists constantsE0>0 and F0>0 depending only on m,k,∥f∥∞ and C2′ such that for all P∈M, for all 1≤i≤m, E0≤σk-1,i(λ(P))≤F0.


## 6.4. Gradient Uniform Estimate

The manifoldM is Riemannian compact and φt∈C2(M), so by the Green's formula
(6.24)φt(P)=1Vol(M)∫Mφt(Q)dvg(Q)+∫MG(P,Q)▵φt(Q)dvg(Q),
where G(P,Q) is the Green's function of the Laplacian ▵. By differentiating locally under the integral sign we obtain ∂uiφt(P)=∫M‍▵φt(Q)(∂ui)PG(P,Q)dvg(Q). We infer then that at P in a holomorphic normal chart, we have
(6.25)|(∇φt)P|≤2m∫M‍|▵φt(Q)||∇PG(P,Q)|dvg(Q).
Now, using the uniform pinching of the eigenvalues, we prove easily the following estimate of the Laplacian.Lemma 6.11.
There exists a constantC3>0 depending on m and C2′ such that ∥▵φt∥∞,M≤C3.Combining Lemma6.11 with (6.25), we deduce that |(∇φt)P|≤2mC3∫M|∇PG(P,Q)|dvg(Q). Besides, classically [13, page 109], there exists constants 𝒞 and 𝒞' such that
(6.26)|∇PG(P,Q)|≤𝒞dg(P,Q)2m-1,∫M‍1dg(P,Q)2m-1dvg(Q)≤𝒞′.
We thus obtain the following result.Proposition 6.12.
There exists a constantC5>0 depending on m,C2′, and (M,g) such that for all P∈M|(∇φt)P|≤C5.Specifically, we can chooseC5=2mC3𝒞𝒞'.


## 6.5. Second Derivatives Estimate

Our equation is of type:(E)F(P,[∂uiujφ]1≤i,j≤2m)=v,P∈M.


### 6.5.1. The Functional

Consider the following functional:(6.27)Φ:UT→ℝ(P,ξ)↦(∇2φt)P(ξ,ξ)+12|(∇φt)P|g2,
where UT is the real unit sphere bundle associated to (TM,g). Φ is continuous on the compact set UT, so it assumes its maximum at a point (P1,ξ1)∈UT.


### 6.5.2. Reduction to Finding a One-Sided Estimate for(∇2φt)P1(ξ1,ξ1)

If we find a uniform upper bound for(∇2φt)P1(ξ1,ξ1), since |∇φt|∞≤C5, we readily deduce that there exists a constant C6>0 such that
(6.28)(∇2φt)P(ξ,ξ)≤C6∀(P,ξ)∈UT.
Fix P∈M. Let (UP,ψP) be a holomorphic g-normal g~-adapted chart centered at P, namely, [gij-(P)]1≤i,j≤m=Im, ∂ℓgij-(P)=0 and [g~ij-(P)]1≤i,j≤m=[diag(λ1(P),…,λm(P))]. Since |∂xj|g=2, we obtain ∂xjxjφt(P)=2(∇2φt)P(∂xj/2,∂xj/2)≤2C6 and similarly ∂yjyjφt(P)=2(∇2φt)P(∂yj/2,∂yj/2)≤2C6 for all 1≤j≤m. Besides, we have ∂xjxjφt(P)+∂yjyjφt(P)=4∂jj-φt(P)=4(λj(P)-1)≥-4[(m-1)C2′+1]; therefore we obtain
(6.29)∂xjxjφt(P)≥-4[(m-1)C2′+1]-2C6=:-C7,∂yjyjφt(P)≥-C7,∀1≤j≤m.
Let us now bound second derivatives of mixed type ∂urusφt(P). Let 1≤r≠s≤2m. Since |∂xr±∂xs|g=2, we have (∇2φt)P((∂xr±∂xs)/2,(∂xr±∂xs)/2)=(1/4)∂xrxrφt(P)+(1/4)∂xsxsφt(P)±(1/2)∂xrxsφt(P)≤C6, which yields ±∂xrxsφt(P)≤2C6-(1/2)∂xrxrφt(P)-(1/2)∂xsxsφt(P), hence as well |∂xrxsφt(P)|≤2C6+C7. Similarly we prove that at P, in the above chart ψP, we have |∂yrysφt(P)|≤2C6+C7 for all 1≤r≠s≤m and |∂xrysφt(P)|≤2C6+C7 for all 1≤r,s≤m. Consequently |∂uiujφt(P)|≤2C6+C7 for all 1≤i, j≤2m. Therefore we deduce that
(6.30)|(∇2φt)(P)|g2=14∑1≤i,j≤2m‍(∂uiujφt(P))2≤m2(2C6+C7)2.Theorem 6.13 (second derivatives uniform estimate).
There exists a constantC8>0 depending only on m,C2′, and C6 such that for all P∈M, |(∇2φt)P|g≤C8.This allows to deduce the needed uniformC2 estimate:
(6.31)∥φ∥C2(M,ℝ)≤C0+C5+C8.


### 6.5.3. Chart Choice

For fixedP∈M, ξ∈UTP↦(∇2φt)P(ξ,ξ) is continuous on the compact subset UTP (the fiber); therefore it assumes its maximum at a unit vector ξP∈UTP. Besides, (∇2φt)P is a symmetric bilinear form on TPM, so by the min-max principle we have (∇2φt)P(ξP,ξP)=maxξ∈TPM,g(ξ,ξ)=1(∇2φt)P(ξ,ξ)=βmax(P), where βmax(P) denotes the largest eigenvalue of (∇2φt)P with respect to gP; furthermore we can choose ξP as the direction of the largest eigenvalue βmax(P). For fixed P, we now have maxξ∈TPM,gP(ξ,ξ)=1Φ(P,ξ)=Φ(P,ξP)=(∇2φt)P(ξP,ξP)+(1/2)|(∇φt)P|g2=βmax(P)+(1/2)|(∇φt)P|g2, consequently max(P,ξ)∈UTΦ(P,ξ)=maxP∈MΦ(P,ξP)=Φ(P1,ξ1)≤Φ(P1,ξP1), hence max(P,ξ)∈UTΦ(P,ξ)=Φ(P1,ξP1)=βmax(P1)+(1/2)|(∇φt)P1|g2.At the pointP1, consider ε1P1,…,ε2mP1 a (real) basis of (TP1M,gP1) that satisfies the following properties: (i)
[gij(P1)]1≤i,j≤2m=I2m,(ii)
[(∇2φt)ij(P0)]1≤i,j≤2m=diag(β1,…,β2m),(iii)
β1=βmax(P1)≥β2≥⋯≥β2m. Let (U1′,ψ1) be a C∞g-normal real chart at P1 obtained from a holomorphic chart z1,…,zm by setting (u1,…,u2m)=(x1,…,xm,y1,…,ym) where zj=xj+iyj (namely, [gij(P1)]1≤i,j≤2m=I2m and ∂uℓgij=0 for all 1≤i,j,ℓ≤2m) satisfying ψ1(P1)=0 and ∂ui|P1=εiP1, so that ∂u1|P1 is the direction of the largest eigenvalue βmax(P1).


### 6.5.4. Auxiliary Local Functional

From now on, we work in thereal chart(U1′,ψ1) so constructed at P1.LetU2⊂U1′ be an open subset such that g11(P)>0 for all P∈U2, and let Φ1 be the functional
(6.32)Φ1:U2→ℝP↦Φ1(P)=(∇2φt)11(P)g11(P)+12|(∇φt)P|g2.
We claim that Φ1 assumes its maximum at P1. Indeed, (∇2φt)11(P)/g11(P)=(∇2φ)P(∂u1,∂u1)/gP(∂u1,∂u1)=(∇2φ)P(∂u1/|∂u1|g,∂u1/|∂u1|g)≤βmax(P), so Φ1(P)≤βmax(P)+(1/2)|(∇φt)P|g2≤βmax(P1)+(1/2)|(∇φt)P1|g2=Φ1(P1) proving our claim.Let us now differentiate twice in thereal direction∂u1 the equation
(E*)F(P,[∂uiujφ]1≤i,j≤2m)=v.
At the point P, in a chart u, we obtain
(6.33)∂u1v=∂F∂u1[φ]+∑i,j=12m‍∂F∂rij[φ]∂u1uiujφ.
Differentiating once again
(6.34)∂u1u1v=∂2F∂u1∂u1[φ]+∑i,j=12m‍∂2F∂rij∂u1[φ]∂u1uiujφ+∑i,j=12m‍[∂2F∂u1∂rij[φ]+∑e,s=12m‍∂2F∂res∂rij[φ]∂u1ueusφ]∂u1uiujφ+∑i,j=12m‍∂F∂rij[φ]∂u1u1uiujφ.
But at the point P1, for our function F(P,r)=Fk[δij+(1/4)gjℓ-(P)(riℓ+r(i+m)(ℓ+m)+iri(ℓ+m)-ir(i+m)ℓ)]1≤i,j≤m, we have (∂2F/∂rij∂u1)[φ]=0 since ∂u1gsq-(P1)=0. Hence, we infer that
(6.35)∂u1u1v=∂2F∂u1∂u1[φ]+∑i,j,e,s=12m‍∂2F∂res∂rij[φ]∂u1ueusφ∂u1uiujφ+∑i,j=12m‍∂F∂rij[φ]∂u1u1uiujφ.


### 6.5.5. Using Concavity

The functionF is concave with respect to the variable r. Indeed
(6.36)F(P,r)=Fk[δij+14gjℓ-(P)(riℓ+r(i+m)(ℓ+m)+iri(ℓ+m)-ir(i+m)ℓ)]1≤i,j≤m=Fk(g-1(P)r~),wherer~=[gij-(P)+14(rij+r(i+m)(j+m)+iri(j+m)-ir(i+m)j)]1≤i,j≤m=Fk(g-1/2(P)r~g-1/2(P)︸)∈ℋm(ℂ)=Fk(ρP(r)),whereρP(r)≔[δij+14∑ℓ,s=1m‍(g-1/2(P))iℓ(g-1/2(P))sj(rℓs+r(ℓ+m)(s+m)+irℓ(s+m)-ir(ℓ+m)s)]1≤i,j≤m
but for a fixed point P the function r∈S2m(ℝ)↦ρP(r)∈ℋm(ℂ) is affine (where S2m(ℝ) denotes the set of symmetric matrices of size 2m); we deduce then that the composition F(P,·)=Fk∘ρP is concave on the set {r∈S2m(ℝ)/ρP(r)∈λ-1(Γk)}=ρP-1(λ-1(Γk)), which proves our claim. Hence, since the matrix [∂u1uiujφ]1≤i,j≤m is symmetric, we obtain that
(6.37)∑i,j,e,s=12m‍∂2F∂res∂rij[φ]∂u1ueusφ∂u1uiujφ≤0.
Consequently
(6.38)∂u1u1v-∂u1u1F[φ]≤∑i,j=12m‍∂F∂rij[φ]∂u1u1uiujφ.
Let us now calculate the quantity ∂u1u1F[φ] (at P1). Since ∂u1gsq-(P1)=0, we have
(6.39)∂2F∂u1∂u1(P1,D2φ(P1))=∑s=1m‍σk-1,s(λ(P1))σk(λ(P1))×∂u1u1gss-(P1)∂ss-φ(P1).
But at P1, ∂u1u1gss-=-gso-gqs-∂u1u1gqo- and [gij-]1≤i,j≤m=2Im, then ∂u1u1gss-=-4∂u1u1gss- so that ∂u1u1gss-=-∂u1u1gss-∂u1u1g(s+m)(s+m). Moreover Γujusur=(1/2)(∂ujgos+∂usgoj-∂uogjs)gor, thus ∂uiΓujusur=(1/2)(∂uiujgrs+∂uiusgrj-∂uiurgjs). Similarly, we have at P1 : ∂uiΓujurus=(1/2)(∂uiujgrs+∂uiurgsj-∂uiusgjr). Consequently, we deduce that ∂uiujgrs=∂uiΓujusur+∂uiΓujurus. Hence, we have at P1: ∂u1u1gss-=-2∂u1Γu1usus-2∂u1Γu1us+mus+m. Besides, ∂ss-φ=(1/4)(∂ususφ+∂us+mus+mφ), which infers that at P1(6.40)∂u1u1F[φ]=∑s=1m‍σk-1,s(λ(P1))σk(λ(P1))×(-2∂u1Γu1usus-2∂u1Γu1us+mus+m)×14(∂ususφ+∂us+mus+mφ).
Consequently, the inequality (6.38) becomes
(6.41)∂u1u1v≤∑i,j=12m‍∂F∂rij[φ]∂u1u1uiujφ-12∑s=1m‍σk-1,s(λ(P1))σk(λ(P1))×(∂u1Γu1usus+∂u1Γu1us+mus+m)×(∂ususφ+∂us+mus+mφ).


### 6.5.6. Differentiation of the FunctionalΦ1

We differentiate twice the functionalΦ1:
(6.42)Φ1(P)=(∇2φ)11(P)g11(P)+12|(∇φ)P|g2,∂ujΦ1(P)=∂uj(∇2φ)11g11(P)-(∇2φ)11∂ujg11(P)g11(P)2+12∂uj|(∇φ)P|g2,∂uiujΦ1(P)=∂uiuj(∇2φ)11g11(P)-∂uj(∇2φ)11∂uig11(P)g11(P)2-∂ui(∇2φ)11∂ujg11(P)+(∇2φ)11(P)∂uiujg11(P)g11(P)2-(∇2φ)11(P)∂ujg11(P)∂ui(1g11(P)2)+12∂uiuj|(∇φ)P|g2.
Hence, at P1 in the chart ψ1, we obtain
(6.43)∂uiujΦ1=∂uiuj(∇2φ)11-(∇2φ)11(P1)∂uiujg11+12∂uiuj|(∇φ)P|g2(P1).
Let us now calculate the different terms of this formula (at P1 in the chart ψ1):
(6.44)∂uiuj(∇2φ)11=∂uiuj(∂u1u1φ-Γu1u1us∂usφ)=∂uiuju1u1φ-∂uiujΓu1u1us∂usφ-∂ujΓu1u1us∂uiusφ-∂uiΓu1u1us∂ujusφ.
Besides, we have Γuju1u1=(1/2)(∂ujgs1+∂u1gsj-∂usgj1)gs1; therefore we deduce that ∂uiΓuju1u1=(1/2)(∂uiujgs1+∂uiu1gsj-∂uiusgj1)gs1+0=(1/2)∂uiujg11; namely, ∂uiujg11=2∂uiΓuju1u1. Moreover, we have at P1(6.45)∂uiuj|(∇φ)P|g2=∂uiuj(∑r,s=12m‍grs∂urφ∂usφ)=∑r,s=12m‍∂uiujgrs∂urφ∂usφ+grs∂uiujurφ∂usφ+grs∂ujurφ∂uiusφ+grs∂uiurφ∂ujusφ+grs∂urφ∂uiujusφ=∑r,s=12m‍∂uiujgrs∂urφ∂usφ+2∑s=12m‍∂uiujusφ∂usφ+2∑s=12m‍∂uiusφ∂ujusφ.
But at P1, ∂uiujgrs=-∂uiujgrs, in addition at this point ∂uiujgrs=∂uiΓujusur+∂uiΓujurus; therefore we obtain at P1 in the chart ψ1(6.46)∂uiuj|(∇φ)P|g2=-2∑r,s=12m∂uiΓujusur∂urφ∂usφ+2∑s=12m∂uiujusφ∂usφ+2∑s=12m∂uiusφ∂ujusφ.
Henceforth, and in order to lighten the notations, we use ∂i instead of ∂ui and Γijs instead of Γuiujus, so we have
(6.47)∂ijΦ1=∂ij11φ-∂ijΓ11s∂sφ-∂jΓ11s∂isφ-∂iΓ11s∂jsφ-2∂iΓj11(∇2φ)11(P1)-∑r,s=12m‍∂iΓjsr∂rφ∂sφ+∑s=12m‍∂ijsφ∂sφ+∑s=12m‍∂isφ∂jsφ.
Let us now consider the second order linear operator:
(6.48)L~=∑i,j=12m∂F∂rij[φ]∂ij.
Since the functional Φ1 assumes its maximum at the point P1, we have L~(Φ1)≤0 at P1 in the chart ψ1. Besides, combining the inequalities (6.41) and (6.47), we obtain
(6.49)2L~Φ1︸≤0-∂11v≥∑i,j=12m∂F∂rij[φ][∑r,s=12m∂iΓjsr∂rφ∂sφ∂ij11φ-∂ijΓ11s∂sφ-∂jΓ11i∂iiφ-∂iΓ11j∂jjφ-2∂iΓj11(∇2φ)11(P1)-∑r,s=12m∂iΓjsr∂rφ∂sφ+∑s=12m∂ijsφ∂sφ+δij(∂iiφ)2-∂11ijφ]+12∑s=1mσk-1,s(λ(P1))σk(λ(P1))(∂1Γ1ss+∂1Γ1(s+m)s+m)(∂ssφ+∂(s+m)(s+m)φ).
The fourth derivatives are simplified. Moreover, we have at P1 : ∂sv=(∂F/∂u1)[φ]+∑i,j=12m‍(∂F/∂rij)[φ]∂sijφ with (∂F/∂u1)(P1,D2φ(P1))=0, consequently:
(6.50)0≥∂11v+∑s=12m∂sv∂sφ+∑i,j=12m∂F∂rij[φ][∑r,s=12m‍-2∂iΓj11(∇2φ)11(P1)-∂iΓ11j∂jjφ-∂jΓ11i∂iiφ-∑s=12m∂ijΓ11s∂sφ-∑r,s=12m‍∂iΓjsr∂rφ∂sφ+δij(∂iiφ)2]+12∑s=1mσk-1,s(λ(P1))σk(λ(P1))(∂1Γ1ss+∂1Γ1(s+m)s+m)(∂ssφ+∂(s+m)(s+m)φ).
Let us now express the quantities ∂iΓj11,∂iΓ11j,∂jΓ11i,∂iΓjsr and ∂ijΓ11s using the components of the Riemann curvature (at the point P1 in the normal chart ψ1):
(6.51)∂iΓj11=13(Rj11i+Rji11︸=0)=13Rj11i,∂iΓ11j=13(R1j1i+R1i1j)=23R1j1i,∂jΓ11i=23R1i1j,∂iΓjsr=13(Rjrsi+Rjisr),∂1Γ1ss=13(R1ss1+R11ss︸=0)=13R1ss1,∂1Γ1(s+m)s+m=13R1(s+m)(s+m)1,∂ijΓ11s=14(∇iR1j1s+∇iR1s1j+∇jR1s1i+∇jR1i1s)-112(∇sR1i1j+∇sR1j1i)=12(∇iR1s1j+∇jR1s1i)-16∇sR1i1j.
We then obtain
(6.52)0≥∂11v+∑s=12m∂sv∂sφ+∑i,j=12m∂F∂rij[φ][∑i,j=12m∂F∂rij[φ]-23Rj11i(∇2φ)11(P1)-23R1j1i∂jjφ-23R1i1j∂iiφ-∑s=12m(12∇iR1s1j+12∇jR1s1i-16∇sR1i1j)∂sφ-∑r,s=12m13(Rjrsi+Rjisr)∂rφ∂sφ+δij(∂iiφ)2]+12∑s=1mσk-1,s(λ(P1))σk(λ(P1))13(R1ss1+R1(s+m)(s+m)1)(∂ssφ+∂(s+m)(s+m)φ).


### 6.5.7. The Uniform Upper Bound ofβ1=(∇2φ)P1(ξ1,ξ1)

By the uniform estimate of the gradient we have|∂jφt|≤C5 for all 1≤j≤2m. Moreover, at P1 in the chart ψ1: [(∇2φ)ij(P1)]1≤i,j≤2m=[∂ijφ(P1)]1≤i,j≤2m=diag(β1,…,β2m). Consequently
(6.53)0≥∂11v+∑s=12m∂sv∂sφ+∑i,j=12m∂F∂rij[φ][δij(βi)2-23Rj11iβ1-23R1j1iβj-23R1i1jβi-13∑r,s=12m‍(Rjrsi+Rjisr)∂rφ∂sφ-12∑s=12m(∇iR1s1j+∇jR1s1i-13∇sR1i1j)∂sφ]+16∑i=1mσk-1,i(λ(P1))σk(λ(P1))×(R1ii1+R1(i+m)(i+m)1)(βi+βi+m).
But for F[φ]=Fk([δij+gjℓ-∂iℓ-φ]1≤i,j≤m) since ∂ss-φ=(1/4)(∂usus+∂us+mus+m), we obtain at P1 in the chart ψ1 that
(6.54)∂F∂rij[φ]=∑s=1m∂Fk∂Bss(diag(λ(P1)))14∂(rss+r(s+m)(s+m))∂rij.
Then
(6.55)∀1≤i≠j≤2m∂F∂rij[φ]=0,∀1≤i≤m∂F∂rii[φ]=∂F∂r(i+m)(i+m)[φ]=14∂Fk∂Bii(diag(λ(P1)))=14σk-1,i(λ(P1))σk(λ(P1))︸>0sinceλ(P1)∈Γk.
Hence
(6.56)0≥∂11v+∑s=12m∂sv∂sφ+∑i-12m∂F∂rii[φ]×[(βi)2+23R1i1i(β1-2βi)+13∑r,s=12m‍Riris∂rφ∂sφ-∑s=12m‍(∇iR1s1i-16∇sR1i1i)∂sφ]+16∑i=1mσk-1,i(λ(P1))σk(λ(P1))(R1ii1+R1(i+m)(i+m)1)(βi+βi+m).
But at P1 in the chart ψ1, ∥R∥g2=gaigbjgcrgdsRabcdRijrs=∑a,b,c,d=12m‍(Rabcd)2; then |Rabcd|≤∥R∥g for all a,b,c,d∈{1,…,2m}, consequently
(6.57)|13∑r,s=12mRiris∂rφ∂sφ|≤13∑r,s=12m∥R∥g(C5)2=13(2m)2∥R∥g(C5)2=43m2(C5)2∥R∥g.
Besides, at P1 in the chart ψ1, we have ∥∇R∥g2=gelgaigbjgcrgds∇eRabcd∇lRijrs=
∑
e
,
a
,
b
,
c
,
d
=
1
2
m
‍
(
∇
e
R
a
b
c
d
)
2, so |∇eRabcd|≤∥∇R∥g for all e,a,b,c,d∈{1,…,2m}, therefore
(6.58)|-∑s=12m(∇iR1s1i-16∇sR1i1i)∂sφ|≤∑s=12m76∥∇R∥gC5=2m76∥∇R∥gC5=73mC5∥∇R∥g.
Hence at P1 in the chart ψ1, we obtain
(6.59)-t∂11f-t∑s=12m∂sf∂sφ≥∑i=12m∂F∂rii[φ][(βi)2+23R1i1i(β1-2βi)]+16∑i=1mσk-1,i(λ(P1))σk(λ(P1))×(R1ii1+R1(i+m)(i+m)1)(βi+βi+m)+(∑i=12m∂F∂rii[φ])×[-43m2(C5)2∥R∥g-73mC5∥∇R∥g].
But |∂11f(P1)|≤∥f∥C2(M), |∂sf(P1)|≤∥f∥C2(M) and |∂sφ|≤C5 for all s then
(6.60)-t∂11f-t∑s=12m∂sf∂sφ≤∥f∥C2(M)(1+2mC5).
Besides
(6.61)∑i=12m‍∂F∂rii[φ]=∑i=1m‍∂F∂rii[φ]+∂F∂r(i+m)(i+m)[φ]=12∑i=1m‍σk-1,i(λ(P1))σk(λ(P1)).
Consequently, we obtain
(6.62)∥f∥C2(M)(1+2mC5)≥∂F∂r11[φ](β1)2+23∑i=12m∂F∂rii[φ]R1i1i(β1-2βi)+16∑i=1mσk-1,i(λ(P1))σk(λ(P1))×(R1ii1+R1(i+m)(i+m)1)(βi+βi+m)+12(∑i=1mσk-1,i(λ(P1))σk(λ(P1)))×[-43m2(C5)2∥R∥g-73mC5∥∇R∥g].
Let us now estimate |βi| for 1≤i≤m using β1. We follow the same method as for the proof of Theorem 6.13. For all (P,ξ)∈UT, we have the inequality (∇2φt)P(ξ,ξ)≤β1+(1/2)(C5)2; then at P in a holomorphic g-normal g~-adapted chart ψP, namely, a chart such that [gij-(P)]1≤i,j≤m=Im, ∂ℓgij-(P)=0 and [g~ij-(P)]1≤i,j≤m=diag(λ1(P),…,λm(P)), we deduce that for all j∈{1,…,m}(6.63)∂xjxjφt(P)=2(∇2φt)P(∂xj2,∂xj2)≤2β1+(C5)2,∂yjyjφt(P)≤2β1+(C5)2.
Since λj(P)≥-(m-1)C2′, we infer the following inequalities:
(6.64)∀j∈{1,…,m}∂xjxjφt(P)≥-4[(m-1)C2′+1]-2β1-(C5)2,∂yjyjφt(P)≥-4[(m-1)C2′+1]-2β1-(C5)2.
Consequently
(6.65)∀1≤i,j≤2m|∂uiujφt(P)|≤4β1+2(C5)2+4[(m-1)C2′+1]︸=:C9,
in the chart ψP.Hence we infer that(6.66)|(∇2φt)P|g2=14∑i,j=12m(∂uiujφt(P))2≤m2[4β1+2(C5)2+C9]2∀P.
But at P1 in the chart ψ1, |(∇2φt)P1|g2=∑i=12m‍(∂uiuiφt(P1))2=∑i=12m‍(βi)2; consequently we obtain
(6.67)∀1≤i≤2m|βi|≤m(4β1+2(C5)2+C9).
Thus
(6.68)|(R1i1i)(β1-2βi)|≤|R1i1i|(|β1|+2|βi|)≤3m∥R∥g(4β1+2(C5)2+C9).
Besides
(6.69)|(R1ii1+R1(i+m)(i+m)1)(βi+βi+m)|≤(|R1ii1|+|R1(i+m)(i+m)1|)(|βi|+|βi+m|)≤4m∥R∥g(4β1+2(C5)2+C9).
Hence
(6.70)∥f∥C2(M)(1+2mC5)≥14σk-1,1(λ(P1))σk(λ(P1))(β1)2+(∑i=1mσk-1,i(λ(P1))σk(λ(P1)))(-m)∥R∥g(4β1+2(C5)2+C9)+(∑i=1mσk-1,i(λ(P1))σk(λ(P1)))(-23m)∥R∥g(4β1+2(C5)2+C9)+(∑i=1mσk-1,i(λ(P1))σk(λ(P1)))[-76mC5∥∇R∥g-23m2(C5)2∥R∥g].
Then
(6.71)∥f∥C2(M)(1+2mC5)≥14σk-1,1(λ(P1))σk(λ(P1))(β1)2+(∑i=1mσk-1,i(λ(P1))σk(λ(P1)))(-53m)∥R∥g(4β1+2(C5)2+C9)+(∑i=1mσk-1,i(λ(P1))σk(λ(P1)))[-76mC5∥∇R∥g-23m2(C5)2∥R∥g].
But using the uniform ellipticity and the inequalities e-2∥f∥∞(mk)≤σk(λ(P))≤e2∥f∥∞(mk), we obtain
(6.72)∑i=1m‍σk-1,i(λ(P1))σk(λ(P1))≤me2∥f∥∞F0(mk),(6.73)σk-1,1(λ(P1))σk(λ(P1))≥e-2∥f∥∞E0(mk).
Then at P1 in the chart ψ1, we have
(6.74)0≥14e-2∥f∥∞E0(mk)(β1)2+me2∥f∥∞F0(mk)(-53m)∥R∥g(4β1+2(C5)2+C9)-me2∥f∥∞F0(mk)[76mC5∥∇R∥g+23m2(C5)2∥R∥g]-∥f∥C2(M)(1+2mC5).
The previous inequality means that some polynomial of second order in the variable β1 is negative:
(6.75)0≥14e-2∥f∥∞E0(mk)(β1)2+me2∥f∥∞F0(mk)(-203m)∥R∥gβ1-me2∥f∥∞F0(mk)[76mC5∥∇R∥g+23m2(C5)2∥R∥g+53m∥R∥g(2(C5)2+C9)]-∥f∥C2(M)(1+2mC5).
Set
(6.76)I:=803m2e4∥f∥∞F0E0∥R∥g>0,J≔4m2e4∥f∥∞F0E0[76C5∥∇R∥g+23m(C5)2∥R∥g+53(2(C5)2+C9)∥R∥g]+4(mk)e2∥f∥∞E0∥f∥C2(M)(1+2mC5)>0.
The previous inequality writes then:
(6.77)(β1)2-Iβ1-J≤0.
The discriminant of this polynomial of second order is equal to ▵=I2+4J>0, which gives an upper bound for β1.


## 6.5.1. The Functional

Consider the following functional:(6.27)Φ:UT→ℝ(P,ξ)↦(∇2φt)P(ξ,ξ)+12|(∇φt)P|g2,
where UT is the real unit sphere bundle associated to (TM,g). Φ is continuous on the compact set UT, so it assumes its maximum at a point (P1,ξ1)∈UT.


## 6.5.2. Reduction to Finding a One-Sided Estimate for(∇2φt)P1(ξ1,ξ1)

If we find a uniform upper bound for(∇2φt)P1(ξ1,ξ1), since |∇φt|∞≤C5, we readily deduce that there exists a constant C6>0 such that
(6.28)(∇2φt)P(ξ,ξ)≤C6∀(P,ξ)∈UT.
Fix P∈M. Let (UP,ψP) be a holomorphic g-normal g~-adapted chart centered at P, namely, [gij-(P)]1≤i,j≤m=Im, ∂ℓgij-(P)=0 and [g~ij-(P)]1≤i,j≤m=[diag(λ1(P),…,λm(P))]. Since |∂xj|g=2, we obtain ∂xjxjφt(P)=2(∇2φt)P(∂xj/2,∂xj/2)≤2C6 and similarly ∂yjyjφt(P)=2(∇2φt)P(∂yj/2,∂yj/2)≤2C6 for all 1≤j≤m. Besides, we have ∂xjxjφt(P)+∂yjyjφt(P)=4∂jj-φt(P)=4(λj(P)-1)≥-4[(m-1)C2′+1]; therefore we obtain
(6.29)∂xjxjφt(P)≥-4[(m-1)C2′+1]-2C6=:-C7,∂yjyjφt(P)≥-C7,∀1≤j≤m.
Let us now bound second derivatives of mixed type ∂urusφt(P). Let 1≤r≠s≤2m. Since |∂xr±∂xs|g=2, we have (∇2φt)P((∂xr±∂xs)/2,(∂xr±∂xs)/2)=(1/4)∂xrxrφt(P)+(1/4)∂xsxsφt(P)±(1/2)∂xrxsφt(P)≤C6, which yields ±∂xrxsφt(P)≤2C6-(1/2)∂xrxrφt(P)-(1/2)∂xsxsφt(P), hence as well |∂xrxsφt(P)|≤2C6+C7. Similarly we prove that at P, in the above chart ψP, we have |∂yrysφt(P)|≤2C6+C7 for all 1≤r≠s≤m and |∂xrysφt(P)|≤2C6+C7 for all 1≤r,s≤m. Consequently |∂uiujφt(P)|≤2C6+C7 for all 1≤i, j≤2m. Therefore we deduce that
(6.30)|(∇2φt)(P)|g2=14∑1≤i,j≤2m‍(∂uiujφt(P))2≤m2(2C6+C7)2.Theorem 6.13 (second derivatives uniform estimate).
There exists a constantC8>0 depending only on m,C2′, and C6 such that for all P∈M, |(∇2φt)P|g≤C8.This allows to deduce the needed uniformC2 estimate:
(6.31)∥φ∥C2(M,ℝ)≤C0+C5+C8.


## 6.5.3. Chart Choice

For fixedP∈M, ξ∈UTP↦(∇2φt)P(ξ,ξ) is continuous on the compact subset UTP (the fiber); therefore it assumes its maximum at a unit vector ξP∈UTP. Besides, (∇2φt)P is a symmetric bilinear form on TPM, so by the min-max principle we have (∇2φt)P(ξP,ξP)=maxξ∈TPM,g(ξ,ξ)=1(∇2φt)P(ξ,ξ)=βmax(P), where βmax(P) denotes the largest eigenvalue of (∇2φt)P with respect to gP; furthermore we can choose ξP as the direction of the largest eigenvalue βmax(P). For fixed P, we now have maxξ∈TPM,gP(ξ,ξ)=1Φ(P,ξ)=Φ(P,ξP)=(∇2φt)P(ξP,ξP)+(1/2)|(∇φt)P|g2=βmax(P)+(1/2)|(∇φt)P|g2, consequently max(P,ξ)∈UTΦ(P,ξ)=maxP∈MΦ(P,ξP)=Φ(P1,ξ1)≤Φ(P1,ξP1), hence max(P,ξ)∈UTΦ(P,ξ)=Φ(P1,ξP1)=βmax(P1)+(1/2)|(∇φt)P1|g2.At the pointP1, consider ε1P1,…,ε2mP1 a (real) basis of (TP1M,gP1) that satisfies the following properties: (i)
[gij(P1)]1≤i,j≤2m=I2m,(ii)
[(∇2φt)ij(P0)]1≤i,j≤2m=diag(β1,…,β2m),(iii)
β1=βmax(P1)≥β2≥⋯≥β2m. Let (U1′,ψ1) be a C∞g-normal real chart at P1 obtained from a holomorphic chart z1,…,zm by setting (u1,…,u2m)=(x1,…,xm,y1,…,ym) where zj=xj+iyj (namely, [gij(P1)]1≤i,j≤2m=I2m and ∂uℓgij=0 for all 1≤i,j,ℓ≤2m) satisfying ψ1(P1)=0 and ∂ui|P1=εiP1, so that ∂u1|P1 is the direction of the largest eigenvalue βmax(P1).


## 6.5.4. Auxiliary Local Functional

From now on, we work in thereal chart(U1′,ψ1) so constructed at P1.LetU2⊂U1′ be an open subset such that g11(P)>0 for all P∈U2, and let Φ1 be the functional
(6.32)Φ1:U2→ℝP↦Φ1(P)=(∇2φt)11(P)g11(P)+12|(∇φt)P|g2.
We claim that Φ1 assumes its maximum at P1. Indeed, (∇2φt)11(P)/g11(P)=(∇2φ)P(∂u1,∂u1)/gP(∂u1,∂u1)=(∇2φ)P(∂u1/|∂u1|g,∂u1/|∂u1|g)≤βmax(P), so Φ1(P)≤βmax(P)+(1/2)|(∇φt)P|g2≤βmax(P1)+(1/2)|(∇φt)P1|g2=Φ1(P1) proving our claim.Let us now differentiate twice in thereal direction∂u1 the equation
(E*)F(P,[∂uiujφ]1≤i,j≤2m)=v.
At the point P, in a chart u, we obtain
(6.33)∂u1v=∂F∂u1[φ]+∑i,j=12m‍∂F∂rij[φ]∂u1uiujφ.
Differentiating once again
(6.34)∂u1u1v=∂2F∂u1∂u1[φ]+∑i,j=12m‍∂2F∂rij∂u1[φ]∂u1uiujφ+∑i,j=12m‍[∂2F∂u1∂rij[φ]+∑e,s=12m‍∂2F∂res∂rij[φ]∂u1ueusφ]∂u1uiujφ+∑i,j=12m‍∂F∂rij[φ]∂u1u1uiujφ.
But at the point P1, for our function F(P,r)=Fk[δij+(1/4)gjℓ-(P)(riℓ+r(i+m)(ℓ+m)+iri(ℓ+m)-ir(i+m)ℓ)]1≤i,j≤m, we have (∂2F/∂rij∂u1)[φ]=0 since ∂u1gsq-(P1)=0. Hence, we infer that
(6.35)∂u1u1v=∂2F∂u1∂u1[φ]+∑i,j,e,s=12m‍∂2F∂res∂rij[φ]∂u1ueusφ∂u1uiujφ+∑i,j=12m‍∂F∂rij[φ]∂u1u1uiujφ.


## 6.5.5. Using Concavity

The functionF is concave with respect to the variable r. Indeed
(6.36)F(P,r)=Fk[δij+14gjℓ-(P)(riℓ+r(i+m)(ℓ+m)+iri(ℓ+m)-ir(i+m)ℓ)]1≤i,j≤m=Fk(g-1(P)r~),wherer~=[gij-(P)+14(rij+r(i+m)(j+m)+iri(j+m)-ir(i+m)j)]1≤i,j≤m=Fk(g-1/2(P)r~g-1/2(P)︸)∈ℋm(ℂ)=Fk(ρP(r)),whereρP(r)≔[δij+14∑ℓ,s=1m‍(g-1/2(P))iℓ(g-1/2(P))sj(rℓs+r(ℓ+m)(s+m)+irℓ(s+m)-ir(ℓ+m)s)]1≤i,j≤m
but for a fixed point P the function r∈S2m(ℝ)↦ρP(r)∈ℋm(ℂ) is affine (where S2m(ℝ) denotes the set of symmetric matrices of size 2m); we deduce then that the composition F(P,·)=Fk∘ρP is concave on the set {r∈S2m(ℝ)/ρP(r)∈λ-1(Γk)}=ρP-1(λ-1(Γk)), which proves our claim. Hence, since the matrix [∂u1uiujφ]1≤i,j≤m is symmetric, we obtain that
(6.37)∑i,j,e,s=12m‍∂2F∂res∂rij[φ]∂u1ueusφ∂u1uiujφ≤0.
Consequently
(6.38)∂u1u1v-∂u1u1F[φ]≤∑i,j=12m‍∂F∂rij[φ]∂u1u1uiujφ.
Let us now calculate the quantity ∂u1u1F[φ] (at P1). Since ∂u1gsq-(P1)=0, we have
(6.39)∂2F∂u1∂u1(P1,D2φ(P1))=∑s=1m‍σk-1,s(λ(P1))σk(λ(P1))×∂u1u1gss-(P1)∂ss-φ(P1).
But at P1, ∂u1u1gss-=-gso-gqs-∂u1u1gqo- and [gij-]1≤i,j≤m=2Im, then ∂u1u1gss-=-4∂u1u1gss- so that ∂u1u1gss-=-∂u1u1gss-∂u1u1g(s+m)(s+m). Moreover Γujusur=(1/2)(∂ujgos+∂usgoj-∂uogjs)gor, thus ∂uiΓujusur=(1/2)(∂uiujgrs+∂uiusgrj-∂uiurgjs). Similarly, we have at P1 : ∂uiΓujurus=(1/2)(∂uiujgrs+∂uiurgsj-∂uiusgjr). Consequently, we deduce that ∂uiujgrs=∂uiΓujusur+∂uiΓujurus. Hence, we have at P1: ∂u1u1gss-=-2∂u1Γu1usus-2∂u1Γu1us+mus+m. Besides, ∂ss-φ=(1/4)(∂ususφ+∂us+mus+mφ), which infers that at P1(6.40)∂u1u1F[φ]=∑s=1m‍σk-1,s(λ(P1))σk(λ(P1))×(-2∂u1Γu1usus-2∂u1Γu1us+mus+m)×14(∂ususφ+∂us+mus+mφ).
Consequently, the inequality (6.38) becomes
(6.41)∂u1u1v≤∑i,j=12m‍∂F∂rij[φ]∂u1u1uiujφ-12∑s=1m‍σk-1,s(λ(P1))σk(λ(P1))×(∂u1Γu1usus+∂u1Γu1us+mus+m)×(∂ususφ+∂us+mus+mφ).


## 6.5.6. Differentiation of the FunctionalΦ1

We differentiate twice the functionalΦ1:
(6.42)Φ1(P)=(∇2φ)11(P)g11(P)+12|(∇φ)P|g2,∂ujΦ1(P)=∂uj(∇2φ)11g11(P)-(∇2φ)11∂ujg11(P)g11(P)2+12∂uj|(∇φ)P|g2,∂uiujΦ1(P)=∂uiuj(∇2φ)11g11(P)-∂uj(∇2φ)11∂uig11(P)g11(P)2-∂ui(∇2φ)11∂ujg11(P)+(∇2φ)11(P)∂uiujg11(P)g11(P)2-(∇2φ)11(P)∂ujg11(P)∂ui(1g11(P)2)+12∂uiuj|(∇φ)P|g2.
Hence, at P1 in the chart ψ1, we obtain
(6.43)∂uiujΦ1=∂uiuj(∇2φ)11-(∇2φ)11(P1)∂uiujg11+12∂uiuj|(∇φ)P|g2(P1).
Let us now calculate the different terms of this formula (at P1 in the chart ψ1):
(6.44)∂uiuj(∇2φ)11=∂uiuj(∂u1u1φ-Γu1u1us∂usφ)=∂uiuju1u1φ-∂uiujΓu1u1us∂usφ-∂ujΓu1u1us∂uiusφ-∂uiΓu1u1us∂ujusφ.
Besides, we have Γuju1u1=(1/2)(∂ujgs1+∂u1gsj-∂usgj1)gs1; therefore we deduce that ∂uiΓuju1u1=(1/2)(∂uiujgs1+∂uiu1gsj-∂uiusgj1)gs1+0=(1/2)∂uiujg11; namely, ∂uiujg11=2∂uiΓuju1u1. Moreover, we have at P1(6.45)∂uiuj|(∇φ)P|g2=∂uiuj(∑r,s=12m‍grs∂urφ∂usφ)=∑r,s=12m‍∂uiujgrs∂urφ∂usφ+grs∂uiujurφ∂usφ+grs∂ujurφ∂uiusφ+grs∂uiurφ∂ujusφ+grs∂urφ∂uiujusφ=∑r,s=12m‍∂uiujgrs∂urφ∂usφ+2∑s=12m‍∂uiujusφ∂usφ+2∑s=12m‍∂uiusφ∂ujusφ.
But at P1, ∂uiujgrs=-∂uiujgrs, in addition at this point ∂uiujgrs=∂uiΓujusur+∂uiΓujurus; therefore we obtain at P1 in the chart ψ1(6.46)∂uiuj|(∇φ)P|g2=-2∑r,s=12m∂uiΓujusur∂urφ∂usφ+2∑s=12m∂uiujusφ∂usφ+2∑s=12m∂uiusφ∂ujusφ.
Henceforth, and in order to lighten the notations, we use ∂i instead of ∂ui and Γijs instead of Γuiujus, so we have
(6.47)∂ijΦ1=∂ij11φ-∂ijΓ11s∂sφ-∂jΓ11s∂isφ-∂iΓ11s∂jsφ-2∂iΓj11(∇2φ)11(P1)-∑r,s=12m‍∂iΓjsr∂rφ∂sφ+∑s=12m‍∂ijsφ∂sφ+∑s=12m‍∂isφ∂jsφ.
Let us now consider the second order linear operator:
(6.48)L~=∑i,j=12m∂F∂rij[φ]∂ij.
Since the functional Φ1 assumes its maximum at the point P1, we have L~(Φ1)≤0 at P1 in the chart ψ1. Besides, combining the inequalities (6.41) and (6.47), we obtain
(6.49)2L~Φ1︸≤0-∂11v≥∑i,j=12m∂F∂rij[φ][∑r,s=12m∂iΓjsr∂rφ∂sφ∂ij11φ-∂ijΓ11s∂sφ-∂jΓ11i∂iiφ-∂iΓ11j∂jjφ-2∂iΓj11(∇2φ)11(P1)-∑r,s=12m∂iΓjsr∂rφ∂sφ+∑s=12m∂ijsφ∂sφ+δij(∂iiφ)2-∂11ijφ]+12∑s=1mσk-1,s(λ(P1))σk(λ(P1))(∂1Γ1ss+∂1Γ1(s+m)s+m)(∂ssφ+∂(s+m)(s+m)φ).
The fourth derivatives are simplified. Moreover, we have at P1 : ∂sv=(∂F/∂u1)[φ]+∑i,j=12m‍(∂F/∂rij)[φ]∂sijφ with (∂F/∂u1)(P1,D2φ(P1))=0, consequently:
(6.50)0≥∂11v+∑s=12m∂sv∂sφ+∑i,j=12m∂F∂rij[φ][∑r,s=12m‍-2∂iΓj11(∇2φ)11(P1)-∂iΓ11j∂jjφ-∂jΓ11i∂iiφ-∑s=12m∂ijΓ11s∂sφ-∑r,s=12m‍∂iΓjsr∂rφ∂sφ+δij(∂iiφ)2]+12∑s=1mσk-1,s(λ(P1))σk(λ(P1))(∂1Γ1ss+∂1Γ1(s+m)s+m)(∂ssφ+∂(s+m)(s+m)φ).
Let us now express the quantities ∂iΓj11,∂iΓ11j,∂jΓ11i,∂iΓjsr and ∂ijΓ11s using the components of the Riemann curvature (at the point P1 in the normal chart ψ1):
(6.51)∂iΓj11=13(Rj11i+Rji11︸=0)=13Rj11i,∂iΓ11j=13(R1j1i+R1i1j)=23R1j1i,∂jΓ11i=23R1i1j,∂iΓjsr=13(Rjrsi+Rjisr),∂1Γ1ss=13(R1ss1+R11ss︸=0)=13R1ss1,∂1Γ1(s+m)s+m=13R1(s+m)(s+m)1,∂ijΓ11s=14(∇iR1j1s+∇iR1s1j+∇jR1s1i+∇jR1i1s)-112(∇sR1i1j+∇sR1j1i)=12(∇iR1s1j+∇jR1s1i)-16∇sR1i1j.
We then obtain
(6.52)0≥∂11v+∑s=12m∂sv∂sφ+∑i,j=12m∂F∂rij[φ][∑i,j=12m∂F∂rij[φ]-23Rj11i(∇2φ)11(P1)-23R1j1i∂jjφ-23R1i1j∂iiφ-∑s=12m(12∇iR1s1j+12∇jR1s1i-16∇sR1i1j)∂sφ-∑r,s=12m13(Rjrsi+Rjisr)∂rφ∂sφ+δij(∂iiφ)2]+12∑s=1mσk-1,s(λ(P1))σk(λ(P1))13(R1ss1+R1(s+m)(s+m)1)(∂ssφ+∂(s+m)(s+m)φ).


## 6.5.7. The Uniform Upper Bound ofβ1=(∇2φ)P1(ξ1,ξ1)

By the uniform estimate of the gradient we have|∂jφt|≤C5 for all 1≤j≤2m. Moreover, at P1 in the chart ψ1: [(∇2φ)ij(P1)]1≤i,j≤2m=[∂ijφ(P1)]1≤i,j≤2m=diag(β1,…,β2m). Consequently
(6.53)0≥∂11v+∑s=12m∂sv∂sφ+∑i,j=12m∂F∂rij[φ][δij(βi)2-23Rj11iβ1-23R1j1iβj-23R1i1jβi-13∑r,s=12m‍(Rjrsi+Rjisr)∂rφ∂sφ-12∑s=12m(∇iR1s1j+∇jR1s1i-13∇sR1i1j)∂sφ]+16∑i=1mσk-1,i(λ(P1))σk(λ(P1))×(R1ii1+R1(i+m)(i+m)1)(βi+βi+m).
But for F[φ]=Fk([δij+gjℓ-∂iℓ-φ]1≤i,j≤m) since ∂ss-φ=(1/4)(∂usus+∂us+mus+m), we obtain at P1 in the chart ψ1 that
(6.54)∂F∂rij[φ]=∑s=1m∂Fk∂Bss(diag(λ(P1)))14∂(rss+r(s+m)(s+m))∂rij.
Then
(6.55)∀1≤i≠j≤2m∂F∂rij[φ]=0,∀1≤i≤m∂F∂rii[φ]=∂F∂r(i+m)(i+m)[φ]=14∂Fk∂Bii(diag(λ(P1)))=14σk-1,i(λ(P1))σk(λ(P1))︸>0sinceλ(P1)∈Γk.
Hence
(6.56)0≥∂11v+∑s=12m∂sv∂sφ+∑i-12m∂F∂rii[φ]×[(βi)2+23R1i1i(β1-2βi)+13∑r,s=12m‍Riris∂rφ∂sφ-∑s=12m‍(∇iR1s1i-16∇sR1i1i)∂sφ]+16∑i=1mσk-1,i(λ(P1))σk(λ(P1))(R1ii1+R1(i+m)(i+m)1)(βi+βi+m).
But at P1 in the chart ψ1, ∥R∥g2=gaigbjgcrgdsRabcdRijrs=∑a,b,c,d=12m‍(Rabcd)2; then |Rabcd|≤∥R∥g for all a,b,c,d∈{1,…,2m}, consequently
(6.57)|13∑r,s=12mRiris∂rφ∂sφ|≤13∑r,s=12m∥R∥g(C5)2=13(2m)2∥R∥g(C5)2=43m2(C5)2∥R∥g.
Besides, at P1 in the chart ψ1, we have ∥∇R∥g2=gelgaigbjgcrgds∇eRabcd∇lRijrs=
∑
e
,
a
,
b
,
c
,
d
=
1
2
m
‍
(
∇
e
R
a
b
c
d
)
2, so |∇eRabcd|≤∥∇R∥g for all e,a,b,c,d∈{1,…,2m}, therefore
(6.58)|-∑s=12m(∇iR1s1i-16∇sR1i1i)∂sφ|≤∑s=12m76∥∇R∥gC5=2m76∥∇R∥gC5=73mC5∥∇R∥g.
Hence at P1 in the chart ψ1, we obtain
(6.59)-t∂11f-t∑s=12m∂sf∂sφ≥∑i=12m∂F∂rii[φ][(βi)2+23R1i1i(β1-2βi)]+16∑i=1mσk-1,i(λ(P1))σk(λ(P1))×(R1ii1+R1(i+m)(i+m)1)(βi+βi+m)+(∑i=12m∂F∂rii[φ])×[-43m2(C5)2∥R∥g-73mC5∥∇R∥g].
But |∂11f(P1)|≤∥f∥C2(M), |∂sf(P1)|≤∥f∥C2(M) and |∂sφ|≤C5 for all s then
(6.60)-t∂11f-t∑s=12m∂sf∂sφ≤∥f∥C2(M)(1+2mC5).
Besides
(6.61)∑i=12m‍∂F∂rii[φ]=∑i=1m‍∂F∂rii[φ]+∂F∂r(i+m)(i+m)[φ]=12∑i=1m‍σk-1,i(λ(P1))σk(λ(P1)).
Consequently, we obtain
(6.62)∥f∥C2(M)(1+2mC5)≥∂F∂r11[φ](β1)2+23∑i=12m∂F∂rii[φ]R1i1i(β1-2βi)+16∑i=1mσk-1,i(λ(P1))σk(λ(P1))×(R1ii1+R1(i+m)(i+m)1)(βi+βi+m)+12(∑i=1mσk-1,i(λ(P1))σk(λ(P1)))×[-43m2(C5)2∥R∥g-73mC5∥∇R∥g].
Let us now estimate |βi| for 1≤i≤m using β1. We follow the same method as for the proof of Theorem 6.13. For all (P,ξ)∈UT, we have the inequality (∇2φt)P(ξ,ξ)≤β1+(1/2)(C5)2; then at P in a holomorphic g-normal g~-adapted chart ψP, namely, a chart such that [gij-(P)]1≤i,j≤m=Im, ∂ℓgij-(P)=0 and [g~ij-(P)]1≤i,j≤m=diag(λ1(P),…,λm(P)), we deduce that for all j∈{1,…,m}(6.63)∂xjxjφt(P)=2(∇2φt)P(∂xj2,∂xj2)≤2β1+(C5)2,∂yjyjφt(P)≤2β1+(C5)2.
Since λj(P)≥-(m-1)C2′, we infer the following inequalities:
(6.64)∀j∈{1,…,m}∂xjxjφt(P)≥-4[(m-1)C2′+1]-2β1-(C5)2,∂yjyjφt(P)≥-4[(m-1)C2′+1]-2β1-(C5)2.
Consequently
(6.65)∀1≤i,j≤2m|∂uiujφt(P)|≤4β1+2(C5)2+4[(m-1)C2′+1]︸=:C9,
in the chart ψP.Hence we infer that(6.66)|(∇2φt)P|g2=14∑i,j=12m(∂uiujφt(P))2≤m2[4β1+2(C5)2+C9]2∀P.
But at P1 in the chart ψ1, |(∇2φt)P1|g2=∑i=12m‍(∂uiuiφt(P1))2=∑i=12m‍(βi)2; consequently we obtain
(6.67)∀1≤i≤2m|βi|≤m(4β1+2(C5)2+C9).
Thus
(6.68)|(R1i1i)(β1-2βi)|≤|R1i1i|(|β1|+2|βi|)≤3m∥R∥g(4β1+2(C5)2+C9).
Besides
(6.69)|(R1ii1+R1(i+m)(i+m)1)(βi+βi+m)|≤(|R1ii1|+|R1(i+m)(i+m)1|)(|βi|+|βi+m|)≤4m∥R∥g(4β1+2(C5)2+C9).
Hence
(6.70)∥f∥C2(M)(1+2mC5)≥14σk-1,1(λ(P1))σk(λ(P1))(β1)2+(∑i=1mσk-1,i(λ(P1))σk(λ(P1)))(-m)∥R∥g(4β1+2(C5)2+C9)+(∑i=1mσk-1,i(λ(P1))σk(λ(P1)))(-23m)∥R∥g(4β1+2(C5)2+C9)+(∑i=1mσk-1,i(λ(P1))σk(λ(P1)))[-76mC5∥∇R∥g-23m2(C5)2∥R∥g].
Then
(6.71)∥f∥C2(M)(1+2mC5)≥14σk-1,1(λ(P1))σk(λ(P1))(β1)2+(∑i=1mσk-1,i(λ(P1))σk(λ(P1)))(-53m)∥R∥g(4β1+2(C5)2+C9)+(∑i=1mσk-1,i(λ(P1))σk(λ(P1)))[-76mC5∥∇R∥g-23m2(C5)2∥R∥g].
But using the uniform ellipticity and the inequalities e-2∥f∥∞(mk)≤σk(λ(P))≤e2∥f∥∞(mk), we obtain
(6.72)∑i=1m‍σk-1,i(λ(P1))σk(λ(P1))≤me2∥f∥∞F0(mk),(6.73)σk-1,1(λ(P1))σk(λ(P1))≥e-2∥f∥∞E0(mk).
Then at P1 in the chart ψ1, we have
(6.74)0≥14e-2∥f∥∞E0(mk)(β1)2+me2∥f∥∞F0(mk)(-53m)∥R∥g(4β1+2(C5)2+C9)-me2∥f∥∞F0(mk)[76mC5∥∇R∥g+23m2(C5)2∥R∥g]-∥f∥C2(M)(1+2mC5).
The previous inequality means that some polynomial of second order in the variable β1 is negative:
(6.75)0≥14e-2∥f∥∞E0(mk)(β1)2+me2∥f∥∞F0(mk)(-203m)∥R∥gβ1-me2∥f∥∞F0(mk)[76mC5∥∇R∥g+23m2(C5)2∥R∥g+53m∥R∥g(2(C5)2+C9)]-∥f∥C2(M)(1+2mC5).
Set
(6.76)I:=803m2e4∥f∥∞F0E0∥R∥g>0,J≔4m2e4∥f∥∞F0E0[76C5∥∇R∥g+23m(C5)2∥R∥g+53(2(C5)2+C9)∥R∥g]+4(mk)e2∥f∥∞E0∥f∥C2(M)(1+2mC5)>0.
The previous inequality writes then:
(6.77)(β1)2-Iβ1-J≤0.
The discriminant of this polynomial of second order is equal to ▵=I2+4J>0, which gives an upper bound for β1.


## 7. AC2,β A Priori Estimate

We infer from theC2 estimate a C2,β estimate using a classical Evans-Trudinger theorem [18, Theorem 17.14 page 461], which achieves the proof of Theorem 1.2. Let us state this Evans-Trudinger theorem; we use Gilbarg and Trudinger's notations for classical norms and seminorms of Hölder spaces (cf. [18] and [9, page 137]).Theorem 7.1.
LetΩ be a bounded domain  (i.e., an open connected set) of ℝn, n≥2. Let one denote by ℝn×n the set of real n×n symmetric matrices. u∈C4(Ω,ℝ) is a solution of
(E′)G[u]=G(x,D2u)=0onΩ,
where G∈C2(Ω×ℝn×n,ℝ) is elliptic with respect to u and satisfies the following hypotheses.(1)
G is uniformly elliptic with respect to u, that is, there exist two real numbers λ,Λ>0 such that
(7.1)∀x∈Ω,∀ξ∈ℝn,λ|ξ|2≤Gij(x,D2u(x))ξiξj≤Λ|ξ|2.(2)
G is concave with respect to u in the variable r. Since G is of class C2, this condition of concavity is equivalent to
(7.2)∀x∈Ω,∀ζ∈ℝn×n,Gij,ks(x,D2u(x))ζijζks≤0.Then for all Ω′⊂⊂Ω, one has the following interior estimate:
(7.3)[D2u]β;Ω′≤C,
where β∈]0,1] depends only on n, λ, and Λ and C>0 depends only on n, λ, Λ, |u|2;Ω′, dist(Ω',∂Ω), Gx, Gr, Gxx et Grx. The notation Grx used here denotes the matrix Grx=[Gij,xℓ]i,j,ℓ=1⋯n evaluated at (x,D2u(x)). It is the same for the notations Gx, Gr, and Gxx [18, page 457].


### 7.1. The Evans-Trudinger Method

Let us suppose that there exists a constantC11>0 such that for all i∈ℕ, we have ∥φti∥C2(M,ℝ)≤C11. In the following, we remove the index i from φti to lighten the notations. In order to construct a C2,β estimate with 0<β<1, we prepare the framework of application of Theorem 7.1.Letℛ=(Uj,ϕj)1≤j≤N be a finite covering of the compact manifold M by charts, and let 𝒫=(θj)1≤j≤N be a partition of unity of class C∞ subordinate to this covering. The family of continuity equations writes in the chart (Us,ϕs) where 1≤s≤N is a fixed integer as follows:
(Ek,t′)Fk([δij+gjℓ-∘ϕs-1(x)∂(φt∘ϕs-1)∂zi∂z-ℓ(x)]1≤i,j≤m)-tf∘ϕs-1(x)-ln(At)=0x∈ϕs(Us)⊂ℝ2m.
Besides, we have ∂/∂za∂zb¯=(1/4)(Dab+D(a+m)(b+m)+iDa(b+m)-iD(a+m)b) where the Dabs denotes real derivatives; thus our equation writes:
(Ek,t′′)G(x,D2(φt∘ϕs-1))=0x∈ϕs(Us)⊂ℝ2mwith,(7.4)G(x,r)=Fk([δij+14gjℓ-(ϕs-1(x))(riℓ+r(i+m)(ℓ+m)+iri(ℓ+m)-ir(i+m)ℓ)]1≤i,j≤m)-tf∘ϕs-1(x)-ln(At).
This map G is concave in the variable r as the map F appearing in the C2 estimate (cf. (6.36)), (namely, for all fixed x of ϕs(Us), G(x,·) is concave on ρϕs-1(x)-1(λ-1(Γk))⊂S2m(ℝ)). For all s∈{1,…,N}, let us consider Ωsa bounded domain of ℝ2m strictly included in ϕs(Us):
(7.5)Ωs⊂⊂ϕs(Us).
The notation S′⊂⊂S means that S′ is strictly included in S, namely, that S′¯⊂S. We will explain later how these domains Ωs are chosen. The map G is of class C2 and the solution ψts:=φt∘ϕs-1∈C4(Ωs,ℝ) since φt∈Cℓ,α(M) with ℓ≥5. The equation (Ek,t′′) on Ωs⊂ϕs(Us) is now written in the form corresponding to the Theorem 7.1; it remains to check the hypotheses of this theorem onΩs, namely, that (1)
G is uniformly elliptic with respect to ψts=φt∘ϕs-1; that is, there exist two real numbers λs,Λs>0 such that
(7.6)∀x∈Ωs,∀ξ∈ℝ2m,λs|ξ|2≤Gij(x,D2(ψts)(x))ξiξj≤Λs|ξ|2.
Moreover, we will impose ourselves to find real numbers λs,Λs independent of t.(2)
G is concave with respect to ψts in the variable r. Since G is of class C2, this concavity condition is equivalent to
(7.7)∀x∈Ωs,∀ζ∈ℝ2m×2m,Gij,kℓ(x,D2(ψts)(x))ζijζkℓ≤0.
This has been checked before.(3)
The derivativesGx, Gr, Gxx, and Grx are controlled (these quantities are evaluated at (x,D2(ψts)(x))). Once these three points checked, and since we have a C2 estimate of φt by C11, Theorem 7.1 allows us to deduce that for all open set Ωs′⊂⊂Ωs there exist two real numbers βs∈]0,1] and Cstes>0 depending only on m, λs, Λs, dist(Ωs′,∂Ωs), on the uniform estimate of |ψts|2;Ωs′, and on the uniform estimates of the quantities Gx, Gr, Gxx, and Grx, so in particular βs and Cstes are independent of t, such that
(7.8)[D2(ψts)]βs;Ωs′≤Cstes.The Choice ofΩs and Ωs′
Let us denote byKs the support of the function θs∘ϕs-1:
(7.9)Ks:=supp(θs∘ϕs-1)=ϕs(suppθs)⊂ϕs(Us).
The set Ks is compact, and it is included in the open set ϕs(Us) of ℝ2m, and ℝ2m is separated locally compact; then by the theorem of intercalation of relatively compact open sets, applied twice, we deduce the existence of two relatively compact open sets Ωs and Ωs′ such that
(7.10)Ks⊂Ωs′⊂⊂Ωs⊂⊂ϕs(Us).
The set Ωs is required to be connected: for this, it suffices that Ks be connected even after restriction to a connected component in Ωs of a point of Ks; indeed, this connected component is an open set of Ωs since Ωs is locally connected (as an open set of ℝ2m); moreover it is bounded since Ωs is bounded.Application of the Theorem
Letβ:=minβs; the norm ∥·∥C2,β is submultiplicative; then
(7.11)∥φt∥C2,β(M)ℛ,𝒫=∑s=1N|(θs∘ϕs-1)×(φs∘ϕs-1)|2,β;Ωs′≤∑s=1N|θs∘ϕs-1|2,β;Ωs′×|ψts|2,β;Ωs′.
But, by (7.8) we have |ψts|2,βs;Ωs′=|ψts|2;Ωs′+[D2(ψts)]βs;Ωs′≤|ψts|2;Ωs′+Cstes≤Cstes′ where Cstes′ depends only on m, λs, Λs, dist(Ωs′,∂Ωs), C11 (the constant of the C2 estimate) and the uniform estimates of the quantities Gx, Gr, Gxx, and Grx. We obtain consequently the needed C2,β estimate:
(7.12)∥φt∥C2,β(M)ℛ,𝒫≤∑s=1N|θs∘ϕs-1|2,β;Ωs′×Cstes′=:C12.
Let us now check the hypotheses 1 and 3 above.


### 7.2. Uniform Ellipticity ofG on Ωs

Letx∈Ωs and ξ∈ℝ2m:
(7.13)∑i,j=12mGij(x,r)ξiξj=d(G(x,·))r(M)withM=[ξiξj]1≤i,j≤m∈S2m(ℝ)=d(Fk∘ρϕs-1(x))r(M)=d(Fk)ρϕs-1(x)(r)·d(ρϕs-1(x))r(M).
Let us recall that ρP(r)=[δij+(1/4)∑ℓ,o=1m‍(g-1/2(P))iℓ(g-1/2(P))oj(rℓo+r(ℓ+m)(o+m)+irℓ(o+m)-ir(ℓ+m)o)]1≤i,j≤m (cf. (6.36)); we consequently obtain
(7.14)∑i,j=12mGij(x,D2(ψts)(x))ξiξj=d(Fk)ρϕs-1(x)(D2(ψts)(x))·[14∑ℓ,o=1m(g-1/2(ϕs-1(x)))iℓ(g-1/2(ϕs-1(x)))oj×(Mℓo+M(ℓ+m)(o+m)+iMℓ(o+m)-iM(ℓ+m)o)∑ℓ,o=1m14]1≤i,j≤m.
In the following, we denote M~:=[(1/4)(Mℓs+M(ℓ+m)(s+m)+iMℓ(s+m)-iM(ℓ+m)s)]1≤ℓ,s≤m. Thus
(7.15)M~=[14(ξℓξs+ξℓ+mξs+m+iξℓξs+m-iξℓ+mξs)]1≤ℓ,s≤m∈ℋm(ℂ)=[14(ξℓ-iξℓ+m)(ξs+iξs+m︸)=:ξ~s]1≤ℓ,s≤m=[14ξ~¯ℓξ~s]1≤ℓ,s≤m.
Besides, let us denote di=σk-1,i[λ(g-1g~φt(ϕs-1(x)))]/σk[λ(g-1g~φt(ϕs-1(x)))] and g-1/2 instead of g-1/2(ϕs-1(x)) in order to lighten the formulas. We obtain by the invariance formula (2.7) that
(7.16)∑i,j=12mGij(x,D2(ψts)(x))ξiξj=d(Fk)[g]-1/2g~φt[g]-1/2·([g]-1/2M~[g]-1/2)=d(Fk)diag(λ1,…,λm)·(U¯t[g]-1/2M~[g]-1/2U)whereU∈Um(ℂ)withU¯t[g]-1/2g~φt[g]-1/2U=diag(λ1,…,λm)we  are  at  the  pointϕs-1(x)=∑i=1mdi(U¯t[g]-1/2M~[g]-1/2U)ii=∑i=1mdi(([g]-1/2U¯)tM~([g]-1/2U))ii=∑i,ℓ,j=1mdi([g]-1/2U¯)ℓiM~ℓj([g]-1/2U)ji=∑i,ℓ,j=1mdi([g]-1/2U¯)ℓi14ξ~¯ℓξ~j([g]-1/2U)ji=14∑i=1mdi(∑j=1m‍ξ~j([g]-1/2U)ji)︸=:αi(∑ℓ=1m‍ξ~¯ℓ([g]-1/2U¯)ℓi)︸=αi¯=14∑i=1mdi|αi|2.
But by Proposition 6.10 and the inequalities e-2∥f∥∞(mk)≤σk(λ(g-1g~φt(P)))≤e2∥f∥∞(mk), we have for (6.72)
(7.17)e-2∥f∥∞E0(mk)≤di≤e2∥f∥∞F0(mk).
Combining (7.16) and (7.17), we obtain
(7.18)14e-2∥f∥∞E0(mk)(∑i=1m|αi|2)≤∑i,j=12mGij(x,D2(ψts)(x))ξiξj≤14e2∥f∥∞F0(mk)(∑i=1m|αi|2).
But
(7.19)∑i=1m‍|αi|2=∑i=1m|∑j=1m‍ξ~j([g]-1/2U)ji|2=∑i=1m(∑j=1m‍ξ~j([g]-1/2U)ji)(∑ℓ=1m‍ξ~¯ℓ([g]-1/2U¯)ℓi)=∑j,ℓ=1m{∑i=1m‍([g]-1/2U)ji([g]-1/2U¯)ℓi}ξ~jξ~¯ℓ=∑j,ℓ=1m(([g]-1/2U)×([g]-1/2U¯)t)jℓξ~jξ~¯ℓ.
And ([g]-1/2U)×([g]-1/2U¯)t=[g]-1/2UtU¯t[g]-1/2¯=[g]-1/2t[g]-1/2¯=[g]-1/2[g]-1/2=[g]-1; then
(7.20)∑i=1m‍|αi|2=∑j,ℓ=1m‍([g]-1)jℓξ~jξ~¯ℓ=∑j,ℓ=1m‍gℓj-(ϕs-1(x))ξ~jξ~¯ℓ.
Consequently, and since |ξ|2=|ξ~|2, the checking of the hypothesis of uniform ellipticity of the Theorem 7.1 is reduced to find two real numbers λso,Λso>0 such that
(7.21)∀x∈Ωs,∀ξ~∈ℂm,λso|ξ~|2≤∑j,ℓ=1mgℓj-(ϕs-1(x))ξ~ℓξ~¯j≤Λso|ξ~|2.
By the min-max principle applied on ℂm to the Hermitian form ≺X,Y≻g(ϕs-1(x))=gab-(ϕs-1(x))XaY¯b relatively to the canonical one, we have
(7.22)λmin[gab-(ϕs-1(x))]1≤a,b≤m|ξ~|2≤∑a,b=1m‍gab-(ϕs-1(x))ξ~aξ~¯b≤λmax[gab-(ϕs-1(x))]1≤a,b≤m|ξ~|2.
But the functions P↦λmin[gab-(P)]1≤a,b≤m and P↦λmax[gab-(P)]1≤a,b≤m are continuous on ϕs-1(Ωs)¯⊂Us which is compact since it is a closed set of the compact manifold M (cf. (7.5) for the choice of the domains Ωs), so these functions are bounded and reach their bounds; thus
(7.23)(minP∈ϕs-1(Ωs)¯λmin[gab-(P)]1≤a,b≤m)︸=:λso×|ξ~|2≤∑a,b=1m‍gab-(ϕs-1(x))ξ~aξ~¯b≤(maxP∈ϕs-1(Ωs)¯λmax[gab-(P)]1≤a,b≤m)︸=:Λso×|ξ~|2.
By the inequalities (7.18) and (7.23), we deduce that
(7.24)λs|ξ~|2≤∑i,j=12m‍Gij(x,D2(ψts)(x))ξiξj≤Λs|ξ~|2withλs:=14e-2∥f∥∞E0(mk)λso,Λs:=14e2∥f∥∞F0(mk)Λso.
The real numbers λs and Λs depend on k, m, ∥f∥∞, E0, F0, g, (Us,ϕs), and Ωs and are independent of t, x and ξ~, which achieves the proof of the global uniform ellipticity.


### 7.3.Uniform Estimate ofGx,Gr,Gxx, andGrx

In this subsection, we estimate uniformly the quantitiesGx, Gr,Gxx, and Grx (recall that these quantities are evaluated at (x,D2(ψts)(x))) by using the same technique as in the previous subsection for the proof of uniform ellipticity (7.24).We have(7.25)|Gx|2=|[Gxi]1≤i≤2m|2=∑i=12m‍|Gxi|2whereGxi=∂G∂xi(x,D2(ψts)(x)).
For (7.14), we obtain
(7.26)Gxi=d(Fk)[g-1g~φt(ϕs-1(x))]·([∑ℓ=1m‍∂(gqℓ-∘ϕs-1)∂xi(x)∂oℓ-φt(ϕs-1(x))]1≤o,q≤m︸)=:Mo-t∂(f∘ϕs-1)∂xi(x)
and for (7.16), we infer then by the invariance formula (2.7) that
(7.27)Gxi=∑j=1m‍dj(U¯tMoU)jj-t∂f∂xi(ϕs-1(x)),
where U∈Um(ℂ) such that (U¯t[g-1g~φt(ϕs-1(x))]U=diag(λ1,…,λm) and di=σk-1,i[λ(g-1g~φt(ϕs-1(x)))]/σk[λ(g-1g~φt(ϕs-1(x)))]. We can then write:
(7.28)Gxi=∑j,p,q=1m‍djUpj¯UqjMpqo-t∂f∂xi(ϕs-1(x))=∑j,p,q=1m‍djUpj¯Uqj(∑ℓ=1m‍∂gqℓ-∂xi(ϕs-1(x))∂pℓ-φt(ϕs-1(x)))-t∂f∂xi(ϕs-1(x)).Thus|Gxi|≤∑j,p,q,ℓ=1m‍e2∥f∥∞F0(mk)|Upj¯||Uqj|×(max1≤a,b≤m,1≤i≤2mmaxP∈ϕs-1(Ωs)¯|∂gab-∂xi(P)|︸)=:Λs1∥φt∥C2(M,ℝ)+∥f∥C1(M,ℝ).
But U∈Um(ℂ); then |Uqj|≤1 for all 1≤q, j≤m, consequently
(7.29)|Gxi|≤m4e2∥f∥∞F0(mk)Λs1∥φt∥C2(M,ℝ)︸≤C11(C2estimate)+∥f∥C1(M,ℝ),
which gives the needed uniform estimate for Gx:
(7.30)|Gx|≤2m(m4e2∥f∥∞F0(mk)Λs1C11+∥f∥C1(M,ℝ)).
Similarly
(7.31)|Gr|2=|[Gpq]1≤p,q≤2m|2=∑p,q=12m‍|Gpq|2,whereGpq=∂G∂rpq(x,D2(ψts)(x)).
And we have
(7.32)Gpq=d(Fk)[g-1g~φt(ϕs-1(x))]·[∑ℓ=1m‍gjℓ-(ϕs-1(x))(Epq~)iℓ-]1≤i,j≤m︸=:M1,
where Epq is the m×m matrix whose all coefficients are equal to zero except the coefficient pq which is equal to 1, and the matrix (Epq~) is obtained from Epq by the formula M~:=[(1/4)(Mℓs+M(ℓ+m)(s+m)+iMℓ(s+m)-iM(ℓ+m)s)]1≤ℓ,s≤m, thus
(7.33)Gpq=∑j=1m‍dj(tU¯M1U)jj,
where  U  and  di  are as before for  Gx.Since|(Epq~)iℓ-|≤1 for all 1≤i,ℓ≤m, we obtain for Gx that
(7.34)|Gpq|≤m4e2∥f∥∞F0(mk)Λs2,
where Λs2=max1≤a,b≤mmaxP∈ϕs-1(Ωs)¯|gab-(P)|, which gives the needed uniform estimate for Gr:
(7.35)|Gr|≤2m5e2∥f∥∞F0(mk)Λs2.
Concerning Gxx, we have
(7.36)|Gxx|2=|[Gxpxq]1≤p,q≤2m|2=∑p,q=12m‍|Gxpxq|2,whereGxpxq=∂2G∂xp∂xq(x,D2(ψts)(x)).
A calculation shows that
(7.37)Gxpxq=-t∂2f∂xp∂xq(ϕs-1(x))+∑i,j,ℓ=1m‍∂Fk∂Bij([g-1g~φt(ϕs-1(x))])∂2gjℓ-∂xp∂xq(ϕs-1(x))∂iℓ-φt(ϕs-1(x))+∑i,j,ℓ,μ,o,ν=1m‍∂2Fk∂Bμo∂Bij([g-1g~φt(ϕs-1(x))])︸=:ℰ×∂goν-∂xp(ϕs-1(x))∂gjℓ-∂xq(ϕs-1(x))∂μν-φt(ϕs-1(x))∂iℓ-φt(ϕs-1(x)).
All the terms are uniformly bounded; it remains to justify that the term in second derivative ℰ is also uniformly bounded:
(7.38)ℰ=d2(Fk)[g-1g~φt(ϕs-1(x))]·(Eμo,Eij)then  by  the  invariance  formula(2.7)=∑a,b,c,d=1m‍∂2Fk∂Bab∂Bcd[diag(λ1,…,λm)](tU¯EμoU)ab(tU¯EijU)cd,whereU∈Um(ℂ)is  like  before.
But we know the second derivatives of Fk at a diagonal matrix by (2.5). Besides, we have 0<σk-1,i(λ)/σk(λ)=di≤e2∥f∥∞F0/(mk) by (7.17), and since e-2∥f∥∞(mk)≤σk(λ), it remains only to control the quantities |σk-2,ij(λ)| with i≠j to prove that ℰ is uniformly bounded. But since λ∈Γk, we have σk-2,ij(λ)>0 [11]. Moreover, by the pinching of the eigenvalues, we deduce automatically that
(7.39)σk-2,ij(λ)≤(m-2k-2)(C2′)k-1=:F1,
which achieves the checking of the fact that Gxx is uniformly bounded.Similarly, we establish a uniform estimate ofGxr using this calculation:
(7.40)Gxo,pq=∂2G∂xo∂rpq(x,D2(ψts)(x))=∑i,j,ℓ=1m‍∂Fk∂Bij([g-1g~φt(ϕs-1(x))])∂gjℓ-∂xo(ϕs-1(x))(Epq~)iℓ-+∑i,j,ℓ,ν,μ,γ=1m‍∂2Fk∂Bνμ∂Bij([g-1g~φt(ϕs-1(x))])×∂gμγ-∂xo(ϕs-1(x))∂νγ-φt(ϕs-1(x))gjℓ-(ϕs-1(x))(Epq~)iℓ-,
which achieves the proof of the C2,β estimate.


## 7.1. The Evans-Trudinger Method

Let us suppose that there exists a constantC11>0 such that for all i∈ℕ, we have ∥φti∥C2(M,ℝ)≤C11. In the following, we remove the index i from φti to lighten the notations. In order to construct a C2,β estimate with 0<β<1, we prepare the framework of application of Theorem 7.1.Letℛ=(Uj,ϕj)1≤j≤N be a finite covering of the compact manifold M by charts, and let 𝒫=(θj)1≤j≤N be a partition of unity of class C∞ subordinate to this covering. The family of continuity equations writes in the chart (Us,ϕs) where 1≤s≤N is a fixed integer as follows:
(Ek,t′)Fk([δij+gjℓ-∘ϕs-1(x)∂(φt∘ϕs-1)∂zi∂z-ℓ(x)]1≤i,j≤m)-tf∘ϕs-1(x)-ln(At)=0x∈ϕs(Us)⊂ℝ2m.
Besides, we have ∂/∂za∂zb¯=(1/4)(Dab+D(a+m)(b+m)+iDa(b+m)-iD(a+m)b) where the Dabs denotes real derivatives; thus our equation writes:
(Ek,t′′)G(x,D2(φt∘ϕs-1))=0x∈ϕs(Us)⊂ℝ2mwith,(7.4)G(x,r)=Fk([δij+14gjℓ-(ϕs-1(x))(riℓ+r(i+m)(ℓ+m)+iri(ℓ+m)-ir(i+m)ℓ)]1≤i,j≤m)-tf∘ϕs-1(x)-ln(At).
This map G is concave in the variable r as the map F appearing in the C2 estimate (cf. (6.36)), (namely, for all fixed x of ϕs(Us), G(x,·) is concave on ρϕs-1(x)-1(λ-1(Γk))⊂S2m(ℝ)). For all s∈{1,…,N}, let us consider Ωsa bounded domain of ℝ2m strictly included in ϕs(Us):
(7.5)Ωs⊂⊂ϕs(Us).
The notation S′⊂⊂S means that S′ is strictly included in S, namely, that S′¯⊂S. We will explain later how these domains Ωs are chosen. The map G is of class C2 and the solution ψts:=φt∘ϕs-1∈C4(Ωs,ℝ) since φt∈Cℓ,α(M) with ℓ≥5. The equation (Ek,t′′) on Ωs⊂ϕs(Us) is now written in the form corresponding to the Theorem 7.1; it remains to check the hypotheses of this theorem onΩs, namely, that (1)
G is uniformly elliptic with respect to ψts=φt∘ϕs-1; that is, there exist two real numbers λs,Λs>0 such that
(7.6)∀x∈Ωs,∀ξ∈ℝ2m,λs|ξ|2≤Gij(x,D2(ψts)(x))ξiξj≤Λs|ξ|2.
Moreover, we will impose ourselves to find real numbers λs,Λs independent of t.(2)
G is concave with respect to ψts in the variable r. Since G is of class C2, this concavity condition is equivalent to
(7.7)∀x∈Ωs,∀ζ∈ℝ2m×2m,Gij,kℓ(x,D2(ψts)(x))ζijζkℓ≤0.
This has been checked before.(3)
The derivativesGx, Gr, Gxx, and Grx are controlled (these quantities are evaluated at (x,D2(ψts)(x))). Once these three points checked, and since we have a C2 estimate of φt by C11, Theorem 7.1 allows us to deduce that for all open set Ωs′⊂⊂Ωs there exist two real numbers βs∈]0,1] and Cstes>0 depending only on m, λs, Λs, dist(Ωs′,∂Ωs), on the uniform estimate of |ψts|2;Ωs′, and on the uniform estimates of the quantities Gx, Gr, Gxx, and Grx, so in particular βs and Cstes are independent of t, such that
(7.8)[D2(ψts)]βs;Ωs′≤Cstes.The Choice ofΩs and Ωs′
Let us denote byKs the support of the function θs∘ϕs-1:
(7.9)Ks:=supp(θs∘ϕs-1)=ϕs(suppθs)⊂ϕs(Us).
The set Ks is compact, and it is included in the open set ϕs(Us) of ℝ2m, and ℝ2m is separated locally compact; then by the theorem of intercalation of relatively compact open sets, applied twice, we deduce the existence of two relatively compact open sets Ωs and Ωs′ such that
(7.10)Ks⊂Ωs′⊂⊂Ωs⊂⊂ϕs(Us).
The set Ωs is required to be connected: for this, it suffices that Ks be connected even after restriction to a connected component in Ωs of a point of Ks; indeed, this connected component is an open set of Ωs since Ωs is locally connected (as an open set of ℝ2m); moreover it is bounded since Ωs is bounded.Application of the Theorem
Letβ:=minβs; the norm ∥·∥C2,β is submultiplicative; then
(7.11)∥φt∥C2,β(M)ℛ,𝒫=∑s=1N|(θs∘ϕs-1)×(φs∘ϕs-1)|2,β;Ωs′≤∑s=1N|θs∘ϕs-1|2,β;Ωs′×|ψts|2,β;Ωs′.
But, by (7.8) we have |ψts|2,βs;Ωs′=|ψts|2;Ωs′+[D2(ψts)]βs;Ωs′≤|ψts|2;Ωs′+Cstes≤Cstes′ where Cstes′ depends only on m, λs, Λs, dist(Ωs′,∂Ωs), C11 (the constant of the C2 estimate) and the uniform estimates of the quantities Gx, Gr, Gxx, and Grx. We obtain consequently the needed C2,β estimate:
(7.12)∥φt∥C2,β(M)ℛ,𝒫≤∑s=1N|θs∘ϕs-1|2,β;Ωs′×Cstes′=:C12.
Let us now check the hypotheses 1 and 3 above.


## 7.2. Uniform Ellipticity ofG on Ωs

Letx∈Ωs and ξ∈ℝ2m:
(7.13)∑i,j=12mGij(x,r)ξiξj=d(G(x,·))r(M)withM=[ξiξj]1≤i,j≤m∈S2m(ℝ)=d(Fk∘ρϕs-1(x))r(M)=d(Fk)ρϕs-1(x)(r)·d(ρϕs-1(x))r(M).
Let us recall that ρP(r)=[δij+(1/4)∑ℓ,o=1m‍(g-1/2(P))iℓ(g-1/2(P))oj(rℓo+r(ℓ+m)(o+m)+irℓ(o+m)-ir(ℓ+m)o)]1≤i,j≤m (cf. (6.36)); we consequently obtain
(7.14)∑i,j=12mGij(x,D2(ψts)(x))ξiξj=d(Fk)ρϕs-1(x)(D2(ψts)(x))·[14∑ℓ,o=1m(g-1/2(ϕs-1(x)))iℓ(g-1/2(ϕs-1(x)))oj×(Mℓo+M(ℓ+m)(o+m)+iMℓ(o+m)-iM(ℓ+m)o)∑ℓ,o=1m14]1≤i,j≤m.
In the following, we denote M~:=[(1/4)(Mℓs+M(ℓ+m)(s+m)+iMℓ(s+m)-iM(ℓ+m)s)]1≤ℓ,s≤m. Thus
(7.15)M~=[14(ξℓξs+ξℓ+mξs+m+iξℓξs+m-iξℓ+mξs)]1≤ℓ,s≤m∈ℋm(ℂ)=[14(ξℓ-iξℓ+m)(ξs+iξs+m︸)=:ξ~s]1≤ℓ,s≤m=[14ξ~¯ℓξ~s]1≤ℓ,s≤m.
Besides, let us denote di=σk-1,i[λ(g-1g~φt(ϕs-1(x)))]/σk[λ(g-1g~φt(ϕs-1(x)))] and g-1/2 instead of g-1/2(ϕs-1(x)) in order to lighten the formulas. We obtain by the invariance formula (2.7) that
(7.16)∑i,j=12mGij(x,D2(ψts)(x))ξiξj=d(Fk)[g]-1/2g~φt[g]-1/2·([g]-1/2M~[g]-1/2)=d(Fk)diag(λ1,…,λm)·(U¯t[g]-1/2M~[g]-1/2U)whereU∈Um(ℂ)withU¯t[g]-1/2g~φt[g]-1/2U=diag(λ1,…,λm)we  are  at  the  pointϕs-1(x)=∑i=1mdi(U¯t[g]-1/2M~[g]-1/2U)ii=∑i=1mdi(([g]-1/2U¯)tM~([g]-1/2U))ii=∑i,ℓ,j=1mdi([g]-1/2U¯)ℓiM~ℓj([g]-1/2U)ji=∑i,ℓ,j=1mdi([g]-1/2U¯)ℓi14ξ~¯ℓξ~j([g]-1/2U)ji=14∑i=1mdi(∑j=1m‍ξ~j([g]-1/2U)ji)︸=:αi(∑ℓ=1m‍ξ~¯ℓ([g]-1/2U¯)ℓi)︸=αi¯=14∑i=1mdi|αi|2.
But by Proposition 6.10 and the inequalities e-2∥f∥∞(mk)≤σk(λ(g-1g~φt(P)))≤e2∥f∥∞(mk), we have for (6.72)
(7.17)e-2∥f∥∞E0(mk)≤di≤e2∥f∥∞F0(mk).
Combining (7.16) and (7.17), we obtain
(7.18)14e-2∥f∥∞E0(mk)(∑i=1m|αi|2)≤∑i,j=12mGij(x,D2(ψts)(x))ξiξj≤14e2∥f∥∞F0(mk)(∑i=1m|αi|2).
But
(7.19)∑i=1m‍|αi|2=∑i=1m|∑j=1m‍ξ~j([g]-1/2U)ji|2=∑i=1m(∑j=1m‍ξ~j([g]-1/2U)ji)(∑ℓ=1m‍ξ~¯ℓ([g]-1/2U¯)ℓi)=∑j,ℓ=1m{∑i=1m‍([g]-1/2U)ji([g]-1/2U¯)ℓi}ξ~jξ~¯ℓ=∑j,ℓ=1m(([g]-1/2U)×([g]-1/2U¯)t)jℓξ~jξ~¯ℓ.
And ([g]-1/2U)×([g]-1/2U¯)t=[g]-1/2UtU¯t[g]-1/2¯=[g]-1/2t[g]-1/2¯=[g]-1/2[g]-1/2=[g]-1; then
(7.20)∑i=1m‍|αi|2=∑j,ℓ=1m‍([g]-1)jℓξ~jξ~¯ℓ=∑j,ℓ=1m‍gℓj-(ϕs-1(x))ξ~jξ~¯ℓ.
Consequently, and since |ξ|2=|ξ~|2, the checking of the hypothesis of uniform ellipticity of the Theorem 7.1 is reduced to find two real numbers λso,Λso>0 such that
(7.21)∀x∈Ωs,∀ξ~∈ℂm,λso|ξ~|2≤∑j,ℓ=1mgℓj-(ϕs-1(x))ξ~ℓξ~¯j≤Λso|ξ~|2.
By the min-max principle applied on ℂm to the Hermitian form ≺X,Y≻g(ϕs-1(x))=gab-(ϕs-1(x))XaY¯b relatively to the canonical one, we have
(7.22)λmin[gab-(ϕs-1(x))]1≤a,b≤m|ξ~|2≤∑a,b=1m‍gab-(ϕs-1(x))ξ~aξ~¯b≤λmax[gab-(ϕs-1(x))]1≤a,b≤m|ξ~|2.
But the functions P↦λmin[gab-(P)]1≤a,b≤m and P↦λmax[gab-(P)]1≤a,b≤m are continuous on ϕs-1(Ωs)¯⊂Us which is compact since it is a closed set of the compact manifold M (cf. (7.5) for the choice of the domains Ωs), so these functions are bounded and reach their bounds; thus
(7.23)(minP∈ϕs-1(Ωs)¯λmin[gab-(P)]1≤a,b≤m)︸=:λso×|ξ~|2≤∑a,b=1m‍gab-(ϕs-1(x))ξ~aξ~¯b≤(maxP∈ϕs-1(Ωs)¯λmax[gab-(P)]1≤a,b≤m)︸=:Λso×|ξ~|2.
By the inequalities (7.18) and (7.23), we deduce that
(7.24)λs|ξ~|2≤∑i,j=12m‍Gij(x,D2(ψts)(x))ξiξj≤Λs|ξ~|2withλs:=14e-2∥f∥∞E0(mk)λso,Λs:=14e2∥f∥∞F0(mk)Λso.
The real numbers λs and Λs depend on k, m, ∥f∥∞, E0, F0, g, (Us,ϕs), and Ωs and are independent of t, x and ξ~, which achieves the proof of the global uniform ellipticity.


## 7.3.Uniform Estimate ofGx,Gr,Gxx, andGrx

In this subsection, we estimate uniformly the quantitiesGx, Gr,Gxx, and Grx (recall that these quantities are evaluated at (x,D2(ψts)(x))) by using the same technique as in the previous subsection for the proof of uniform ellipticity (7.24).We have(7.25)|Gx|2=|[Gxi]1≤i≤2m|2=∑i=12m‍|Gxi|2whereGxi=∂G∂xi(x,D2(ψts)(x)).
For (7.14), we obtain
(7.26)Gxi=d(Fk)[g-1g~φt(ϕs-1(x))]·([∑ℓ=1m‍∂(gqℓ-∘ϕs-1)∂xi(x)∂oℓ-φt(ϕs-1(x))]1≤o,q≤m︸)=:Mo-t∂(f∘ϕs-1)∂xi(x)
and for (7.16), we infer then by the invariance formula (2.7) that
(7.27)Gxi=∑j=1m‍dj(U¯tMoU)jj-t∂f∂xi(ϕs-1(x)),
where U∈Um(ℂ) such that (U¯t[g-1g~φt(ϕs-1(x))]U=diag(λ1,…,λm) and di=σk-1,i[λ(g-1g~φt(ϕs-1(x)))]/σk[λ(g-1g~φt(ϕs-1(x)))]. We can then write:
(7.28)Gxi=∑j,p,q=1m‍djUpj¯UqjMpqo-t∂f∂xi(ϕs-1(x))=∑j,p,q=1m‍djUpj¯Uqj(∑ℓ=1m‍∂gqℓ-∂xi(ϕs-1(x))∂pℓ-φt(ϕs-1(x)))-t∂f∂xi(ϕs-1(x)).Thus|Gxi|≤∑j,p,q,ℓ=1m‍e2∥f∥∞F0(mk)|Upj¯||Uqj|×(max1≤a,b≤m,1≤i≤2mmaxP∈ϕs-1(Ωs)¯|∂gab-∂xi(P)|︸)=:Λs1∥φt∥C2(M,ℝ)+∥f∥C1(M,ℝ).
But U∈Um(ℂ); then |Uqj|≤1 for all 1≤q, j≤m, consequently
(7.29)|Gxi|≤m4e2∥f∥∞F0(mk)Λs1∥φt∥C2(M,ℝ)︸≤C11(C2estimate)+∥f∥C1(M,ℝ),
which gives the needed uniform estimate for Gx:
(7.30)|Gx|≤2m(m4e2∥f∥∞F0(mk)Λs1C11+∥f∥C1(M,ℝ)).
Similarly
(7.31)|Gr|2=|[Gpq]1≤p,q≤2m|2=∑p,q=12m‍|Gpq|2,whereGpq=∂G∂rpq(x,D2(ψts)(x)).
And we have
(7.32)Gpq=d(Fk)[g-1g~φt(ϕs-1(x))]·[∑ℓ=1m‍gjℓ-(ϕs-1(x))(Epq~)iℓ-]1≤i,j≤m︸=:M1,
where Epq is the m×m matrix whose all coefficients are equal to zero except the coefficient pq which is equal to 1, and the matrix (Epq~) is obtained from Epq by the formula M~:=[(1/4)(Mℓs+M(ℓ+m)(s+m)+iMℓ(s+m)-iM(ℓ+m)s)]1≤ℓ,s≤m, thus
(7.33)Gpq=∑j=1m‍dj(tU¯M1U)jj,
where  U  and  di  are as before for  Gx.Since|(Epq~)iℓ-|≤1 for all 1≤i,ℓ≤m, we obtain for Gx that
(7.34)|Gpq|≤m4e2∥f∥∞F0(mk)Λs2,
where Λs2=max1≤a,b≤mmaxP∈ϕs-1(Ωs)¯|gab-(P)|, which gives the needed uniform estimate for Gr:
(7.35)|Gr|≤2m5e2∥f∥∞F0(mk)Λs2.
Concerning Gxx, we have
(7.36)|Gxx|2=|[Gxpxq]1≤p,q≤2m|2=∑p,q=12m‍|Gxpxq|2,whereGxpxq=∂2G∂xp∂xq(x,D2(ψts)(x)).
A calculation shows that
(7.37)Gxpxq=-t∂2f∂xp∂xq(ϕs-1(x))+∑i,j,ℓ=1m‍∂Fk∂Bij([g-1g~φt(ϕs-1(x))])∂2gjℓ-∂xp∂xq(ϕs-1(x))∂iℓ-φt(ϕs-1(x))+∑i,j,ℓ,μ,o,ν=1m‍∂2Fk∂Bμo∂Bij([g-1g~φt(ϕs-1(x))])︸=:ℰ×∂goν-∂xp(ϕs-1(x))∂gjℓ-∂xq(ϕs-1(x))∂μν-φt(ϕs-1(x))∂iℓ-φt(ϕs-1(x)).
All the terms are uniformly bounded; it remains to justify that the term in second derivative ℰ is also uniformly bounded:
(7.38)ℰ=d2(Fk)[g-1g~φt(ϕs-1(x))]·(Eμo,Eij)then  by  the  invariance  formula(2.7)=∑a,b,c,d=1m‍∂2Fk∂Bab∂Bcd[diag(λ1,…,λm)](tU¯EμoU)ab(tU¯EijU)cd,whereU∈Um(ℂ)is  like  before.
But we know the second derivatives of Fk at a diagonal matrix by (2.5). Besides, we have 0<σk-1,i(λ)/σk(λ)=di≤e2∥f∥∞F0/(mk) by (7.17), and since e-2∥f∥∞(mk)≤σk(λ), it remains only to control the quantities |σk-2,ij(λ)| with i≠j to prove that ℰ is uniformly bounded. But since λ∈Γk, we have σk-2,ij(λ)>0 [11]. Moreover, by the pinching of the eigenvalues, we deduce automatically that
(7.39)σk-2,ij(λ)≤(m-2k-2)(C2′)k-1=:F1,
which achieves the checking of the fact that Gxx is uniformly bounded.Similarly, we establish a uniform estimate ofGxr using this calculation:
(7.40)Gxo,pq=∂2G∂xo∂rpq(x,D2(ψts)(x))=∑i,j,ℓ=1m‍∂Fk∂Bij([g-1g~φt(ϕs-1(x))])∂gjℓ-∂xo(ϕs-1(x))(Epq~)iℓ-+∑i,j,ℓ,ν,μ,γ=1m‍∂2Fk∂Bνμ∂Bij([g-1g~φt(ϕs-1(x))])×∂gμγ-∂xo(ϕs-1(x))∂νγ-φt(ϕs-1(x))gjℓ-(ϕs-1(x))(Epq~)iℓ-,
which achieves the proof of the C2,β estimate.



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*Source: 350183-2012-12-27.xml*