Complex Hessian Equations on Some Compact Kähler Manifolds

Abstract

On a compact connected 2𝑚-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 𝜔^𝑘∧𝜔^𝑚−𝑘=𝑒^𝑓𝜔^𝑚, relying on the notion of 𝑘-positivity for 𝜔∈[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.

1. The Theorem

All manifolds considered in this paper are connected.

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 𝑇^1,0 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 𝑇^1,0 satisfying ℎ(𝑈,𝑉)=ℎ(𝑈,𝐴𝑉)∀𝑈,𝑉∈𝑇^1,0. (1.1) Calculations infer that the endomorphism 𝐴 writes 𝐴∶𝑇^1,0→𝑇^1,0,𝑈^𝑖𝜕_𝑖↦𝐴^𝑗_𝑖𝑈^𝑖𝜕_𝑗=𝑔^𝑗ℓ𝑔_𝑖ℓ𝑈^𝑖𝜕_𝑗. (1.2)𝐴 is a self-adjoint/Hermitian endomorphism of the Hermitian space (𝑇^1,0,ℎ), therefore 𝜆(𝑔⁻¹𝑔)∈ℝ^𝑚. Let us consider the following cone: Γ_𝑘={𝜆∈ℝ^𝑚/∀1≤𝑗≤𝑘,𝜎_𝑗(𝜆)>0}, where 𝜎_𝑗 denotes the 𝑗th elementary symmetric function.

Definition 1.1. 𝜑is said to be 𝑘-admissible if and only if 𝜆(𝑔⁻¹𝑔)∈Γ_𝑘.

In this paper, we prove the following theorem.

Theorem 1.2 (the 𝜎_𝑘 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 ∫_𝑀𝑒^𝑓𝜔^𝑚=(^𝑚_𝑘)∫_𝑀𝜔^𝑚. There exists a unique function 𝜑∶𝑀→ℝ of class 𝐶^∞ such that (1)_𝑀𝜑𝜔^𝑚=0,(2)𝜔^𝑘∧𝜔^𝑚−𝑘=𝑒^𝑓(^𝑚_𝑘)𝜔^𝑚. (1.3)𝐸_𝑘
Moreover the solution 𝜑 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 𝐸_𝑘 on ℂ^𝑚, 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 function 𝑓 appearing in the second member of 𝐸_𝑘 satisfies necessarily the normalisation condition ∫_𝑀𝑒^𝑓𝜔^𝑚=(^𝑚_𝑘)∫_𝑀𝜔^𝑚. Indeed, this results from the following lemma.

Lemma 1.3. Consider ∫_𝑀𝜔^𝑘∧𝜔^𝑚−𝑘=∫_𝑀𝜔^𝑚.

Proof. See [9, page 44].

Let us write 𝐸_𝑘 differently.

Lemma 1.4. Consider 𝜔^𝑘∧𝜔^𝑚−𝑘=(𝜎_𝑘(𝜆(𝑔⁻¹𝑔))/(^𝑚_𝑘))𝜔^𝑚.

Proof. Let 𝑃∈𝑀. It suffices to prove the equality at 𝑃 in a 𝑔-normal 𝑔-adapted chart 𝑧 centered at 𝑃. In such a chart 𝑔_𝑖𝑗(0)=𝛿_𝑖𝑗 and 𝑔_𝑖𝑗(0)=𝛿_𝑖𝑗𝜆_𝑖(0), so at 𝑧=0, 𝜔=𝑖𝑑𝑧^𝑎∧𝑑𝑧^𝑎 and 𝜔=𝑖𝜆_𝑎(0)𝑑𝑧^𝑎∧𝑑𝑧^𝑎. Thus 𝜔^𝑘∧𝜔^𝑚−𝑘=_𝑎𝑖𝜆_𝑎(0)𝑑𝑧^𝑎∧𝑑𝑧^𝑎𝑘∧_𝑏𝑖𝑑𝑧^𝑏∧𝑑𝑧^{𝑏𝑚−𝑘}=_{𝑎1,…,𝑎𝑘∈{}1,…,𝑚distinctintegers𝑏1,…,𝑏𝑚−𝑘∈{}⧵𝑎1,…,𝑚1,…,𝑎𝑘distinctintegers}𝑖^𝑚𝜆_𝑎1(0)⋯𝜆_𝑎𝑘(0)𝑑𝑧^𝑎1∧𝑑𝑧^𝑎1∧⋯∧𝑑𝑧^𝑎𝑘∧𝑑𝑧^𝑎𝑘∧𝑑𝑧^𝑏1∧𝑑𝑧^𝑏1∧⋯∧𝑑𝑧^{𝑏𝑚−𝑘}∧𝑑𝑧^{𝑏𝑚−𝑘}. (1.4) Now 𝑎₁,…,𝑎_𝑘,𝑏₁,…,𝑏_𝑚−𝑘 are 𝑚 distinct integers of {1,…,𝑚} and 2-forms commute therefore, 𝜔^𝑘∧𝜔^𝑚−𝑘=_{𝑎1,…,𝑎𝑘∈{}1,…,𝑚distinctintegers(𝑏1,…,𝑏𝑚−𝑘)∈{1,…,𝑚}⧵{𝑎1,…,𝑎𝑘}distinctintegers}𝜆_𝑎1(0)⋯𝜆_𝑎𝑘(0)𝑖^𝑚𝑑𝑧¹∧𝑑𝑧¹∧⋯∧𝑑𝑧^𝑚∧𝑑𝑧^𝑚₌𝜔^𝑚=𝑚!_{𝑎1,…,𝑎𝑘∈{}1,…,𝑚distinctintegers}(𝑚−𝑘)!𝜆_𝑎1(0)⋯𝜆_𝑎𝑘(0)𝜔^𝑚𝜔𝑚!_𝑘∧𝜔^𝑚−𝑘=(𝑚−𝑘)!𝑚!𝑘!𝜎_𝑘𝜆₁(0),…,𝜆_𝑚(0)𝜔^𝑚=𝜎_𝑘𝜆𝑔⁻¹𝑔(^𝑚_𝑘)𝜔^𝑚. (1.5)

Consequently, 𝐸_𝑘 writes: 𝜎_𝑘𝜆𝑔⁻¹𝑔=𝑒^𝑓. 𝐸^′_𝑘 Let us remark that 𝐸_𝑚 corresponds to the Calabi-Yau equation det(𝑔)/det(𝑔)=𝑒^𝑓, 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 𝑇^1,0 of eigenvectors 𝑒₁,…,𝑒_𝑚: 𝐴𝑒_𝑖=𝜆_𝑖𝑒_𝑖, 𝜆=(𝜆₁,…,𝜆_𝑚)∈Γ_𝑘. At 𝑃∈𝑀 in a chart 𝑧, we have Mat_{𝜕1,…,𝜕𝑚}𝐴_𝑃=[𝐴^𝑖_𝑗(𝑧)]_{1≤𝑖,𝑗≤𝑚}, thus 𝜎_𝑘(𝜆(𝐴_𝑃))=𝜎_𝑘(𝜆([𝐴^𝑖_𝑗(𝑧)]_{1≤𝑖,𝑗≤𝑚})). In addition, 𝐴^𝑗_𝑖=𝑔^𝑗ℓ𝑔_𝑖ℓ=𝑔^𝑗ℓ(𝑔_𝑖ℓ+𝜕_𝑖ℓ𝜑)=𝛿^𝑗_𝑖+𝑔^𝑗ℓ𝜕_𝑖ℓ𝜑, so the equation writes locally: 𝜎_𝑘𝜆𝛿^𝑗_𝑖+𝑔^𝑗ℓ𝜕_𝑖ℓ𝜑_{1≤𝑖,𝑗≤𝑚}=𝑒^𝑓. 𝐸_𝑘^′′ Let us notice that a solution of this equation 𝐸_𝑘^′′ is necessarily 𝑘-admissible [9, page 46]. Let us define 𝑓_𝑘(𝐵)=𝜎_𝑘(𝜆(𝐵)) and 𝐹_𝑘(𝐵)=ln𝜎_𝑘(𝜆(𝐵)) where 𝐵=[𝐵^𝑗_𝑖]_{1≤𝑖,𝑗≤𝑚} is a Hermitian matrix. The function 𝑓_𝑘 is a polynomial in the variables 𝐵^𝑗_𝑖, specifically 𝑓_𝑘(𝐵)=∑_|𝐼|=𝑘𝐵_𝐼𝐼 (sum of the principal minors of order 𝑘 of the matrix 𝐵). Equivalently 𝐸_𝑘^′′ writes: 𝐹_𝑘𝛿^𝑗_𝑖+𝑔^𝑗ℓ𝜕_𝑖ℓ𝜑_{1≤𝑖,𝑗≤𝑚}=𝑓. 𝐸_𝑘^′′′ 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 of 𝐹_𝑘

2.1. Calculation of the Derivatives at a Diagonal Matrix

The first derivatives of the symmetric polynomial 𝜎_𝑘 are given by the following: for all 1≤𝑖≤𝑚, (𝜕𝜎_𝑘/𝜕𝜆_𝑖)(𝜆)=𝜎_{𝑘−1,𝑖}(𝜆) where 𝜎_{𝑘−1,𝑖}(𝜆)∶=𝜎_𝑘−1|_𝜆𝑖=0. For 1≤𝑖≠𝑗≤𝑚, let us denote 𝜎_{𝑘−2,𝑖𝑗}(𝜆)∶=𝜎_𝑘−2|_{𝜆𝑖=𝜆𝑗=0} and 𝜎_{𝑘−2,𝑖𝑖}(𝜆)=0. The second derivatives of the polynomial 𝜎_𝑘 are given by (𝜕²𝜎_𝑘/𝜕𝜆_𝑖𝜕𝜆_𝑗)(𝜆)=𝜎_{𝑘−2,𝑖𝑗}(𝜆). We calculate the derivatives of the function 𝑓_𝑘∶ℋ_𝑚(ℂ)→ℝ, where ℋ_𝑚(ℂ) denotes the set of Hermitian matrices, at diagonal matrices using the formula: 𝑓_𝑘(𝐵)=_{1≤𝑖1<⋯<𝑖𝑘≤𝑚𝜎∈𝑆𝑘}𝜀(𝜎)𝐵^𝑖𝜎(1)_𝑖1⋯𝐵^{𝑖𝜎(𝑘)}_𝑖𝑘=1𝑘!_{1≤𝑖1,…,𝑖𝑘,𝑗1,…,𝑗𝑘≤𝑚}𝜀^{𝑖1⋯𝑖𝑘}_{𝑗1⋯𝑗𝑘}𝐵^𝑗1_𝑖1⋯𝐵^𝑗𝑘_𝑖𝑘, (2.1) where 𝜀^{𝑖1⋯𝑖𝑘}_{𝑗1⋯𝑗𝑘}=1if𝑖₁,…,𝑖_𝑘distinctand𝑗₁,…,𝑗_𝑘evenpermutationof𝑖₁,…,𝑖_𝑘,−1if𝑖₁,…,𝑖_𝑘distinctand𝑗₁,…,𝑗_𝑘oddpermutationof𝑖₁,…,𝑖_𝑘,0else. (2.2) These derivatives are given by [9, page 48] 𝜕𝑓_𝑘𝜕𝐵^𝑗_𝑖diag𝑏₁,…,𝑏_𝑚=𝜎0if𝑖≠𝑗,_{𝑘−1,𝑖}𝑏₁,…,𝑏_𝑚𝜕if𝑖=𝑗,if𝑖≠𝑗²𝑓_𝑘𝜕𝐵^𝑗_𝑗𝜕𝐵^𝑖_𝑖diag𝑏₁,…,𝑏_𝑚=𝜎_{𝑘−2,𝑖𝑗}𝑏₁,…,𝑏_𝑚𝜕²𝑓_𝑘𝜕𝐵^𝑖_𝑗𝜕𝐵^𝑗_𝑖diag𝑏₁,…,𝑏_𝑚=−𝜎_{𝑘−2,𝑖𝑗}𝑏₁,…,𝑏_𝑚, (2.3) and all the other second derivatives of 𝑓_𝑘 at diag(𝑏₁,…,𝑏_𝑚) vanish.

Consequently, the derivatives of the function 𝐹_𝑘=ln𝑓_𝑘∶𝜆⁻¹(Γ_𝑘)⊂ℋ_𝑚(ℂ)→ℝ at diagonal matrices diag(𝜆₁,…,𝜆_𝑚) with 𝜆=(𝜆₁,…,𝜆_𝑚)∈Γ_𝑘, where 𝜆⁻¹(Γ_𝑘)={𝐵∈ℋ_𝑚(ℂ)/𝜆(𝐵)∈Γ_𝑘}, are given by 𝜕𝐹_𝑘𝜕𝐵^𝑗_𝑖diag𝜆₁,…,𝜆_𝑚=𝜎0if𝑖≠𝑗,_{𝑘−1,𝑖}(𝜆)𝜎_𝑘(𝜆)if𝑖=𝑗,𝜕if𝑖≠𝑗²𝐹_𝑘𝜕𝐵^𝑖_𝑗𝜕𝐵^𝑗_𝑖diag𝜆₁,…,𝜆_𝑚𝜎=−^{𝑘−2,𝑖𝑗}(𝜆)𝜎_𝑘(𝜆)𝜕²𝐹_𝑘𝜕𝐵^𝑗_𝑗𝜕𝐵^𝑖_𝑖diag𝜆₁,…,𝜆_𝑚=𝜎_{𝑘−2,𝑖𝑗}(𝜆)𝜎_𝑘(𝜆)−𝜎_{𝑘−1,𝑖}(𝜆)𝜎_{𝑘−1,𝑗}(𝜆)𝜎_𝑘(𝜆)²𝜕²𝐹_𝑘𝜕𝐵^𝑖_𝑖𝜕𝐵^𝑖_𝑖diag𝜆₁,…,𝜆_𝑚𝜎=−^{𝑘−1,𝑖}(𝜆)²𝜎_𝑘(𝜆)² (2.4)(2.5) and all the other second derivatives of 𝐹_𝑘 at diag(𝜆₁,…,𝜆_𝑚) vanish.

2.2. The Invariance of 𝐹_𝑘 and of Its First and Second Differentials

The function 𝐹_𝑘∶𝜆⁻¹(Γ_𝑘)→ℝ is invariant under unitary similitudes: ∀𝐵∈𝜆⁻¹Γ_𝑘,∀𝑈∈𝑈_𝑚(ℂ),𝐹_𝑘(𝐵)=𝐹_𝑘^𝑡𝑈𝐵𝑈. (2.6) Differentiating the previous invariance formula (2.6), we show that the first and second differentials of 𝐹_𝑘 are also invariant under unitary similitudes: ∀𝐵∈𝜆⁻¹Γ_𝑘,∀𝜁∈ℋ_𝑚(ℂ),∀𝑈∈𝑈_𝑚(ℂ),𝑑𝐹_𝑘𝐵⋅𝜁=𝑑𝐹_𝑘𝑡𝑈_𝐵𝑈⋅^𝑡𝑈𝜁𝑈,∀𝐵∈𝜆⁻¹Γ_𝑘,∀𝜁∈ℋ_𝑚(ℂ),∀Θ∈ℋ_𝑚(ℂ),∀𝑈∈𝑈_𝑚(ℂ),𝑑²𝐹_𝑘𝐵⋅(𝜁,Θ)=𝑑²𝐹_𝑘𝑡𝑈_𝐵𝑈⋅^𝑡𝑈𝜁𝑈,^𝑡𝑈Θ𝑈. (2.7)(2.8) These invariance formulas are allowed to come down to the diagonal case, when it is useful.

2.3. Concavity of 𝐹_𝑘

We prove in [9] the concavity of the functions 𝑢∘𝜆 and more generally 𝑢∘𝜆_𝐵 when 𝑢∈Γ₀(ℝ^𝑚) and is symmetric [9, Theorem  VII.4.2], which in particular gives the concavity of the functions 𝐹_𝑘=ln𝜎_𝑘𝜆 [9, Corollary VII.4.30] and more generally ln𝜎_𝑘𝜆_𝐵 [9, Theorem VII.4.29]. In this section, let us show by an elementary calculation the concavity of the function 𝐹_𝑘.

Proposition 2.1. The function 𝐹_𝑘∶𝜆⁻¹(Γ_𝑘)→ℝ,𝐵↦𝐹_𝑘(𝐵)=ln𝜎_𝑘(𝜆(𝐵)) is concave (this holds for all 𝑘∈{1,…,𝑚}).

Proof. The function 𝐹_𝑘 is of class 𝐶², so its concavity is equivalent to the following inequality: ∀𝐵∈𝜆⁻¹Γ_𝑘,∀𝜁∈ℋ_𝑚(ℂ)^𝑚_{𝑖,𝑗,𝑟,𝑠=1}𝜕²𝐹_𝑘𝜕𝐵^𝑠_𝑟𝜕𝐵^𝑗_𝑖(𝐵)𝜁^𝑗_𝑖𝜁^𝑠_𝑟≤0. (2.9) Let 𝐵∈𝜆⁻¹(Γ_𝑘), 𝜁∈ℋ_𝑚(ℂ), and 𝑈∈𝑈_𝑚(ℂ) such that ^𝑡𝑈𝐵𝑈=diag(𝜆₁,…,𝜆_𝑚). We have 𝜆=(𝜆₁,…,𝜆_𝑚)∈Γ_𝑘. Let us denote 𝜁̃=^𝑡𝑈𝜁𝑈∈ℋ_𝑚(ℂ): 𝑆∶=^𝑚_{𝑖,𝑗,𝑟,𝑠=1}𝜕²𝐹_𝑘𝜕𝐵^𝑠_𝑟𝜕𝐵^𝑗_𝑖(𝐵)𝜁^𝑗_𝑖𝜁^𝑠_𝑟=𝑑²𝐹_𝑘𝐵⋅()()=𝑑𝜁,𝜁sobytheinvarianceformula2.8₂𝐹_𝑘𝑡𝑈_𝐵𝑈⋅^𝑡𝑈𝜁𝑈,^𝑡=𝑈𝜁𝑈_{𝑚𝑖,𝑗,𝑟,𝑠=1}𝜕²𝐹_𝑘𝜕𝐵^𝑠_𝑟𝜕𝐵^𝑗_𝑖diag𝜆₁,…,𝜆_𝑚𝜁̃^𝑗_𝑖𝜁̃^𝑠_𝑟=^𝑚_{𝑖≠𝑗=1}−𝜎_{𝑘−2,𝑖𝑗}(𝜆)𝜎_𝑘(𝜆)𝜁̃^𝑗_𝑖𝜁̃^𝑖_{𝑗=̃𝜁𝑗𝑖}+^𝑚_{𝑖≠𝑗=1}𝜎_{𝑘−2,𝑖𝑗}(𝜆)𝜎_𝑘(𝜆)−𝜎_{𝑘−1,𝑖}(𝜆)𝜎_{𝑘−1,𝑗}(𝜆)𝜎_𝑘(𝜆)²_{=∶𝑐𝑖𝑗}𝜁̃^𝑖_𝑖𝜁̃^𝑗_𝑗+^𝑚_𝑖=1−𝜎_{𝑘−1,𝑖}(𝜆)²𝜎_𝑘(𝜆)²𝜁̃^𝑖_𝑖²=^𝑚_𝑖,𝑗=1−𝜎_{𝑘−2,𝑖𝑗}(𝜆)𝜎_𝑘(𝜆)𝜁̃^𝑗_𝑖²+^𝑚_𝑖,𝑗=1𝑐_𝑖𝑗𝜁̃^𝑖_𝑖𝜁̃^𝑗_𝑗. (2.10) But 𝑐_𝑖𝑗=(𝜕²(ln𝜎_𝑘)/𝜕𝜆_𝑖𝜕𝜆_𝑗)(𝜆), and 𝜁̃^𝑖_𝑖∈ℝ, so ∑^𝑚_𝑖,𝑗=1𝑐_𝑖𝑗𝜁̃^𝑖_𝑖𝜁̃^𝑗_𝑗≤0 by concavity of ln𝜎_𝑘 at 𝜆∈Γ_𝑘 [10, page 269]. In addition, 𝜎_{𝑘−2,𝑖𝑗}(𝜆)>0 since 𝜆∈Γ_𝑘 [11], consequently ∑^𝑚_𝑖,𝑗=1−(𝜎_{𝑘−2,𝑖𝑗}(𝜆)/𝜎_𝑘(𝜆))|𝜁̃^𝑗_𝑖|²≤0, which shows that 𝑆≤0 and achieves the proof.

3. The Proof of Uniqueness

Let 𝜑₀ and 𝜑₁ be two smooth 𝑘-admissible solutions of 𝐸_𝑘^′′′ such that ∫_𝑀𝜑₀𝜔^𝑚=∫_𝑀𝜑₁𝜔^𝑚=0. For all 𝑡∈[0,1], let us consider the function 𝜑_𝑡=𝑡𝜑₁+(1−𝑡)𝜑₀=𝜑₀+𝑡𝜑 with 𝜑=𝜑₁−𝜑₀. Let 𝑃∈𝑀, and let us denote ℎ^𝑃_𝑘(𝑡)=𝑓_𝑘([𝛿^𝑗_𝑖+𝑔^𝑗ℓ(𝑃)𝜕_𝑖ℓ𝜑_𝑡(𝑃)]). We have ℎ^𝑃_𝑘(1)−ℎ^𝑃_𝑘(0)=0 which is equivalent to ∫¹₀ℎ^𝑃′_𝑘(𝑡)𝑑𝑡=0. But ℎ^𝑃′_𝑘(𝑡)=^𝑚_𝑖,𝑗=1^𝑚_ℓ=1𝜕𝑓_𝑘𝜕𝐵^ℓ_𝑖𝛿^𝑗_𝑖+𝑔^𝑗ℓ(𝑃)𝜕_𝑖ℓ𝜑_𝑡(𝑃)𝑔^ℓ𝑗(𝑃)_{=∶𝛼𝑡𝑖𝑗(𝑃)}𝜕_𝑖𝑗𝜑(𝑃). (3.1) Therefore we obtain ℒ𝜑(𝑃)∶=^𝑚_𝑖,𝑗=1𝑎_𝑖𝑗(𝑃)𝜕_𝑖𝑗𝜑(𝑃)=0with𝑎_𝑖𝑗(𝑃)=¹₀𝛼^𝑡_𝑖𝑗(𝑃)𝑑𝑡. (3.2) We show easily that the matrix [𝑎_𝑖𝑗(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, [𝜕_𝑖𝑗𝜑(𝑚₀)]_{1≤𝑖,𝑗≤2𝑚}, is positive-semidefinite.

Lemma 3.1. For all 𝑡∈[0,1], 𝜆(𝑔⁻¹𝑔_𝜑𝑡)(𝑚₀)∈Γ_𝑘; namely, the functions (𝜑_𝑡)_𝑡∈[0,1] are 𝑘-admissible at 𝑚₀.

Proof. Let us denote 𝒲∶={𝑡∈[0,1]/𝜆(𝑔⁻¹𝑔_𝜑𝑡)(𝑚₀)∈Γ_𝑘}. 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 𝜎_𝑞(𝜆(𝑔⁻¹𝑔_𝜑𝑡)(𝑚₀))−𝜎_𝑞(𝜆(𝑔⁻¹𝑔_𝜑0)(𝑚₀))=ℎ^𝑚0_𝑞(𝑡)−ℎ^𝑚0_𝑞(0)=∫^𝑡₀ℎ^𝑚′0_𝑞(𝑠)𝑑𝑠. Let us prove that ℎ^𝑚′0_𝑞(𝑠)≥0 for all 𝑠∈[0,𝑡[. Fix 𝑠∈[0,𝑡[; the quantity ℎ^𝑚′0_𝑞(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 𝑚₀ℎ^𝑚′0_𝑞(𝑠)=^𝑚_{𝑖,𝑗,ℓ=1}𝜕𝑓_𝑞𝜕𝐵^𝑗_𝑖𝛿^𝑗_𝑖+𝑔^𝑗ℓ𝑚₀𝜕_𝑖ℓ𝜑_𝑠𝑚₀𝑔^𝑗ℓ𝑚₀𝜕_𝑖ℓ𝜑𝑚₀=^𝑚_𝑖=1𝜕𝜎_𝑞𝜕𝜆_𝑖𝜆𝑔⁻¹𝑔_𝜑𝑠𝑚₀𝜕_𝑖𝑖𝜑𝑚₀. (3.3) But 𝜆(𝑔⁻¹𝑔_𝜑𝑠)(𝑚₀)∈Γ_𝑘⊂Γ_𝑞 since 𝑠∈[0,𝑡[⊂𝒲, then (𝜕𝜎_𝑞/𝜕𝜆_𝑖)(𝜆(𝑔⁻¹𝑔_𝜑𝑠)(𝑚₀))>0 for all 1≤𝑖≤𝑚. Besides, 𝜕_𝑖𝑖𝜑(𝑚₀)≥0 since the matrix [𝜕_𝑖𝑗𝜑(𝑚₀)]_{1≤𝑖,𝑗≤𝑚} is positive-semidefinite. Therefore, we infer that ℎ^𝑚′0_𝑞(𝑠)≥0. Consequently, we obtain that 𝜎_𝑞(𝜆(𝑔⁻¹𝑔_𝜑𝑡)(𝑚₀))≥𝜎_𝑞(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)∈Γ_𝑘 which proves that 𝑡∈𝒲. This is a contradiction; we infer then that 𝒲=[0,1].

We check easily that the Hermitian matrix [𝑎_𝑖𝑗(𝑚₀)]_{1≤𝑖,𝑗≤𝑚} is positive definite [9, page 54] and deduce then the following lemma since the map 𝑃↦𝑎_𝑖𝑗(𝑃)=∫¹₀(∑^𝑚_ℓ=1(𝜕𝑓_𝑘/𝜕𝐵^ℓ_𝑖)([𝛿^𝑗_𝑖+𝑔^𝑗ℓ(𝑃)𝜕_𝑖ℓ𝜑_𝑡(𝑃)])𝑔^ℓ𝑗(𝑃))𝑑𝑡 is continuous on a neighbourhood of 𝑚₀.

Lemma 3.2. There exists an open ball 𝐵_𝑚0 centered at 𝑚₀ such that for all 𝑃∈𝐵_𝑚0 the Hermitian matrix [𝑎_𝑖𝑗(𝑃)]_{1≤𝑖,𝑗≤𝑚} is positive definite.

Consequently, the operator ℒ is elliptic on the open set 𝐵_𝑚0. But the map 𝜑 is 𝐶^∞, assumes its minimum at 𝑚₀∈𝐵_𝑚0, and satisfies ℒ𝜑=0; then by the Hopf maximum principle [12], we deduce that 𝜑(𝑃)=𝜑(𝑚₀) for all 𝑃∈𝐵_𝑚0. 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 𝐵_𝑚 centered at 𝑚 such that for all 𝑃∈𝐵_𝑚𝜑(𝑃)=𝜑(𝑚) so for all 𝑃∈𝐵_𝑚𝜑(𝑃)=𝜑(𝑚₀) then 𝐵_𝑚⊂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 ∫_𝑀𝜑𝜔^𝑚=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 𝐸_𝑘,𝑡, 𝑡∈[0,1]ℱ_𝑘𝜑_𝑡∶=𝐹_𝑘𝛿^𝑗_𝑖+𝑔^𝑗ℓ𝜕_𝑖ℓ𝜑_{𝑡1≤𝑖,𝑗≤𝑚}(=𝑡𝑓+ln^𝑚𝑘)∫_𝑀𝜔^𝑚∫_𝑀𝑒^𝑡𝑓𝜔^𝑚_𝐴𝑡. 𝐸_𝑘,𝑡 The function 𝜑₀≡0 is a 𝑘-admissible solution of (𝐸_𝑘,0): 𝜎_𝑘(𝜆([𝛿^𝑗_𝑖+𝑔^𝑗ℓ𝜕_𝑖ℓ𝜑₀]_{1≤𝑖,𝑗≤𝑚}))=(^𝑚_𝑘) and satisfies ∫_𝑀𝜑₀𝜔^𝑚=0. For 𝑡=1, 𝐴₁=1 so (𝐸_𝑘,1) corresponds to 𝐸_𝑘^′′′.

Let us fix 𝑙∈ℕ, 𝑙≥5 and 0<𝛼<1, and let us consider the nonempty set (containing 0): 𝒯_𝑙,𝛼∶=𝑡∈[0,1]/𝐸_𝑘,𝑡havea𝑘-admissiblesolution𝜑∈𝐶^𝑙,𝛼(𝑀)suchthat_𝑀𝜑𝜔^𝑚=0. (4.1) The aim is to prove that 1∈𝒯_𝑙,𝛼. For this we prove, using the connectedness of [0,1], that 𝒯_𝑙,𝛼=[0,1].

4.1. 𝒯_𝑙,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: 𝑆_𝑙,𝛼∶=𝜑∈𝐶^𝑙,𝛼(𝑀),_𝑀𝜑𝜔^𝑚,𝑆=0_𝑙,𝛼∶=𝜑∈𝑆_𝑙,𝛼,𝑘-admissiblefor𝑔, (4.2) where 𝑆_𝑙,𝛼 is a vector space and 𝑆_𝑙,𝛼 is an open set of 𝑆_𝑙,𝛼. Using these notations, the set 𝒯_𝑙,𝛼 writes 𝒯_𝑙,𝛼∶={𝑡∈[0,1]/∃𝜑∈𝑆_𝑙,𝛼 solution of 𝐸_𝑘,𝑡}.

Lemma 4.1. The operator ℱ_𝑘∶𝑆_𝑙,𝛼→𝐶^{𝑙−2,𝛼}(𝑀), 𝜑↦ℱ_𝑘[𝜑]=𝐹_𝑘([𝛿^𝑗_𝑖+𝑔^𝑗ℓ𝜕_𝑖ℓ𝜑]_{1≤𝑖,𝑗≤𝑚}), is differentiable, and its differential at a point 𝜑∈𝑆_𝑙,𝛼, 𝑑ℱ_𝑘𝜑∈ℒ(𝑆_𝑙,𝛼,𝐶^{𝑙−2,𝛼}(𝑀)) is equal to 𝑑ℱ_𝑘𝜑⋅𝜓=^𝑚_𝑖,𝑗=1𝜕𝐹_𝑘𝜕𝐵^𝑗_𝑖𝛿^𝑗_𝑖+𝑔^𝑗ℓ𝜕_𝑖ℓ𝜑𝑔^𝑗ℓ𝜕_𝑖ℓ𝜓∀𝜓∈𝑆_𝑙,𝛼. (4.3)

Proof. See [9, page 60].

Proposition 4.2. The nonlinear operator ℱ_𝑘 is elliptic on 𝑆_𝑙,𝛼.

Proof. Let us fix a function 𝜑∈𝑆_𝑙,𝛼 and check that the nonlinear operator ℱ_𝑘 is elliptic for this function 𝜑. This goes back to show that the linearization at 𝜑 of the nonlinear operator ℱ_𝑘 is elliptic. By Lemma 4.1, this linearization is the following linear operator: 𝑑ℱ_𝑘𝜑⋅𝑣=^𝑚_𝑖,𝑜=1^𝑚_𝑗=1𝜕𝐹_𝑘𝜕𝐵^𝑗_𝑖𝛿^𝑗_𝑖+𝑔^𝑗𝑜𝜕_𝑖𝑜𝜑_{1≤𝑖,𝑗≤𝑚}×𝑔^𝑗𝑜𝜕_𝑖𝑜𝑣. (4.4) In order to prove that this linear operator is elliptic, it suffices to check the ellipticity in a particular chart, for example, at the center of a 𝑔-normal 𝑔_𝜑-adapted chart. At the center of such a chart, 𝑑ℱ_𝑘𝜑⋅𝑣=^𝑚_𝑖,𝑜=1𝜕𝐹_𝑘𝜕𝐵^𝑜_𝑖diag𝜆𝑔⁻¹𝑔𝜕_𝑖𝑜𝑣=^𝑚_𝑖=1𝜎_{𝑘−1,𝑖}𝜆𝑔⁻¹𝑔𝜎_𝑘𝜆𝑔⁻¹𝑔𝜕_𝑖𝑖𝑣. (4.5) But for all 𝑖∈{1,…,𝑚} we have 𝜎_{𝑘−1,𝑖}𝜆(𝑔⁻¹𝑔)/𝜎_𝑘𝜆(𝑔⁻¹𝑔)>0 on 𝑀 since 𝜆(𝑔⁻¹𝑔)∈Γ_𝑘 [11], which proves that the linearization is elliptic and achieves the proof.

Let us denote 𝔉_𝑘 the operator 𝔉_𝑘𝜑∶=𝑓_𝑘𝛿^𝑗_𝑖+𝑔^𝑗ℓ𝜕_𝑖ℓ𝜑_{1≤𝑖,𝑗≤𝑚}. (4.6) As ℱ_𝑘, the operator 𝔉_𝑘∶𝑆_𝑙,𝛼→𝐶^{𝑙−2,𝛼}(𝑀) is differentiable and elliptic on 𝑆_𝑙,𝛼 of differential 𝑑𝔉_𝑘𝜑⋅𝜓=^𝑚_𝑖,𝑗=1𝜕𝑓_𝑘𝜕𝐵^𝑗_𝑖𝛿^𝑗_𝑖+𝑔^𝑗ℓ𝜕_𝑖ℓ𝜑𝑔^𝑗ℓ𝜕_𝑖ℓ𝜓∀𝜓∈𝑆_𝑙,𝛼. (4.7) Let us denote 𝑎_𝜑 the matrix [𝛿^𝑗_𝑖+𝑔^𝑗ℓ𝜕_𝑖ℓ𝜑]_{1≤𝑖,𝑗≤𝑚} and calculate this linearization in a different way, by using the expression (2.1) of 𝑓_𝑘: 𝔉_𝑘𝜑=𝑓_𝑘𝑎_𝜑=1𝑘!_{1≤𝑖1,…,𝑖𝑘,𝑗1,…,𝑗𝑘≤𝑚}𝜀^{𝑖1⋯𝑖𝑘}_{𝑗1⋯𝑗𝑘}𝑎_𝜑^𝑗1_𝑖1⋯𝑎_𝜑^𝑗𝑘_𝑖𝑘. (4.8) Thus 𝑑𝔉_𝑘𝜑𝑑⋅𝑣=𝔉𝑑𝑡^𝑘𝜑+𝑡𝑣_|𝑡=0=𝑑1𝑑𝑡𝑘!_{1≤𝑖1,…,𝑖𝑘,𝑗1,…,𝑗𝑘≤𝑚}𝜀^{𝑖1⋯𝑖𝑘}_{𝑗1⋯𝑗𝑘}𝑎_{𝜑+𝑡𝑣}^𝑗1_𝑖1⋯𝑎_{𝜑+𝑡𝑣}^𝑗𝑘_{𝑖𝑘|𝑡=0}=1𝑘!_{1≤𝑖1,…,𝑖𝑘,𝑗1,…,𝑗𝑘≤𝑚}𝜀^{𝑖1⋯𝑖𝑘}_{𝑗1...𝑗𝑘}𝑔^𝑗1𝑠𝜕_𝑖1𝑠𝑣𝑎_𝜑^𝑗2_𝑖2⋯𝑎_𝜑^𝑗𝑘_𝑖𝑘+1𝑘!_{1≤𝑖1,…,𝑖𝑘,𝑗1,…,𝑗𝑘≤𝑚}𝜀^{𝑖1⋯𝑖𝑘}_{𝑗1⋯𝑗𝑘}𝑎_𝜑^𝑗1_𝑖1𝑔^𝑗2𝑠𝜕_𝑖2𝑠𝑣⋯𝑎_𝜑^𝑗𝑘_𝑖𝑘1+⋯+𝑘!_{1≤𝑖1,…,𝑖𝑘,𝑗1,…,𝑗𝑘≤𝑚}𝜀^{𝑖1⋯𝑖𝑘}_{𝑗1⋯𝑗𝑘}𝑎_𝜑^𝑗1_𝑖1⋯𝑎_𝜑^𝑗𝑘−1_𝑖𝑘−1𝑔^𝑗𝑘𝑠𝜕_𝑖𝑘𝑠𝑣=1(𝑘−1)!_{1≤𝑖1,…,𝑖𝑘,𝑗1,…,𝑗𝑘≤𝑚}𝜀^{𝑖1⋯𝑖𝑘}_{𝑗1⋯𝑗𝑘}𝑎_𝜑^𝑗1_𝑖1⋯𝑎_𝜑^𝑗𝑘−1_𝑖𝑘−1𝑔^𝑗𝑘𝑠𝜕_𝑖𝑘𝑠𝑣=bysymmetry_{𝑚𝑖,𝑗=1}1(𝑘−1)!_{1≤𝑖1,…,𝑖𝑘−1,𝑗1,…,𝑗𝑘−1≤𝑚}𝜀^{𝑖1⋯𝑖𝑘−1𝑖}_{𝑗1⋯𝑗𝑘−1𝑗}𝑎_𝜑^𝑗1_𝑖1⋯𝑎_𝜑^𝑗𝑘−1_{𝑖𝑘−1=∶𝒞𝑖𝑗𝑎𝜑}∇^𝑗_𝑖𝑣. (4.9) We infer then the following proposition.

Proposition 4.3. The linearization 𝑑𝔉_𝑘 of the operator 𝔉_𝑘 is of divergence type: 𝑑𝔉_𝑘𝜑=∇_𝑖𝒞^𝑖_𝑗𝑎_𝜑∇^𝑗. (4.10)

Proof. By (4.9) we have 𝑑𝔉_𝑘𝜑⋅𝑣=^𝑚_𝑖,𝑗=1𝒞^𝑖_𝑗𝑎_𝜑∇^𝑗_𝑖𝑣=^𝑚_𝑖=1∇_𝑖^𝑚_𝑗=1𝒞^𝑖_𝑗𝑎_𝜑∇^𝑗𝑣−^𝑚_𝑗=1^𝑚_𝑖=1∇_𝑖𝒞^𝑖_𝑗𝑎_𝜑∇^𝑗𝑣. (4.11) Moreover ^𝑚_𝑖=1∇_𝑖𝒞^𝑖_𝑗𝑎_𝜑=1(𝑘−2)!^𝑚_{𝑖=11≤𝑖1,…,𝑖𝑘−1,𝑗1,…,𝑗𝑘−1≤𝑚}𝜀^{𝑖1⋯𝑖𝑘−1𝑖}_{𝑗1⋯𝑗𝑘−1𝑗}𝑎_𝜑^𝑗1_𝑖1⋯𝑎_𝜑^𝑗𝑘−2_𝑖𝑘−2∇_𝑖𝑎_𝜑^𝑗𝑘−1_𝑖𝑘−1. (4.12) But ∇_𝑖((𝑎_𝜑)^𝑗𝑘−1_𝑖𝑘−1)=∇_𝑖(𝛿^𝑗𝑘−1_𝑖𝑘−1+∇^𝑗𝑘−1_𝑖𝑘−1𝜑)=∇^𝑗𝑘−1_{𝑖𝑖𝑘−1}𝜑, then ^𝑚_𝑖=1∇_𝑖𝒞^𝑖_𝑗𝑎_𝜑=1(𝑘−2)!^𝑚_{𝑖=11≤𝑖1,…,𝑖𝑘−1,𝑗1,…,𝑗𝑘−1≤𝑚}𝜀^{𝑖1⋯𝑖𝑘−1𝑖}_{𝑗1⋯𝑗𝑘−1𝑗}𝑎_𝜑^𝑗1_𝑖1⋯𝑎_𝜑^𝑗𝑘−2_𝑖𝑘−2∇^𝑗𝑘−1_{𝑖𝑖𝑘−1}𝜑. (4.13) Besides, the quantity ∇^𝑗𝑘−1_{𝑖𝑖𝑘−1}𝜑 is symmetric in 𝑖,𝑖_𝑘−1 (indeed, ∇^𝑗𝑘−1_{𝑖𝑖𝑘−1}𝜑−∇^𝑗𝑘−1_{𝑖𝑘−1𝑖}𝜑=𝑅^𝑗𝑘−1_{𝑠𝑖𝑖𝑘−1}∇^𝑠𝜑 and 𝑅^𝑗𝑘−1_{𝑠𝑖𝑖𝑘−1}=0 since 𝑔 is Kähler), and 𝜀^{𝑖1⋯𝑖𝑘−1𝑖}_{𝑗1⋯𝑗𝑘−1𝑗} is antisymmetric in 𝑖,𝑖_𝑘−1; it follows then that ∑^𝑚_𝑖=1∇_𝑖(𝒞^𝑖_𝑗(𝑎_𝜑))=0, consequently 𝑑𝔉_𝑘𝜑⋅𝑣=∑^𝑚_𝑖=1∇_𝑖(∑^𝑚_𝑗=1𝒞^𝑖_𝑗(𝑎_𝜑)∇^𝑗𝑣).

From Proposition 4.3, we infer easily [9, page 62] the following corollary.

Corollary 4.4. The map 𝐹∶𝑆_𝑙,𝛼→𝑆_{𝑙−2,𝛼},𝜑↦𝐹(𝜑)=𝔉_𝑘[𝜑]−(^𝑚_𝑘) is well defined and differentiable and its differential equals 𝑑𝐹_𝜑=𝑑𝔉_𝑘𝜑=∇_𝑖(𝒞^𝑖_𝑗(𝑎_𝜑)∇^𝑗)∈ℒ(𝑆_𝑙,𝛼,𝑆_{𝑙−2,𝛼}).

Now, let 𝑡₀∈𝒯_𝑙,𝛼 and let 𝜑₀∈𝑆_𝑙,𝛼 be a solution of the corresponding equation (𝐸_𝑘,𝑡0): 𝐹(𝜑₀)=𝑒^𝑡0𝑓𝐴_𝑡0−(^𝑚_𝑘).

Lemma 4.5. 𝑑𝐹_𝜑0∶𝑆_𝑙,𝛼→𝑆_{𝑙−2,𝛼} is an isomorphism.

Proof. Let 𝜓∈𝐶^{𝑙−2,𝛼}(𝑀) with ∫_𝑀𝜓𝑣_𝑔=0. Let us consider the equation ∇_𝑖𝒞^𝑖_𝑗𝑎_𝜑0∇^𝑗𝑢=𝜓. (4.14) We have 𝒞^𝑖_𝑗(𝑎_𝜑0)∈𝐶^{𝑙−2,𝛼}(𝑀) and the matrix [𝒞^𝑖_𝑗(𝑎_𝜑0)]_{1≤𝑖,𝑗≤𝑚}=[(𝜕𝑓_𝑘/𝜕𝐵^𝑗_𝑖)([𝛿^𝑗_𝑖+𝑔^𝑗ℓ𝜕_𝑖ℓ𝜑₀])]_{1≤𝑖,𝑗≤𝑚} is positive definite (since 𝔉_𝑘 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∈𝐶^𝑙,𝛼(𝑀) satisfying ∫_𝑀𝑢𝑣_𝑔=0 which is solution of (4.14) and then solution of 𝑑𝐹_𝜑0𝑢=𝜓. Thus, the linear continuous map 𝑑𝐹_𝜑0∶𝑆_𝑙,𝛼→𝑆_{𝑙−2,𝛼} is bijective, and its inverse is continuous by the open map theorem, which achieves the proof.

We deduce then by the local inverse mapping theorem that there exists an open set 𝑈 of 𝑆_𝑙,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 𝒯_𝑙,𝛼: if |𝑡−𝑡₀|≤𝜀 is sufficiently small then ‖(𝑒^𝑡𝑓𝐴_𝑡−(^𝑚_𝑘))−(𝑒^𝑡0𝑓𝐴_𝑡0−(^𝑚_𝑘))‖_{𝐶𝑙−2,𝛼(𝑀)} is small enough so that 𝑒^𝑡𝑓𝐴_𝑡−(^𝑚_𝑘)∈𝑉, thus there exists 𝜑∈𝑈⊂𝑆_𝑙,𝛼 such that 𝐹(𝜑)=𝑒^𝑡𝑓𝐴_𝑡−(^𝑚_𝑘) and consequently there exists 𝜑∈𝐶^𝑙,𝛼(𝑀) of vanishing integral for 𝑔 which is solution of 𝐸_𝑘,𝑡. Hence 𝑡∈𝒯_𝑙,𝛼. We conclude therefore that 𝒯_𝑙,𝛼 is an open set of [0,1].

4.2. 𝒯_𝑙,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 (𝑡_𝑠)_𝑠∈ℕ be a sequence of elements of 𝒯_𝑙,𝛼 that converges to 𝜏∈[0,1], and let (𝜑_𝑡𝑠)_𝑠∈ℕ be the corresponding sequence of functions: 𝜑_𝑡𝑠 is 𝐶^𝑙,𝛼, 𝑘-admissible, has a vanishing integral, and is a solution of 𝐹_𝑘𝛿^𝑗_𝑖+𝑔^𝑗ℓ𝜕_𝑖ℓ𝜑_{𝑡𝑠1≤𝑖,𝑗≤𝑚}=𝑡_𝑠𝑓+ln𝐴_𝑡𝑠. 𝐸_{𝑘,𝑡𝑠} Let us prove that 𝜏∈𝒯_𝑙,𝛼. Here is the scheme of the proof.(1)Reduction to a 𝐶^2,𝛽(𝑀) estimate: if (𝜑_𝑡𝑠)_𝑠∈ℕ is bounded in a 𝐶^2,𝛽(𝑀) with 0<𝛽<1, the inclusion 𝐶^2,𝛽(𝑀)⊂𝐶²(𝑀,ℝ) being compact, we deduce that after extraction (𝜑_𝑡𝑠)_𝑠∈ℕ converges in 𝐶²(𝑀,ℝ) to 𝜑_𝜏∈𝐶²(𝑀,ℝ). We show by tending to the limit that 𝜑_𝜏 is a solution of (𝐸_𝑘,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 𝜑_𝜏∈𝐶^∞(𝑀,ℝ), which allows us to deduce that 𝜏∈𝒯_𝑙,𝛼 (see [9, pages 64–67] for details). (2)We show that (𝜑_𝑡𝑠)_𝑠∈ℕ is bounded in 𝐶⁰(𝑀,ℝ): first of all we prove a positivity Lemma 5.4 for 𝐸_𝑘,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, a 𝐶² a priori estimate (Section 6). (4)With the uniform ellipticity at hand (consequence of the previous step), we obtain the needed 𝐶^2,𝛽(𝑀) estimate by the Evans-Trudinger theory (Section 7).

5. The 𝐶⁰ A Priori Estimate

5.1. The Positivity Lemma

Our first three lemmas are based on the ideas of [7, Section 2].

Lemma 5.1. Let 𝜋 be a real (1−1)-form, it then writes 𝜋=𝑖𝑝_𝑎𝑏𝑑𝑧^𝑎∧𝑑𝑧^𝑏, with 𝑝_𝑎𝑏=𝑝(𝜕_𝑎,𝜕_𝑏) where 𝑝 is the symmetric tensor 𝑝(𝑈,𝑉)=𝜋(𝑈,𝐽𝑉); hence ∀ℓ≤𝑚𝜋^ℓ∧𝜔^𝑚−ℓ=ℓ!(𝑚−ℓ)!𝜎𝑚!^ℓ𝜆𝑔⁻¹𝑝𝜔^𝑚. (5.1)

Proof. The same proof as Lemma 1.4.

We consider for 1≤ℓ≤𝑚 the map 𝑓_ℓ=𝜎_ℓ∘𝜆∶ℋ_𝑚→ℝ where ℋ_𝑚 denotes the ℝ-vector space of Hermitian square matrices of size 𝑚. 𝑓_ℓ 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 𝑓_ℓ(𝐼)=𝜎_ℓ(1,…,1)=(^𝑚_ℓ)>0. Let 𝑓_ℓ be the totally polarized form of 𝑓_ℓ. This polarized form 𝑓_ℓ∶ℋ_𝑚×⋯×ℋ_𝑚ℓtimes→ℝ is characterized by the following properties: (i)𝑓_ℓ is ℓ-linear. (ii)𝑓_ℓ is symmetric. (iii)For all 𝐵∈ℋ_𝑚, 𝑓_ℓ(𝐵,…,𝐵)=𝑓_ℓ(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 𝜋^ℓ∧𝜔^𝑚−ℓ=ℓ!(𝑚−ℓ)!𝑓𝑚!^ℓ𝑔⁻¹𝑝𝜔^𝑚. (5.2) By transition to the polarized form in this equality we obtain the following lemma.

Lemma 5.2. Let 1≤ℓ≤𝑚 and 𝜋₁,…,𝜋_ℓ be real (1−1)-forms. These forms write 𝜋_𝛼=𝑖(𝑝_𝛼)_𝑎𝑏𝑑𝑧^𝑎∧𝑑𝑧^𝑏, with (𝑝_𝛼)_𝑎𝑏=𝑝_𝛼(𝜕_𝑎,𝜕_𝑏) where 𝑝_𝛼 is the symmetric tensor 𝑝_𝛼(𝑈,𝑉)=𝜋_𝛼(𝑈,𝐽𝑉). Then, at the center of a 𝑔-unitary chart we have 𝜋₁∧⋯∧𝜋_ℓ∧𝜔^𝑚−ℓ=ℓ!(𝑚−ℓ)!𝑓𝑚!^ℓ𝑔⁻¹𝑝₁,…,𝑔⁻¹𝑝_ℓ𝜔^𝑚. (5.3)

Proof. See [9, page 71].

Theorem 5 of Gårding [16] applies to 𝑓_ℓ with 2≤ℓ≤𝑚.

Lemma 5.3 (the Gårding inequality for 𝑓_ℓ). Let 2≤ℓ≤𝑚, for all 𝑦¹,…,𝑦^ℓ∈Γ(𝑓_ℓ,𝐼), 𝑓_ℓ𝑦¹,…,𝑦^ℓ≥𝑓_ℓ𝑦^11/ℓ⋯𝑓_ℓ𝑦^ℓ1/ℓ. (5.4)

Let us recall that Γ(𝑓_ℓ,𝐼) is the connected component of {𝑦∈ℋ_𝑚/𝑓_ℓ(𝑦)>0} containing 𝐼. The same proof as [17, pages 129, 130] implies that Γ𝑓_ℓ,𝐼=𝑦∈ℋ_𝑚/∀1≤𝑖≤ℓ𝑓_𝑖𝑦>0=𝑦∈ℋ_𝑚/𝜆𝑦∈Γ_ℓ=𝜆⁻¹Γ_ℓ. (5.5)

Note that the Gårding inequality (Lemma 5.3) holds for Γ(𝑓_ℓ,𝐼)={𝑦∈ℋ_𝑚/∀1≤𝑖≤ℓ𝑓_𝑖(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). Let 𝛼 be a real 1-form on 𝑀 and 𝑗∈{1,…,𝑘−1}, then the function 𝑓∶𝑀→ℝ defined by 𝑓𝜔^𝑚=^𝑡𝐽𝛼∧𝛼∧𝜔^{𝑚−1−𝑗}∧𝜔^𝑗 is nonnegative.

Proof. Let 1≤𝑗≤𝑘−1, then 2≤ℓ=𝑗+1≤𝑘. Let 𝛼 be a real 1-form, it then writes 𝛼=𝛼_𝑎𝑑𝑧^𝑎+𝛼_𝑎𝑑𝑧^𝑎. Let 𝜋₁=^𝑡𝐽𝛼∧𝛼, hence 𝜋₁(𝜕_𝑎,𝜕_𝑏)=𝛼(𝐽𝜕_𝑎)𝛼(𝜕_𝑏)−𝛼(𝐽𝜕_𝑏)𝛼(𝜕_𝑎)=𝑖𝛼_𝑎𝛼_𝑏−(−𝑖)𝛼_𝑏𝛼_𝑎=2𝑖𝛼_𝑎𝛼_𝑏. Similarly, we prove that 𝜋₁(𝜕_𝑎,𝜕_𝑏)=𝜋₁(𝜕_𝑎,𝜕_𝑏)=0, consequently 𝜋₁=𝑖2𝛼_𝑎𝛼_{𝑏=∶𝑝𝑎𝑏}𝑑𝑧^𝑎∧𝑑𝑧^𝑏. Besides, set 𝜋₂=⋯=𝜋_𝑗+1=𝜔=𝑖𝑔_𝑎𝑏𝑑𝑧^𝑎∧𝑑𝑧^𝑏. 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: ^𝑡𝐽𝛼∧𝛼∧𝜔^𝑗∧𝜔^{𝑚−(𝑗+1)}=𝜋₁∧⋯∧𝜋_𝑗+1∧𝜔^{𝑚−(𝑗+1)}=𝑚−𝑗−1!𝑗+1!𝑓𝑚!^𝑗+1𝑔⁻¹𝑝,𝑔⁻¹𝑔,…,𝑔⁻¹𝑔𝜔^𝑚. (5.6) But at 𝑥, 𝑔⁻¹𝑔=𝑔∈Γ(𝑓_𝑗+1,𝐼) and 𝑔⁻¹𝑝=𝑝∈Γ(𝑓_𝑗+1,𝐼). Indeed, 𝜆(𝑔⁻¹𝑔)∈Γ_𝑘 since 𝜑 is 𝑘-admissible and Γ_𝑘⊂Γ_𝑗+1. Moreover, the Hermitian matrix [2𝛼_𝑎𝛼_𝑏]_{1≤𝑎,𝑏≤𝑚} is positive-semidefinite since for all 𝜉∈ℂ^𝑚, we have ∑^𝑚_𝑎,𝑏=12𝛼_𝑎𝛼_𝑏𝜉_𝑎𝜉_𝑏=2|∑^𝑚_𝑎=1𝛼_𝑎𝜉_𝑎|²≥0; we then deduce that for all 1≤𝑖≤𝑗+1, we have at 𝑥, 𝑓_𝑖(𝑔⁻¹𝑝)=𝜎_𝑖(𝜆(𝑔⁻¹𝑝))≥0. Finally, we infer by the Gårding inequality that at 𝑥 in the chart 𝜙 we have 𝑓_𝑗+1𝑔⁻¹𝑝,𝑔⁻¹𝑔,…,𝑔⁻¹𝑔≥𝑓_𝑗+1𝑔⁻¹𝑝^1/(𝑗+1)𝑓_𝑗+1𝑔⁻¹𝑔^{𝑗/(𝑗+1)}≥0 (5.7) which proves the positivity lemma.

5.2. The Fundamental Inequality

The 𝐶⁰ 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. Let ℎ(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: _𝑀𝑚𝑘−𝑓_𝑘𝑔⁻¹𝑔ℎ𝜑𝜔^𝑚≥1𝑚𝑘2𝑚_𝑀ℎ^′𝜑∇𝜑²_𝑔𝜔^𝑚. (5.8)

Proof. We have the equality ∫_𝑀[(^𝑚_𝑘)−𝑓_𝑘(𝑔⁻¹𝑔)]ℎ(𝜑)𝜔^𝑚=(^𝑚_𝑘)∫_𝑀ℎ(𝜑)(𝜔^𝑚−𝜔^𝑘∧𝜔^𝑚−𝑘). Besides, since Λ²𝑀 is commutative 𝜔^𝑚−𝜔^𝑘∧𝜔^𝑚−𝑘=(𝜔−𝜔)∧(𝜔^𝑚−1+𝜔^𝑚−2∧𝜔+⋯+𝜔^𝑚−𝑘∧𝜔^𝑘−1)_=∶Ω, namely, 𝜔^𝑚−𝜔^𝑘∧𝜔^𝑚−𝑘=−(1/2)𝑑𝑑^𝑐𝜑∧Ω, then ∫_𝑀[(^𝑚_𝑘)−𝑓_𝑘(𝑔⁻¹𝑔)]ℎ(𝜑)𝜔^𝑚=−(1/2)(^𝑚_𝑘)∫_𝑀𝑑𝑑^𝑐𝜑∧(ℎ(𝜑)Ω). But 𝑑(𝑑^𝑐𝜑∧ℎ(𝜑)Ω)=𝑑𝑑^𝑐𝜑∧ℎ(𝜑)Ω+(−1)¹𝑑^𝑐𝜑∧𝑑(ℎ(𝜑)Ω), and 𝑑(ℎ(𝜑)Ω)=ℎ^′(𝜑)𝑑𝜑∧Ω+(−1)⁰ℎ(𝜑)𝑑Ω_{=0since𝜔and𝜔areclosed} so 𝑑𝑑^𝑐𝜑∧ℎ(𝜑)Ω=𝑑^𝑐𝜑∧ℎ^′(𝜑)𝑑𝜑∧Ω+𝑑(something). In addition by Stokes' theorem, ∫_𝑀𝑑(something)=0; therefore, _𝑀𝑚𝑘−𝑓_𝑘𝑔⁻¹𝑔ℎ𝜑𝜔^𝑚1=−2𝑚𝑘^𝑀ℎ^′𝜑𝑑^𝑐=1𝜑∧𝑑𝜑∧Ω2𝑚𝑘^𝑀ℎ^′𝜑−𝑑^𝑐𝜑∧𝑑𝜑∧𝜔^𝑚−1_𝑇1+^𝑘−1_𝑗=1𝑀ℎ^′𝜑−𝑑^𝑐𝜑∧𝑑𝜑∧𝜔^{𝑚−1−𝑗}∧𝜔^𝑗_𝑇2. (5.9) Let us prove that 𝑇₂≥0 (using the positivity lemma) and that 𝑇₁=(1/𝑚)∫_𝑀ℎ^′(𝜑)|∇𝜑|²_𝑔𝜔^𝑚. Let us apply the positivity lemma to 𝑑𝜑: the function 𝑓∶𝑀→ℝ defined by 𝑓𝜔^𝑚=^𝑡𝐽𝑑𝜑∧𝑑𝜑∧𝜔^{𝑚−1−𝑗}∧𝜔^𝑗 is nonnegative for all 1≤𝑗≤𝑘−1. But ^𝑡𝐽𝑑𝜑=−𝑑^𝑐𝜑 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=(𝑑𝑧^𝑚+𝑑𝑧^𝑚)/√2 at 𝑥. We have then 𝜔=𝑖𝑑𝑧^𝑎∧𝑑𝑧^𝑎 at 𝑥 and ^𝑡𝐽𝑑𝜑∧𝑑𝜑=𝑖|𝑑𝜑|²_𝑔𝑑𝑧^𝑚∧𝑑𝑧^𝑚; therefore, ^𝑡𝐽𝑑𝜑∧𝑑𝜑∧𝜔^𝑚−1=_{𝑎1,…,𝑎𝑚−12∈{1,…,𝑚−1}by2≠}𝑖^𝑚𝑑𝜑²_𝑔𝑑𝑧^𝑚∧𝑑𝑧^𝑚∧𝑑𝑧^𝑎1∧𝑑𝑧^𝑎1∧⋯∧𝑑𝑧^𝑎𝑚−1∧𝑑𝑧^𝑎𝑚−1=_{𝑎1,…,𝑎𝑚−12∈{1,…,𝑚−1}by2≠}1𝑑𝜑²_𝑔𝑖^𝑚𝑑𝑧¹∧𝑑𝑧¹∧⋯∧𝑑𝑧^𝑚∧𝑑𝑧^𝑚=(𝑚−1)!𝑑𝜑²_𝑔𝜔^𝑚=1𝑚!𝑚∇𝜑²_𝑔𝜔^𝑚. (5.10) Thus 𝑇₁=(1/𝑚)∫_𝑀ℎ^′(𝜑)|∇𝜑|²_𝑔𝜔^𝑚, consequently ∫_𝑀[(^𝑚_𝑘)−𝑓_𝑘(𝑔⁻¹𝑔)]ℎ(𝜑)𝜔^𝑚≥(1/2)(^𝑚_𝑘)𝑇₁=(1/2𝑚)(^𝑚_𝑘)∫_𝑀ℎ^′(𝜑)|∇𝜑|²_𝑔𝜔^𝑚, 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 to 𝜑_𝑡𝑠 in order to obtain a crucial inequality (the inequality (IN1) from which we will infer the a priori estimate of ‖𝜑_𝑡𝑠‖_𝐶0. Let 𝑝≥2 be a real number. The function 𝜑_𝑡𝑠 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 𝑝−1𝑚𝑘2𝑚_𝑀𝜑_𝑡𝑠^𝑝−2∇𝜑_𝑡𝑠²𝑣_𝑔≤_𝑀𝑚𝑘−𝑓_𝑘𝑔⁻¹𝑔𝜑_𝑡𝑠𝜑_𝑡𝑠^𝑝−2𝑣_𝑔. (5.11) Besides, |∇|𝜑_𝑡𝑠|^𝑝/2|²=2𝑔^𝑎𝑏𝜕_𝑎|𝜑_𝑡𝑠|^𝑝/2𝜕_𝑏|𝜑_𝑡𝑠|^𝑝/2=2𝑔^𝑎𝑏((𝑝/2)𝜑_𝑡𝑠|𝜑_𝑡𝑠|^𝑝/2−2)²𝜕_𝑎𝜑_𝑡𝑠𝜕_𝑏𝜑_𝑡𝑠=(𝑝²/4)|𝜑_𝑡𝑠|^𝑝−2|∇𝜑_𝑡𝑠|², so the previous inequality writes: _𝑀∇𝜑_𝑡𝑠^𝑝/22𝑣_𝑔≤𝑚𝑝²2𝑝−1(^𝑚_𝑘)_𝑀𝑚𝑘−𝑓_𝑘𝑔⁻¹𝑔𝜑_𝑡𝑠𝜑_𝑡𝑠^𝑝−2𝑣_𝑔. (IN1) 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 𝐻²₁(𝑀)⊂𝐿^{2𝑚/(𝑚−1)}(𝑀) that 𝜑_𝑡𝑠^𝑝_{𝑚/(𝑚−1)}=𝜑_𝑡𝑠^𝑝/22_{()2𝑚/𝑚−1}≤𝐶𝑠𝑡𝑒_𝑀∇𝜑_𝑡𝑠^𝑝/22+_𝑀𝜑_𝑡𝑠^(𝑝/2)⋅2, (IN2) where 𝐶𝑠𝑡𝑒 is independent of 𝑝. Besides, 𝑓_𝑘(𝑔⁻¹𝑔) is uniformly bounded; indeed, 𝑓_𝑘𝑔⁻¹𝑔=𝑒^𝑡𝑠𝑓(^𝑚_𝑘)Vol(𝑀)∫_𝑀𝑒^𝑡𝑠𝑓𝑣_𝑔≤𝑚𝑘𝑒^{2𝑡𝑠‖𝑓‖∞}≤𝑚𝑘𝑒^{2‖𝑓‖∞}. (IN3) Using the inequalities (IN1), (IN2), (IN3), and 𝑝²/2(𝑝−1)≤𝑝 we obtain 𝜑_𝑡𝑠^𝑝_{𝑚/(𝑚−1)}≤𝐶𝑠𝑡𝑒^′×𝑝_𝑀𝜑_𝑡𝑠^𝑝−1+_𝑀𝜑_𝑡𝑠^𝑝𝑝≥2, (IN4) 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 𝐿_𝑞 estimates.

Lemma 5.6. There exists a constant 𝜇 such that for all 1≤𝑞≤2𝑚/(𝑚−1), 𝜑_𝑡𝑠𝑞≤𝜇. (5.12)

Proof. 𝑀 is a compact Riemannian manifold and 𝜑_𝑡𝑠∈𝐶², so by the Green's formula 𝜑_𝑡𝑠(𝑥)=(1/Vol(𝑀))_𝑀𝜑_𝑡𝑠𝑑𝑣₌₀+∫_𝑀𝐺(𝑥,𝑦)▵𝜑_𝑡𝑠(𝑦)𝑑𝑣(𝑦), where 𝐺(𝑥,𝑦)≥0 and ∫_𝑀𝐺(𝑥,𝑦)𝑑𝑣(𝑦) is independent of 𝑥. Here ▵𝜑_𝑡𝑠 denotes the real Laplacian. Then, we infer that ‖𝜑_𝑡𝑠‖₁≤𝐶‖▵𝜑_𝑡𝑠‖₁. But ‖▵𝜑_𝑡𝑠‖₁=∫_𝑀▵𝜑⁺_𝑡𝑠+▵𝜑⁻_𝑡𝑠 and ∫_𝑀▵𝜑_𝑡𝑠=∫_𝑀▵𝜑⁺_𝑡𝑠−▵𝜑⁻_𝑡𝑠=0; then ‖▵𝜑_𝑡𝑠‖₁=2∫_𝑀▵𝜑⁺_𝑡𝑠. Besides ▵𝜑_𝑡𝑠<2𝑚 since 𝜑_𝑡𝑠 is 𝑘-admissible: indeed, at 𝑥 in a 𝑔-normal 𝑔-adapted chart, namely, a chart satisfying 𝑔_𝑎𝑏=𝛿_𝑎𝑏, 𝑔_𝑎𝑏=𝛿_𝑎𝑏𝜆_𝑎 and 𝜕_𝜈𝑔_𝑎𝑏=0 for all 1≤𝑎, 𝑏≤𝑚, 𝜈∈{1,…,𝑚,1,…,𝑚}, we have 𝜆(𝑔⁻¹𝑔)=(𝜆₁,…,𝜆_𝑚) so 𝜆=(𝜆₁…,𝜆_𝑚)∈Γ_𝑘 since 𝜑_𝑡𝑠 is 𝑘-admissible; consequently ▵𝜑_𝑡𝑠=−2𝑔^𝑎𝑏𝜕_𝑎𝑏𝜑_𝑡𝑠=−2∑_𝑎𝜕_𝑎𝑎𝜑_𝑡𝑠=2∑_𝑎(1−𝜆_𝑎)=2𝑚−2𝜎₁(𝜆), but 𝜎₁(𝜆)>0 since 𝜆∈Γ_𝑘 which proves that ▵𝜑_𝑡𝑠<2𝑚. Therefore ▵𝜑⁺_𝑡𝑠<2𝑚 and ‖▵𝜑_𝑡𝑠‖₁≤4𝑚Vol(𝑀). We infer then that ‖𝜑_𝑡𝑠‖₁≤4𝑚𝐶Vol(𝑀). Now let us take 𝑝=2 in the inequality (IN2): ‖𝜑_𝑡𝑠‖²_{2𝑚/(𝑚−1)}≤𝐶𝑠𝑡𝑒(∫_𝑀|∇|𝜑_𝑡𝑠||²+∫_𝑀|𝜑_𝑡𝑠|²). Besides, 𝜑_𝑡𝑠∈𝐻²₁(𝑀) and has a vanishing integral; then by the Sobolev-Poincaré inequality we infer ‖𝜑_𝑡𝑠‖₂≤𝒜‖∇𝜑_𝑡𝑠‖₂. But |∇|𝜑_𝑡𝑠||=|∇𝜑_𝑡𝑠| almost everywhere; therefore ‖𝜑_𝑡𝑠‖_{2𝑚/(𝑚−1)}≤𝐶𝑠𝑡𝑒‖∇|𝜑_𝑡𝑠|‖₂. Using the inequality (IN1) with 𝑝=2 and the fact that 𝑓_𝑘(𝑔⁻¹𝑔) is uniformly bounded, we obtain that ‖∇|𝜑_𝑡𝑠|‖²₂≤𝐶𝑠𝑡𝑒‖𝜑_𝑡𝑠‖₁≤𝐶𝑠𝑡𝑒^′. Consequently, we infer that ‖𝜑_𝑡𝑠‖_{2𝑚/(𝑚−1)}≤𝐶𝑠𝑡𝑒.
Let 1≤𝑞≤2𝑚/(𝑚−1)=∶2𝛿. By the Hölder inequality we have ‖𝜑_𝑡𝑠‖^𝑞_𝑞=∫_𝑀|𝜑_𝑡𝑠|^𝑞⋅1≤(∫_𝑀|𝜑_𝑡𝑠|^{𝑞⋅(2𝛿/𝑞)})^𝑞/2𝛿Vol(𝑀)^{1−𝑞/2𝛿}. Therefore ‖𝜑_𝑡𝑠‖_𝑞≤Vol(𝑀)^{(1/𝑞)−(1/2𝛿)}‖𝜑_𝑡𝑠‖_2𝛿. But Vol(𝑀)^{1/𝑞−1/2𝛿}=𝑒^{(1/𝑞−1/2𝛿)ln(Vol(𝑀))}≤(𝑀)(𝑀)1ifVol≤1,Vol^1−1/2𝛿ifVol(𝑀)≥1 (5.13) and ‖𝜑_𝑡𝑠‖_2𝛿≤𝐶𝑠𝑡𝑒, thus ‖𝜑_𝑡𝑠‖_𝑞≤𝜇∶=𝐶𝑠𝑡𝑒×Max(1,Vol(𝑀)^1−1/2𝛿).

Suppose without limitation of generality that 𝜇≥1. Now, we deduce from the previous lemma and the inequality (IN4), by induction, these more general 𝐿_𝑝 estimates using the same method as [13].

Lemma 5.7. There exists a constant 𝐶₀ such that for all 𝑝≥2, 𝜑_𝑡𝑠𝑝≤𝐶₀𝛿^𝑚−1𝐶𝑝^−𝑚/𝑝, (5.14) 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 for 2≤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=𝜇𝛿^{𝑚(𝑚−1)}𝐶^𝑚𝑒^𝑚/𝑒. For 2≤𝑝≤2𝛿 we have ‖𝜑_𝑡𝑠‖_𝑝≤𝜇, so it suffices to check that 𝜇≤𝐶₀(𝛿^𝑚−1𝐶𝑝)^−𝑚/𝑝. This inequality is equivalent to 𝛿^{𝑚(𝑚−1)}𝐶^𝑚𝑒^𝑚/𝑒(𝛿^𝑚−1𝐶𝑝)^−𝑚/𝑝≥1; then (𝛿^{𝑚(𝑚−1)}𝐶^𝑚)𝑒^𝑚/𝑒≥(𝛿^{𝑚(𝑚−1)}𝐶^𝑚)^1/𝑝𝑝^𝑚/𝑝. But if 𝑥≥1, then 𝑥≥𝑥^1/𝑝 (since 𝑝≥1), and 𝛿^{𝑚(𝑚−1)}𝐶^𝑚≥1 (since 𝐶≥1,𝑚≥1 and 𝛿≥1); therefore 𝛿^{𝑚(𝑚−1)}𝐶^𝑚≥(𝛿^{𝑚(𝑚−1)}𝐶^𝑚)^1/𝑝. Besides, 𝑝^𝑚/𝑝=𝑒^{𝑚(ln𝑝/𝑝)}≤𝑒^𝑚/𝑒, which proves the inequality for 2≤𝑝≤2𝛿. Now let us fix 𝑝≥2. Suppose that ‖𝜑_𝑡𝑠‖_𝑝≤𝐶₀(𝛿^𝑚−1𝐶𝑝)^−𝑚/𝑝 and prove that ‖𝜑_𝑡𝑠‖_𝛿𝑝≤𝐶₀(𝛿^𝑚−1𝐶𝛿𝑝)^{−𝑚/𝛿𝑝}. The inequality (IN4) proved previously writes: 𝜑_𝑡𝑠^𝑝_𝛿≤𝐶𝑠𝑡𝑒^′×𝑝_𝑀𝜑_𝑡𝑠^𝑝−1+_𝑀𝜑_𝑡𝑠^𝑝𝑝≥2, IN4^′ where 𝐶𝑠𝑡𝑒^′ is independent of 𝑝, namely, ‖𝜑_𝑡𝑠‖^𝑝_𝛿𝑝≤𝐶𝑠𝑡𝑒^′×𝑝(‖𝜑_𝑡𝑠‖^𝑝−1_𝑝−1+‖𝜑_𝑡𝑠‖^𝑝_𝑝). But since 1≤𝑝−1≤𝑝, we have by the Hölder inequality that ‖𝜑_𝑡𝑠‖_𝑝−1≤Vol(𝑀)^{1/(𝑝−1)−1/𝑝}‖𝜑_𝑡𝑠‖_𝑝; therefore ‖𝜑_𝑡𝑠‖^𝑝_𝛿𝑝≤𝐶𝑠𝑡𝑒^′×𝑝(Vol(𝑀)^1/𝑝‖𝜑_𝑡𝑠‖_𝑝^𝑝−1+‖𝜑_𝑡𝑠‖^𝑝_𝑝).(i)If ‖𝜑_𝑡𝑠‖_𝑝≤1, then ‖𝜑_𝑡𝑠‖^𝑝_𝛿𝑝≤𝐶×𝑝; therefore ‖𝜑_𝑡𝑠‖_𝛿𝑝≤(𝐶𝑝)^1/𝑝. Let us check that (𝐶𝑝)^1/𝑝≤𝐶₀(𝛿^𝑚−1𝐶𝛿𝑝)^{−𝑚/𝛿𝑝}. This inequality is equivalent to 𝑝^{(1/𝑝)(1+𝑚/𝛿)}≤𝜇𝛿^{𝑚(𝑚−1)(1−1/𝑝)}𝑒^𝑚/𝑒×𝐶^{𝑚−𝑚/𝛿𝑝−1/𝑝}, but 1+𝑚/𝛿=𝑚 so it is equivalent to 𝑝^𝑚/𝑝≤𝜇𝛿^{𝑚(𝑚−1)(1−1/𝑝)}𝑒^𝑚/𝑒×𝐶^{𝑚(1−1/𝑝)}. Besides 𝑝^𝑚/𝑝≤𝑒^𝑚/𝑒 and 𝜇𝛿^{𝑚(𝑚−1)(1−1/𝑝)}≥1, then it suffices to have 𝐶^{𝑚(1−1/𝑝)}≥1, and this is satisfied since 𝐶≥1. (ii)If ‖𝜑_𝑡𝑠‖_𝑝≥1, we infer that ‖𝜑_𝑡𝑠‖^𝑝_𝛿𝑝≤𝐶×𝑝‖𝜑_𝑡𝑠‖^𝑝_𝑝, therefore ‖𝜑_𝑡𝑠‖_𝛿𝑝≤𝐶^1/𝑝×𝑝^1/𝑝‖𝜑_𝑡𝑠‖_𝑝≤(𝐶𝑝)^1/𝑝𝐶₀(𝛿^𝑚−1𝐶𝑝)^−𝑚/𝑝 by the induction hypothesis. But (1−𝑚)/𝑝=−𝑚/𝛿𝑝; then we obtain the required inequality ‖𝜑_𝑡𝑠‖_𝛿𝑝≤𝐶₀𝛿^{−𝑚2/𝛿𝑝}(𝐶𝑝)^{−𝑚/𝛿𝑝}=𝐶₀(𝛿^𝑚−1𝐶𝛿𝑝)^{−𝑚/𝛿𝑝}.

By tending to the limit 𝑝→+∞ in the inequality of the previous lemma, we obtain the needed 𝐶⁰ a priori estimate.

Corollary 5.8. Consider 𝜑_{𝑡𝑠𝐶0}≤𝐶₀. (5.15)

6. The 𝐶² A Priori Estimate

6.1. Strategy for a 𝐶² Estimate

First, we will look for a uniform upper bound on the eigenvalues 𝜆([𝛿^𝑗_𝑖+𝑔^𝑗ℓ𝜕_𝑖ℓ𝜑_𝑡]_{1≤𝑖,𝑗≤𝑚}. Secondly, we will infer from it the uniform ellipticity of the continuity equation 𝐸_𝑘,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

Let 𝑡∈𝒯_𝑙,𝛼, and let 𝜑_𝑡 : 𝑀→ℝ be a 𝐶^𝑙,𝛼  𝑘-admissible solution of 𝐸_𝑘,𝑡 satisfying ∫_𝑀𝜑_𝑡𝜔^𝑚=0. Consider the following functional: 𝐵∶𝑈𝑇^1,0()()=ℎ→ℝ𝑃,𝜉↦𝐵𝑃,𝜉_𝑃(𝜉,𝜉)−𝜑_𝑡(𝑃), (6.1) where 𝑈𝑇^1,0 is the unit sphere bundle associated to (𝑇^1,0,ℎ) and 𝑔 is related to 𝑔 by: 𝜔=𝜔+𝑖𝜕𝜕𝜑_𝑡. 𝐵 is continuous on the compact set 𝑈𝑇^1,0, so it assumes its maximum at a point (𝑃₀,𝜉₀)∈𝑈𝑇^1,0. In addition, for fixed 𝑃∈𝑀, 𝜉∈𝑈𝑇_𝑃^1,0↦ℎ_𝑃(𝜉,𝜉) is continuous on the compact subset 𝑈𝑇_𝑃^1,0 (the fiber); therefore it attains its maximum at a unit vector 𝜉_𝑃∈𝑈𝑇_𝑃^1,0, and by the min-max principle we can choose 𝜉_𝑃 as the direction of the largest eigenvalue of 𝐴_𝑃, 𝜆_max(𝐴_𝑃). Specifically, we have the following.

Lemma 6.1 (min-max principle). Consider ℎ_𝑃𝜉_𝑃,𝜉_𝑃=max_{𝜉∈𝑇𝑃1,0,ℎ𝑃()𝜉,𝜉=1}ℎ_𝑃(𝜉,𝜉)=𝜆_max𝐴_𝑃. (6.2)

For fixed 𝑃, we have max_{ℎ𝑃(𝜉,𝜉)=1}𝐵(𝑃,𝜉)=𝐵(𝑃,𝜉_𝑃)=𝜆_max(𝐴_𝑃)−𝜑_𝑡(𝑃); therefore max_{(𝑃,𝜉)∈𝑈𝑇}^1,0𝐵(𝑃,𝜉)=max_𝑃∈𝑀𝐵(𝑃,𝜉_𝑃)=𝐵(𝑃₀,𝜉₀)≤𝐵(𝑃₀,𝜉_𝑃0); hence, max_{()𝑃,𝜉∈𝑈𝑇1,0}𝐵(𝑃,𝜉)=𝐵𝑃₀,𝜉_𝑃0=𝜆_max𝐴_𝑃0−𝜑_𝑡𝑃₀. (6.3) At the point 𝑃₀, consider 𝑒^𝑃0₁,…,𝑒^𝑃0_𝑚 an ℎ_𝑃0-orthonormal basis of (𝑇_𝑃^1,0₀,ℎ_𝑃0) made of eigenvectors of 𝐴_𝑃0 that satisfies the following properties: (1)ℎ_𝑃0-orthonormal: [ℎ_𝑖𝑗(𝑃₀)]_{1≤𝑖,𝑗≤𝑚}=𝐼_𝑚. (2)ℎ_𝑃0-diagonal: [ℎ_𝑖𝑗(𝑃₀)]_{1≤𝑖,𝑗≤𝑚}=Mat𝐴_𝑃0=diag(𝜆₁,…,𝜆_𝑚), 𝜆∈Γ_𝑘. (3)𝜆_max(𝐴_𝑃0) is achieved in the direction 𝑒^𝑃0₁=𝜉_𝑃0: 𝐴_𝑃0(𝜉_𝑃0)=𝜆_max(𝐴_𝑃0)𝜉_𝑃0=𝜆₁𝜉_𝑃0 and 𝜆₁≥⋯≥𝜆_𝑚. In other words, it is a basis satisfying(1)[𝑔_𝑖𝑗(𝑃₀)]_{1≤𝑖,𝑗≤𝑚}=𝐼_𝑚,(2)[𝑔_𝑖̃𝑗(𝑃₀)]_{1≤𝑖,𝑗≤𝑚}=Mat𝐴_𝑃0=diag(𝜆₁,…,𝜆_𝑚), 𝜆∈Γ_𝑘,(3)𝜆_max(𝐴_𝑃0)=𝜆₁≥⋯≥𝜆_𝑚. Let us consider a holomorphic normal chart (𝑈₀,𝜓₀) centered at 𝑃₀ such that 𝜓₀(𝑃₀)=0 and 𝜕_𝑖|_𝑃0=𝑒^𝑃0_𝑖 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 𝐵₁∶𝑈₁→ℝ𝑃↦𝐵₁(𝑃)=𝑔₁₁(𝑃)𝑔₁₁(𝑃)−𝜑_𝑡(𝑃). (6.4) We claim that 𝐵₁ assumes a local maximum at 𝑃₀. Indeed, we have at each 𝑃∈𝑈₁: 𝑔₁₁(𝑃)/𝑔₁₁(𝑃)=𝑔_𝑃(𝜕₁,𝜕₁)/𝑔_𝑃(𝜕₁,𝜕₁)=ℎ_𝑃(𝜕₁,𝜕₁)/ℎ_𝑃(𝜕₁,𝜕₁)=ℎ_𝑃(𝜕₁/|𝜕₁|_ℎ𝑃,𝜕₁/|𝜕₁|_ℎ𝑃)≤𝜆_max(𝐴_𝑃) (see Lemma 6.1); thus 𝐵₁(𝑃)≤𝜆_max(𝐴_𝑃)−𝜑_𝑡(𝑃)≤𝜆_max(𝐴_𝑃0)−𝜑_𝑡(𝑃₀)=𝐵₁(𝑃₀).

6.2.3. Differentiating the Equation

For short, we drop henceforth the subscript 𝑡 of 𝜑_𝑡. Let us differentiate 𝐸_𝑘,𝑡 at 𝑃, in a chart 𝑧: 𝑡𝜕₁𝑓=𝑑𝐹_𝑘[𝛿^𝑗_{𝑖+𝑔𝑗ℓ(𝑃)𝜕𝑖ℓ𝜑(𝑃)]1≤𝑖,𝑗≤𝑚}⋅𝜕₁𝑔^𝑗ℓ𝜕_𝑖ℓ𝜑_{1≤𝑖,𝑗≤𝑚}=^𝑚_𝑖,𝑗=1𝜕𝐹_𝑘𝜕𝐵^𝑗_𝑖𝛿^𝑗_𝑖+𝑔^𝑗ℓ𝜕_𝑖ℓ𝜑𝜕₁𝑔^𝑗ℓ𝜕_𝑖ℓ𝜑+𝑔^𝑗ℓ𝜕_1𝑖ℓ𝜑. (6.5) Differentiating once again, we find 𝑡𝜕₁₁𝑓=^𝑚_{𝑖,𝑗,𝑟,𝑠=1}𝜕²𝐹_𝑘𝜕𝐵^𝑠_𝑟𝜕𝐵^𝑗_𝑖𝛿^𝑗_𝑖+𝑔^𝑗ℓ𝜕_𝑖ℓ𝜑𝜕₁𝑔^𝑠𝑜𝜕_𝑟𝑜𝜑+𝑔^𝑠𝑜𝜕_1𝑟𝑜𝜑×𝜕₁𝑔^𝑗ℓ𝜕_𝑖ℓ𝜑+𝑔^𝑗ℓ𝜕_1𝑖ℓ𝜑+^𝑚_𝑖,𝑗=1𝜕𝐹_𝑘𝜕𝐵^𝑗_𝑖𝛿^𝑗_𝑖+𝑔^𝑗ℓ𝜕_𝑖ℓ𝜑×𝜕₁₁𝑔^𝑗ℓ𝜕_𝑖ℓ𝜑+𝜕₁𝑔^𝑗ℓ𝜕_1𝑖ℓ𝜑+𝜕₁𝑔^𝑗ℓ𝜕_1𝑖ℓ𝜑+𝑔^𝑗ℓ𝜕_11𝑖ℓ𝜑. (6.6) Using the above chart (𝑈₁,𝜓₀) at the point 𝑃₀, normality yields 𝑔^𝑗ℓ=𝛿^𝑗ℓ, 𝜕_𝛼𝑔_𝑖ℓ=0 and 𝜕_𝛼𝑔^𝑖ℓ=0. Furthermore [𝛿^𝑗_𝑖+𝑔^𝑗ℓ𝜕_𝑖ℓ𝜑]=[𝛿^𝑗_𝑖+𝜕_𝑖𝑗𝜑]=[𝑔_𝑖𝑗]=diag(𝜆₁,…,𝜆_𝑚). In this chart, we can simplify the previous expression; we get then at 𝑃₀, 𝑡𝜕₁₁𝑓=^𝑚_{𝑖,𝑗,𝑟,𝑠=1}𝜕²𝐹_𝑘𝜕𝐵^𝑠_𝑟𝜕𝐵^𝑗_𝑖diag𝜆₁,…,𝜆_𝑚𝜕_1𝑟𝑠𝜑𝜕_1𝑖𝑗𝜑+^𝑚_𝑖,𝑗=1𝜕𝐹_𝑘𝜕𝐵^𝑗_𝑖diag𝜆₁,…,𝜆_𝑚𝜕₁₁𝑔^𝑗𝑖𝜕_𝑖𝑖𝜑+𝜕_11𝑖𝑗𝜑. (6.7) Besides, 𝜕₁₁𝑔^𝑗𝑖=𝜕₁(−𝑔^𝑗𝑠𝑔^𝑜𝑖𝜕₁𝑔_𝑜𝑠), so still by normality, we obtain at 𝑃₀ that 𝜕₁₁𝑔^𝑗𝑖=−𝑔^𝑗𝑠𝑔^𝑜𝑖𝜕₁₁𝑔_𝑜𝑠=−𝜕₁₁𝑔_𝑖𝑗−𝑅_11𝑖𝑗. Therefore we get 𝑡𝜕₁₁𝑓=^𝑚_𝑖,𝑗=1𝜕𝐹_𝑘𝜕𝐵^𝑗_𝑖diag𝜆₁,…,𝜆_𝑚𝜕_11𝑖𝑗𝜑−𝑅_11𝑖𝑗𝜕_𝑖𝑖𝜑+^𝑚_{𝑖,𝑗,𝑟,𝑠=1}𝜕²𝐹_𝑘𝜕𝐵^𝑠_𝑟𝜕𝐵^𝑗_𝑖diag𝜆₁,…,𝜆_𝑚𝜕_1𝑟𝑠𝜑𝜕_1𝑖𝑗𝜑. (6.8)

6.2.4. Using Concavity

Now, using the concavity of ln𝜎_𝑘 [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 𝐹_𝑘 since the matrix [𝜕_1𝑖𝑗𝜑]_{1≤𝑖,𝑗≤𝑚} is not Hermitian.

Lemma 6.2. Consider 𝑆∶=^𝑚_{𝑖,𝑗,𝑟,𝑠=1}𝜕²𝐹_𝑘𝜕𝐵^𝑠_𝑟𝜕𝐵^𝑗_𝑖diag𝜆₁,…,𝜆_𝑚𝜕_1𝑟𝑠𝜑𝜕_1𝑖𝑗𝜑≤0. (6.9)

Hence, from (6.8) combined with Lemma 6.2 we infer 𝑡𝜕₁₁𝑓≤^𝑚_𝑖=1𝜎_{𝑘−1,𝑖}(𝜆)𝜎_𝑘(𝜆)𝜕_11𝑖𝑖𝜑−𝑅_11𝑖𝑖𝜕_𝑖𝑖𝜑. (6.10)

6.2.5. Differentiation of the Functional 𝐵₁

Let us differentiate twice the functional 𝐵₁: 𝐵₁(𝑃)=𝑔₁₁(𝑃)𝑔₁₁(𝑃)(𝑃),𝜕−𝜑_𝑖𝐵₁=𝜕_𝑖𝑔₁₁𝑔₁₁−𝑔₁₁𝜕_𝑖𝑔₁₁𝑔₁₁²−𝜕_𝑖𝜕𝜑,_𝑖𝑖𝐵₁=𝜕_𝑖𝑖𝑔₁₁𝑔₁₁−𝜕_𝑖𝑔₁₁𝜕_𝑖𝑔₁₁+𝜕_𝑖𝑔₁₁𝜕_𝑖𝑔₁₁+𝑔₁₁𝜕_𝑖𝑖𝑔₁₁𝑔₁₁²+2𝑔₁₁𝜕_𝑖𝑔₁₁𝜕_𝑖𝑔₁₁𝑔₁₁³−𝜕_𝑖𝑖𝜑. (6.11) Therefore at 𝑃₀, in the above chart (𝑈₁,𝜓₀) we find 𝜕_𝑖𝑖𝐵₁=𝜕_𝑖𝑖(𝑔₁₁+𝜕₁₁𝜑)−𝜆₁𝜕_𝑖𝑖𝑔₁₁−𝜕_𝑖𝑖𝜑=𝑅_11𝑖𝑖+𝜕_11𝑖𝑖𝜑−𝜆₁𝑅_11𝑖𝑖−𝜕_𝑖𝑖𝜑. Let us define the operator: 𝐿∶=^𝑚_𝑖,𝑗=1𝜕𝐹_𝑘𝜕𝐵^𝑗_𝑖𝛿^𝑗_𝑖+𝑔^𝑗ℓ𝜕_𝑖ℓ𝜑_{1≤𝑖,𝑗≤𝑚}∇^𝑗_𝑖. (6.12) Thus, we have at 𝑃₀𝐿𝐵₁=^𝑚_𝑖=1𝜎_{𝑘−1,𝑖}(𝜆)𝜎_𝑘(𝜆)𝜕_11𝑖𝑖𝜑+1−𝜆₁𝑅_11𝑖𝑖−𝜕_𝑖𝑖𝜑. (6.13) Combining (6.13) with (6.10), we get rid of the fourth derivatives: 𝑡𝜕₁₁𝑓−𝐿𝐵₁≤^𝑚_𝑖=1𝜎_{𝑘−1,𝑖}(𝜆)𝜎_𝑘(𝜆)𝑅_11𝑖𝑖𝜆₁−1−𝜆_𝑖++1_𝑚𝑖=1𝜎_{𝑘−1,𝑖}(𝜆)𝜎_𝑘(𝜆)𝜆_𝑖−1. (6.14) 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: 0≥^𝑚_𝑖=2𝜎_{𝑘−1,𝑖}(𝜆)𝜎_𝑘(𝜆)−𝑅_11𝑖𝑖𝜆₁−𝜆_𝑖−^𝑚_𝑖=1𝜎_{𝑘−1,𝑖}(𝜆)𝜎_𝑘(𝜆)𝜆_𝑖+^𝑚_𝑖=1𝜎_{𝑘−1,𝑖}(𝜆)𝜎_𝑘(𝜆)+𝑡𝜕₁₁𝑓. (6.15)

Curvature Assumption
Henceforth, we will suppose that the holomorphic bisectional curvature is nonnegative at any 𝑃∈𝑀. Thus in a holomorphic normal chart centered at 𝑃 we have 𝑅_{𝑎𝑎𝑏𝑏}(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 𝜎_𝑘(𝜆)>0 and 𝜎_{𝑘−1,𝑖}(𝜆)>0 since 𝜆∈Γ_𝑘, and under our curvature assumption (−𝑅_11𝑖𝑖)≥0 for all 𝑖≥2. Besides, 𝜆_𝑖≤𝜆₁ for all 𝑖; therefore ∑^𝑚_𝑖=2(𝜎_{𝑘−1,𝑖}(𝜆)/𝜎_𝑘(𝜆))(−𝑅_11𝑖𝑖)(𝜆₁−𝜆_𝑖)≥0. So we can get rid of the curvature terms in (6.15) and infer from it the inequality 0≥−^𝑚_𝑖=1𝜎_{𝑘−1,𝑖}(𝜆)𝜎_𝑘(𝜆)𝜆_𝑖+^𝑚_𝑖=1𝜎_{𝑘−1,𝑖}(𝜆)𝜎_𝑘(𝜆)+𝑡𝜕₁₁𝑓. (6.16)

6.2.6. A 𝜆₁’s Upper Bound

Here, we require elementary identities satisfied by the 𝜎_ℓ's [11], namely: ∀1≤ℓ≤𝑚𝜎_ℓ(𝜆)=𝜎_ℓ,𝑖(𝜆)+𝜆_𝑖𝜎_ℓ−1,𝑖(𝜆),∀1≤ℓ≤𝑚^𝑚_𝑖=1𝜎_ℓ−1,𝑖(𝜆)𝜆_𝑖=ℓ𝜎_ℓ(𝜆),soinparticular^𝑚_𝑖=1𝜎_{𝑘−1,𝑖}(𝜆)𝜎_𝑘(𝜆)𝜆_𝑖=𝑘,∀1≤ℓ≤𝑚^𝑚_𝑖=1𝜎_ℓ,𝑖(𝜆)=(𝑚−ℓ)𝜎_ℓ(𝜆),soinparticular^𝑚_𝑖=1𝜎_{𝑘−1,𝑖}(𝜆)𝜎_𝑘(𝜆)=()𝜎𝑚−𝑘+1^𝑘−1(𝜆)𝜎_𝑘(𝜆). (6.17) Consequently, (6.16) writes: 𝑞_𝑘()∶=𝑚−𝑘+1𝑘𝜎_𝑘−1(𝜆)𝜎_𝑘(𝜆)𝑡≤1−𝑘𝜕₁₁𝑓. (6.18) So 𝑞_𝑘≤1+(1/𝑘)|𝜕₁₁𝑓|. But at 𝑃₀, |∇²𝑓|²_𝑔=2𝑔^𝑎𝑐𝑔^𝑑𝑏(∇_𝑎𝑏𝑓∇_𝑐𝑑𝑓+∇_𝑎𝑑𝑓∇_𝑐𝑏𝑓)=2∑^𝑚_𝑎,𝑏=1(|𝜕_𝑎𝑏𝑓|²+|𝜕_𝑎𝑏𝑓|²), then |𝜕₁₁𝑓|≤|∇²𝑓|_𝑔, and consequently 𝑞_𝑘≤1+(1/𝑘)‖𝑓‖_𝐶2(𝑀)=∶𝐶₁. In other words, there exists a constant 𝐶₁ independent of 𝑡∈[0,1] such that 𝑞_𝑘≤𝐶₁. (6.19) To proceed further, we recall the following

Lemma 6.3 (Newton inequalities). For all ℓ≥2, 𝜆∈ℝ^𝑚: 𝜎_ℓ(𝜆)𝜎_ℓ−2(𝜆)≤(ℓ−1)(𝑚−ℓ+1)ℓ()𝜎𝑚−ℓ+2^ℓ−1(𝜆)². (6.20)

Let us use Newton inequalities to relate 𝑞_𝑘 to 𝜎₁. Since for 2≤ℓ≤𝑘 and 𝜆∈Γ_𝑘 we have 𝜎_ℓ(𝜆)>0,𝜎_ℓ−1(𝜆)>0 and 𝜎_ℓ−2(𝜆)>0 (𝜎₀(𝜆)=1 by convention), Newton inequalities imply then that ((𝑚−ℓ+2)/(ℓ−1))(𝜎_ℓ−2(𝜆)/𝜎_ℓ−1(𝜆))≤((𝑚−ℓ+1)/ℓ)(𝜎_ℓ−1(𝜆)/𝜎_ℓ(𝜆)), or else 𝑞_ℓ−1≤𝑞_ℓ, consequently 𝑞_𝑘≥𝑞_𝑘−1≥⋯≥𝑞₂=(𝑚−1)𝜎₁(𝜆)/2𝜎₂(𝜆). By induction, we get 𝜎₁(𝜆)≤(ℓ!(𝑚−ℓ)!/(𝑚−1)!)𝜎_ℓ(𝜆)(𝑞_ℓ)^ℓ−1 for all 2≤ℓ≤𝑘. In particular 𝜎₁(𝜆)≤𝑘!(𝑚−𝑘)!()!𝜎𝑚−1^𝑘(𝜆)𝑞_𝑘^𝑘−1. (6.21) But 𝜎_𝑘(𝜆)≤𝑒^{2‖𝑓‖∞}(^𝑚_𝑘); combining this with (6.19) and (6.21) we obtain at 𝑃₀ that 𝜎₁(𝜆)≤𝑚𝑒^{2‖𝑓‖∞}𝐶₁^𝑘−1=∶𝐶₂. (6.22) Hence we may state the following.

Theorem 6.4. There exists a constant 𝐶₂>0 depending only on 𝑚,𝑘,‖𝑓‖_∞ and ‖𝑓‖_𝐶2 such that for all 1≤𝑖≤𝑚𝜆_𝑖(𝑃₀)≤𝐶₂.

Combining this result with the 𝐶⁰ a priori estimate ‖𝜑_𝑡‖_𝐶0≤𝐶₀ immediately yields the following.

Theorem 6.5. There exists a constant 𝐶^′₂>0 depending only on 𝑚,𝑘,‖𝑓‖_𝐶2 and 𝐶₀ such that for all 𝑃∈𝑀, for all 1≤𝑖≤𝑚, 𝜆_𝑖(𝑃)≤𝐶₂+2𝐶₀=∶𝐶^′₂.

6.2.7. Uniform Pinching of the Eigenvalues

We infer automatically the following pinchings of the eigenvalues.

Proposition 6.6. For all 1≤𝑖≤𝑚, −(𝑚−1)𝐶₂≤𝜆_𝑖(𝑃₀)≤𝐶₂.

Proposition 6.7. For all 𝑃∈𝑀, for all 1≤𝑖≤𝑚,−(𝑚−1)𝐶^′₂≤𝜆_𝑖(𝑃)≤𝐶^′₂.

6.3. Uniform Ellipticity of the Continuity Equation

To prove the next proposition on uniform ellipticity, we require some inequalities satisfied by the 𝜎_ℓ’s.

Lemma 6.8 (Maclaurin inequalities). For all 1≤ℓ≤𝑠  for all 𝜆∈Γ_𝑠,(𝜎_𝑠(𝜆)/(^𝑚_𝑠))^1/𝑠≤(𝜎_ℓ(𝜆)/(^𝑚_ℓ))^1/ℓ.

Proposition 6.9 (uniform ellipticity). There exist constants 𝐸>0 and 𝐹>0 depending only on 𝑚,𝑘,‖𝑓‖_∞ and 𝐶₂ such that: 𝐸≤𝜎_𝑘−1,1(𝜆)≤⋯≤𝜎_{𝑘−1,𝑚}(𝜆)≤𝐹 where 𝜆=𝜆(𝑃₀).

Proof. We have 𝜕𝜎_𝑘/𝜕𝜆₁=𝜎_𝑘−1,1(𝜆)≤⋯≤𝜎_{𝑘−1,𝑚}(𝜆)≤^𝑚−1_𝑘−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 𝜎_𝑘−1,1(𝜆), using the identity 𝜎_𝑘(𝜆)=𝜆₁𝜎_𝑘−1,1(𝜆)+𝜎_𝑘,1(𝜆). We distinguish two cases.
Case  1. (𝜎_𝑘,1(𝜆)≤0). When so, we have 𝜎_𝑘(𝜆)≤𝜆₁𝜎_𝑘−1,1(𝜆); therefore 𝜎_𝑘−1,1(𝜆)≥𝜎_𝑘(𝜆)/𝜆₁. But 𝜎_𝑘(𝜆)≥𝑒^{−2‖𝑓‖∞}(^𝑚_𝑘) and 0<𝜆₁≤𝐶₂; hence 𝜎_𝑘−1,1(𝜆)≥𝑒^{−2‖𝑓‖∞}(^𝑚_𝑘)/𝐶₂.
Case  2. (𝜎_𝑘,1(𝜆)>0). For 1≤𝑗≤𝑘−1,𝜎_𝑗(𝜆₂,…,𝜆_𝑚)=𝜎_𝑗,1(𝜆)>0 since 𝑗+1≤𝑘 and 𝜆∈Γ_𝑘. Besides 𝜎_𝑘(𝜆₂,…,𝜆_𝑚)=𝜎_𝑘,1(𝜆)>0 by hypothesis, therefore (𝜆₂,…,𝜆_𝑚)∈Γ_𝑘,1={𝛽∈ℝ^𝑚−1/∀1≤𝑗≤𝑘,𝜎_𝑗(𝛽)>0}. From the latter, we infer by Maclaurin inequalities (𝜎_𝑘(𝜆₂,…,𝜆_𝑚)/_𝑘^𝑚−1)^1/𝑘≤(𝜎_𝑘−1(𝜆₂,…,𝜆_𝑚)/^𝑚−1_𝑘−1)^1/(𝑘−1) or else (𝜎_𝑘,1(𝜆)/_𝑘^𝑚−1)^1/𝑘≤(𝜎_𝑘−1,1(𝜆)/^𝑚−1_𝑘−1)^1/(𝑘−1); thus we have 𝜎_𝑘,1(𝜆)≤_𝑘^𝑚−1(𝜎_𝑘−1,1(𝜆)/^𝑚−1_𝑘−1)^{1+1/(𝑘−1)}, consequently 𝜎_𝑘(𝜆)=𝜆₁𝜎_𝑘−1,1(𝜆)+𝜎_𝑘,1(𝜆)≤𝜆₁𝜎_𝑘−1,1(𝜆)+𝑘𝑚−1𝜎_𝑘−1,1(𝜆)_{𝑚−1𝑘−1}^{1+1/(𝑘−1)}≤𝜎_𝑘−1,1(𝜆)𝜆₁+^𝑘𝑚−1_{𝑚−1𝑘−1}𝜎_𝑘−1,1(𝜆)_{𝑚−1𝑘−1}^1/(𝑘−1). (6.23) Here, let us distinguish two subcases of Case 2.(i) If 𝜎_𝑘−1,1(𝜆)>^𝑚−1_𝑘−1, then we have the uniform lower bound that we look for. (ii) Else 𝜎_𝑘−1,1(𝜆)≤^𝑚−1_𝑘−1, thus (𝜎_𝑘−1,1(𝜆)/^𝑚−1_𝑘−1)^1/(𝑘−1)≤1, therefore 𝜎_𝑘(𝜆)≤𝜎_𝑘−1,1(𝜆)[𝜆₁+_𝑘^𝑚−1/^𝑚−1_𝑘−1]=𝜎_𝑘−1,1(𝜆)(𝜆₁+𝑚/𝑘−1); then we get 𝜎_𝑘−1,1(𝜆)≥𝜎_𝑘(𝜆)/(𝜆₁+𝑚/𝑘−1)≥𝑒^{−2‖𝑓‖∞}(^𝑚_𝑘)/(𝐶₂+𝑚/𝑘−1). Consequently 𝜎_𝑘−1,1(𝜆)≥min(𝑒^{−2‖𝑓‖∞}(^𝑚_𝑘)/𝐶₂,^𝑚−1_𝑘−1,𝑒^{−2‖𝑓‖∞}(^𝑚_𝑘)/(𝐶₂+𝑚/𝑘−1)) or finally 𝜎_𝑘−1,1(𝜆)≥min(^𝑚−1_𝑘−1,𝑒^{−2‖𝑓‖∞}(^𝑚_𝑘)/(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 constants 𝐸₀>0 and 𝐹₀>0 depending only on 𝑚,𝑘,‖𝑓‖_∞ and 𝐶^′₂ such that for all 𝑃∈𝑀, for all 1≤𝑖≤𝑚, 𝐸₀≤𝜎_{𝑘−1,𝑖}(𝜆(𝑃))≤𝐹₀.

6.4. Gradient Uniform Estimate

The manifold 𝑀 is Riemannian compact and 𝜑_𝑡∈𝐶²(𝑀), so by the Green's formula 𝜑_𝑡(𝑃)=1Vol(𝑀)_𝑀𝜑_𝑡(𝑄)𝑑𝑣_𝑔(𝑄)+_𝑀𝐺(𝑃,𝑄)▵𝜑_𝑡(𝑄)𝑑𝑣_𝑔(𝑄), (6.24) where 𝐺(𝑃,𝑄) is the Green's function of the Laplacian ▵. By differentiating locally under the integral sign we obtain 𝜕_𝑢𝑖𝜑_𝑡(𝑃)=∫_𝑀▵𝜑_𝑡(𝑄)(𝜕_𝑢𝑖)_𝑃𝐺(𝑃,𝑄)𝑑𝑣_𝑔(𝑄). We infer then that at 𝑃 in a holomorphic normal chart, we have ∇𝜑_𝑡𝑃≤√2𝑚_𝑀▵𝜑_𝑡(𝑄)∇_𝑃𝐺(𝑃,𝑄)𝑑𝑣_𝑔(𝑄). (6.25) Now, using the uniform pinching of the eigenvalues, we prove easily the following estimate of the Laplacian.

Lemma 6.11. There exists a constant 𝐶₃>0 depending on 𝑚 and 𝐶^′₂ such that ‖▵𝜑_𝑡‖_∞,𝑀≤𝐶₃.

Combining Lemma 6.11 with (6.25), we deduce that |(∇𝜑_𝑡)_𝑃|≤√2𝑚𝐶₃∫_𝑀|∇_𝑃𝐺(𝑃,𝑄)|𝑑𝑣_𝑔(𝑄). Besides, classically [13, page 109], there exists constants 𝒞 and 𝒞^′ such that ∇_𝑃𝐺(≤𝒞𝑃,𝑄)𝑑_𝑔(𝑃,𝑄)^2𝑚−1,_𝑀1𝑑_𝑔(𝑃,𝑄)^2𝑚−1𝑑𝑣_𝑔(𝑄)≤𝒞^′. (6.26) We thus obtain the following result.

Proposition 6.12. There exists a constant 𝐶₅>0 depending on 𝑚,𝐶^′₂, and (𝑀,𝑔) such that for all 𝑃∈𝑀|(∇𝜑_𝑡)_𝑃|≤𝐶₅.

Specifically, we can choose 𝐶₅=√2𝑚𝐶₃𝒞𝒞^′.

6.5. Second Derivatives Estimate

Our equation is of type: 𝐹𝑃,𝜕_{𝑢𝑖𝑢𝑗}𝜑_{1≤𝑖,𝑗≤2𝑚}=𝑣,𝑃∈𝑀. (𝐸)

6.5.1. The Functional

Consider the following functional: ()↦∇Φ∶𝑈𝑇→ℝ𝑃,𝜉²𝜑_𝑡𝑃()+1𝜉,𝜉2∇𝜑_𝑡𝑃²_𝑔, (6.27) 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 (∇²𝜑_𝑡)_𝑃1(𝜉₁,𝜉₁)

If we find a uniform upper bound for (∇²𝜑_𝑡)_𝑃1(𝜉₁,𝜉₁), since |∇𝜑_𝑡|_∞≤𝐶₅, we readily deduce that there exists a constant 𝐶₆>0 such that ∇²𝜑_𝑡𝑃(𝜉,𝜉)≤𝐶₆∀(𝑃,𝜉)∈𝑈𝑇. (6.28) Fix 𝑃∈𝑀. Let (𝑈_𝑃,𝜓_𝑃) be a holomorphic 𝑔-normal 𝑔-adapted chart centered at 𝑃, namely, [𝑔_𝑖𝑗(𝑃)]_{1≤𝑖,𝑗≤𝑚}=𝐼_𝑚, 𝜕_ℓ𝑔_𝑖𝑗(𝑃)=0 and [𝑔_𝑖𝑗(𝑃)]_{1≤𝑖,𝑗≤𝑚}=[diag(𝜆₁(𝑃),…,𝜆_𝑚(𝑃))]. Since |𝜕_𝑥𝑗|_𝑔=√2, we obtain 𝜕_{𝑥𝑗𝑥𝑗}𝜑_𝑡(𝑃)=2(∇²𝜑_𝑡)_𝑃(𝜕_𝑥𝑗/√2,𝜕_𝑥𝑗/√2)≤2𝐶₆ and similarly 𝜕_{𝑦𝑗𝑦𝑗}𝜑_𝑡(𝑃)=2(∇²𝜑_𝑡)_𝑃(𝜕_𝑦𝑗/√2,𝜕_𝑦𝑗/√2)≤2𝐶₆ for all 1≤𝑗≤𝑚. Besides, we have 𝜕_{𝑥𝑗𝑥𝑗}𝜑_𝑡(𝑃)+𝜕_{𝑦𝑗𝑦𝑗}𝜑_𝑡(𝑃)=4𝜕_𝑗𝑗𝜑_𝑡(𝑃)=4(𝜆_𝑗(𝑃)−1)≥−4[(𝑚−1)𝐶^′₂+1]; therefore we obtain 𝜕_{𝑥𝑗𝑥𝑗}𝜑_𝑡(𝑃)≥−4(𝑚−1)𝐶^′₂+1−2𝐶₆=∶−𝐶₇,𝜕_{𝑦𝑗𝑦𝑗}𝜑_𝑡(𝑃)≥−𝐶₇,∀1≤𝑗≤𝑚. (6.29) Let us now bound second derivatives of mixed type 𝜕_{𝑢𝑟𝑢𝑠}𝜑_𝑡(𝑃). Let 1≤𝑟≠𝑠≤2𝑚. Since |𝜕_𝑥𝑟±𝜕_𝑥𝑠|_𝑔=2, we have (∇²𝜑_𝑡)_𝑃((𝜕_𝑥𝑟±𝜕_𝑥𝑠)/2,(𝜕_𝑥𝑟±𝜕_𝑥𝑠)/2)=(1/4)𝜕_{𝑥𝑟𝑥𝑟}𝜑_𝑡(𝑃)+(1/4)𝜕_{𝑥𝑠𝑥𝑠}𝜑_𝑡(𝑃)±(1/2)𝜕_{𝑥𝑟𝑥𝑠}𝜑_𝑡(𝑃)≤𝐶₆, which yields ±𝜕_{𝑥𝑟𝑥𝑠}𝜑_𝑡(𝑃)≤2𝐶₆−(1/2)𝜕_{𝑥𝑟𝑥𝑟}𝜑_𝑡(𝑃)−(1/2)𝜕_{𝑥𝑠𝑥𝑠}𝜑_𝑡(𝑃), hence as well |𝜕_{𝑥𝑟𝑥𝑠}𝜑_𝑡(𝑃)|≤2𝐶₆+𝐶₇. Similarly we prove that at 𝑃, in the above chart 𝜓_𝑃, we have |𝜕_{𝑦𝑟𝑦𝑠}𝜑_𝑡(𝑃)|≤2𝐶₆+𝐶₇ for all 1≤𝑟≠𝑠≤𝑚 and |𝜕_{𝑥𝑟𝑦𝑠}𝜑_𝑡(𝑃)|≤2𝐶₆+𝐶₇ for all 1≤𝑟,𝑠≤𝑚. Consequently |𝜕_{𝑢𝑖𝑢𝑗}𝜑_𝑡(𝑃)|≤2𝐶₆+𝐶₇ for all 1≤𝑖, 𝑗≤2𝑚. Therefore we deduce that ∇²𝜑_𝑡(𝑃)²_𝑔=14_{1≤𝑖,𝑗≤2𝑚}𝜕_{𝑢𝑖𝑢𝑗}𝜑_𝑡(𝑃)²≤𝑚²2𝐶₆+𝐶₇². (6.30)

Theorem 6.13 (second derivatives uniform estimate). There exists a constant 𝐶₈>0 depending only on 𝑚,𝐶^′₂, and 𝐶₆ such that for all 𝑃∈𝑀, |(∇²𝜑_𝑡)_𝑃|_𝑔≤𝐶₈.

This allows to deduce the needed uniform 𝐶² estimate: 𝜑_{𝐶2(𝑀,ℝ)}≤𝐶₀+𝐶₅+𝐶₈. (6.31)

6.5.3. Chart Choice

For fixed 𝑃∈𝑀, 𝜉∈𝑈𝑇_𝑃↦(∇²𝜑_𝑡)_𝑃(𝜉,𝜉) is continuous on the compact subset 𝑈𝑇_𝑃 (the fiber); therefore it assumes its maximum at a unit vector 𝜉^𝑃∈𝑈𝑇_𝑃. Besides, (∇²𝜑_𝑡)_𝑃 is a symmetric bilinear form on 𝑇_𝑃𝑀, so by the min-max principle we have (∇²𝜑_𝑡)_𝑃(𝜉^𝑃,𝜉^𝑃)=max_{𝜉∈𝑇𝑃𝑀,𝑔(𝜉,𝜉)=1}(∇²𝜑_𝑡)_𝑃(𝜉,𝜉)=𝛽_max(𝑃), where 𝛽_max(𝑃) denotes the largest eigenvalue of (∇²𝜑_𝑡)_𝑃 with respect to 𝑔_𝑃; furthermore we can choose 𝜉^𝑃 as the direction of the largest eigenvalue 𝛽_max(𝑃). For fixed 𝑃, we now have max_{𝜉∈𝑇𝑃𝑀,𝑔𝑃(𝜉,𝜉)=1}Φ(𝑃,𝜉)=Φ(𝑃,𝜉^𝑃)=(∇²𝜑_𝑡)_𝑃(𝜉^𝑃,𝜉^𝑃)+(1/2)|(∇𝜑_𝑡)_𝑃|²_𝑔=𝛽_max(𝑃)+(1/2)|(∇𝜑_𝑡)_𝑃|²_𝑔, consequently max_{(𝑃,𝜉)∈𝑈𝑇}Φ(𝑃,𝜉)=max_𝑃∈𝑀Φ(𝑃,𝜉^𝑃)=Φ(𝑃₁,𝜉₁)≤Φ(𝑃₁,𝜉^𝑃1), hence max_{(𝑃,𝜉)∈𝑈𝑇}Φ(𝑃,𝜉)=Φ(𝑃₁,𝜉^𝑃1)=𝛽_max(𝑃₁)+(1/2)|(∇𝜑_𝑡)_𝑃1|²_𝑔.

At the point 𝑃₁, consider 𝜀^𝑃1₁,…,𝜀^𝑃1_2𝑚 a (real) basis of (𝑇_𝑃1𝑀,𝑔_𝑃1) that satisfies the following properties: (i)[𝑔_𝑖𝑗(𝑃₁)]_{1≤𝑖,𝑗≤2𝑚}=𝐼_2𝑚,(ii)[(∇²𝜑_𝑡)_𝑖𝑗(𝑃₀)]_{1≤𝑖,𝑗≤2𝑚}=diag(𝛽₁,…,𝛽_2𝑚),(iii)𝛽₁=𝛽_max(𝑃₁)≥𝛽₂≥⋯≥𝛽_2𝑚. Let (𝑈^′₁,𝜓₁) be a 𝐶^∞𝑔-normal real chart at 𝑃₁ obtained from a holomorphic chart 𝑧¹,…,𝑧^𝑚 by setting (𝑢¹,…,𝑢^2𝑚)=(𝑥¹,…,𝑥^𝑚,𝑦¹,…,𝑦^𝑚) where 𝑧^𝑗=𝑥^𝑗+𝑖𝑦^𝑗 (namely, [𝑔_𝑖𝑗(𝑃₁)]_{1≤𝑖,𝑗≤2𝑚}=𝐼_2𝑚 and 𝜕_𝑢ℓ𝑔_𝑖𝑗=0 for all 1≤𝑖,𝑗,ℓ≤2𝑚) satisfying 𝜓₁(𝑃₁)=0 and 𝜕_𝑢𝑖|_𝑃1=𝜀^𝑃1_𝑖, so that 𝜕_𝑢1|_𝑃1 is the direction of the largest eigenvalue 𝛽_max(𝑃₁).

6.5.4. Auxiliary Local Functional

From now on, we work in the real chart (𝑈^′₁,𝜓₁) so constructed at 𝑃₁.

Let 𝑈₂⊂𝑈^′₁ be an open subset such that 𝑔₁₁(𝑃)>0 for all 𝑃∈𝑈₂, and let Φ₁ be the functional Φ₁∶𝑈₂→ℝ𝑃↦Φ₁(𝑃)=∇²𝜑_𝑡11(𝑃)𝑔₁₁(𝑃)+12∇𝜑_𝑡𝑃²_𝑔. (6.32) We claim that Φ₁ assumes its maximum at 𝑃₁. Indeed, (∇²𝜑_𝑡)₁₁(𝑃)/𝑔₁₁(𝑃)=(∇²𝜑)_𝑃(𝜕_𝑢1,𝜕_𝑢1)/𝑔_𝑃(𝜕_𝑢1,𝜕_𝑢1)=(∇²𝜑)_𝑃(𝜕_𝑢1/|𝜕_𝑢1|_𝑔,𝜕_𝑢1/|𝜕_𝑢1|_𝑔)≤𝛽_max(𝑃), so Φ₁(𝑃)≤𝛽_max(𝑃)+(1/2)|(∇𝜑_𝑡)_𝑃|²_𝑔≤𝛽_max(𝑃₁)+(1/2)|(∇𝜑_𝑡)_𝑃1|²_𝑔=Φ₁(𝑃₁) proving our claim.

Let us now differentiate twice in the real direction 𝜕_𝑢1 the equation 𝐹𝑃,𝜕_{𝑢𝑖𝑢𝑗}𝜑_{1≤𝑖,𝑗≤2𝑚}=𝑣. 𝐸^∗ At the point 𝑃, in a chart 𝑢, we obtain 𝜕_𝑢1𝑣=𝜕𝐹𝜕𝑢¹𝜑+^2𝑚_𝑖,𝑗=1𝜕𝐹𝜕𝑟_𝑖𝑗𝜑𝜕_{𝑢1𝑢𝑖𝑢𝑗}𝜑. (6.33) Differentiating once again 𝜕_𝑢1𝑢1𝜕𝑣=²𝐹𝜕𝑢¹𝜕𝑢¹𝜑+^2𝑚_𝑖,𝑗=1𝜕²𝐹𝜕𝑟_𝑖𝑗𝜕𝑢¹𝜑𝜕_{𝑢1𝑢𝑖𝑢𝑗}𝜑+^2𝑚_𝑖,𝑗=1𝜕²𝐹𝜕𝑢¹𝜕𝑟_𝑖𝑗𝜑+^2𝑚_𝑒,𝑠=1𝜕²𝐹𝜕𝑟_𝑒𝑠𝜕𝑟_𝑖𝑗𝜑𝜕_{𝑢1𝑢𝑒𝑢𝑠}𝜑𝜕_{𝑢1𝑢𝑖𝑢𝑗}𝜑+^2𝑚_𝑖,𝑗=1𝜕𝐹𝜕𝑟_𝑖𝑗𝜑𝜕_{𝑢1𝑢1𝑢𝑖𝑢𝑗}𝜑. (6.34) But at the point 𝑃₁, for our function 𝐹(𝑃,𝑟)=𝐹_𝑘[𝛿^𝑗_𝑖+(1/4)𝑔^𝑗ℓ(𝑃)(𝑟_𝑖ℓ+𝑟_{(𝑖+𝑚)(ℓ+𝑚)}+𝑖𝑟_{𝑖(ℓ+𝑚)}−𝑖𝑟_{(𝑖+𝑚)ℓ})]_{1≤𝑖,𝑗≤𝑚}, we have (𝜕²𝐹/𝜕𝑟_𝑖𝑗𝜕𝑢¹)[𝜑]=0 since 𝜕_𝑢1𝑔^𝑠𝑞(𝑃₁)=0. Hence, we infer that 𝜕_𝑢1𝑢1𝜕𝑣=²𝐹𝜕𝑢¹𝜕𝑢¹𝜑+^2𝑚_{𝑖,𝑗,𝑒,𝑠=1}𝜕²𝐹𝜕𝑟_𝑒𝑠𝜕𝑟_𝑖𝑗𝜑𝜕_{𝑢1𝑢𝑒𝑢𝑠}𝜑𝜕_{𝑢1𝑢𝑖𝑢𝑗}𝜑+^2𝑚_𝑖,𝑗=1𝜕𝐹𝜕𝑟_𝑖𝑗𝜑𝜕_{𝑢1𝑢1𝑢𝑖𝑢𝑗}𝜑. (6.35)

6.5.5. Using Concavity

The function 𝐹 is concave with respect to the variable 𝑟. Indeed 𝐹(𝑃,𝑟)=𝐹_𝑘𝛿^𝑗_𝑖+14𝑔^𝑗ℓ(𝑃)𝑟_𝑖ℓ+𝑟_{(𝑖+𝑚)(ℓ+𝑚)}+𝑖𝑟_{𝑖(ℓ+𝑚)}−𝑖𝑟_{(𝑖+𝑚)ℓ1≤𝑖,𝑗≤𝑚}=𝐹_𝑘𝑔⁻¹(𝑃)𝑟̃̃𝑔,where𝑟=_𝑖𝑗(𝑃)+14𝑟_𝑖𝑗+𝑟_{(𝑖+𝑚)(𝑗+𝑚)}+𝑖𝑟_{𝑖(𝑗+𝑚)}−𝑖𝑟_{(𝑖+𝑚)𝑗1≤𝑖,𝑗≤𝑚}=𝐹_𝑘𝑔^−1/2(𝑃)𝑟𝑔̃^−1/2(𝑃)_{∈ℋ𝑚(ℂ)}=𝐹_𝑘𝜌_𝑃(𝑟𝜌),where_𝑃(𝑟)∶=𝛿^𝑗_𝑖+14^𝑚_ℓ,𝑠=1𝑔^−1/2(𝑃)_𝑖ℓ𝑔^−1/2(𝑃)_𝑠𝑗𝑟_ℓ𝑠+𝑟_{(ℓ+𝑚)(𝑠+𝑚)}+𝑖𝑟_{ℓ(𝑠+𝑚)}−𝑖𝑟_{(ℓ+𝑚)𝑠1≤𝑖,𝑗≤𝑚} (6.36) but for a fixed point 𝑃 the function 𝑟∈𝑆_2𝑚(ℝ)↦𝜌_𝑃(𝑟)∈ℋ_𝑚(ℂ) is affine (where 𝑆_2𝑚(ℝ) denotes the set of symmetric matrices of size 2𝑚); we deduce then that the composition 𝐹(𝑃,⋅)=𝐹_𝑘∘𝜌_𝑃 is concave on the set {𝑟∈𝑆_2𝑚(ℝ)/𝜌_𝑃(𝑟)∈𝜆⁻¹(Γ_𝑘)}=𝜌_𝑃⁻¹(𝜆⁻¹(Γ_𝑘)), which proves our claim. Hence, since the matrix [𝜕_{𝑢1𝑢𝑖𝑢𝑗}𝜑]_{1≤𝑖,𝑗≤𝑚} is symmetric, we obtain that ^2𝑚_{𝑖,𝑗,𝑒,𝑠=1}𝜕²𝐹𝜕𝑟_𝑒𝑠𝜕𝑟_𝑖𝑗𝜑𝜕_{𝑢1𝑢𝑒𝑢𝑠}𝜑𝜕_{𝑢1𝑢𝑖𝑢𝑗}𝜑≤0. (6.37) Consequently 𝜕_𝑢1𝑢1𝑣−𝜕_𝑢1𝑢1𝐹𝜑≤^2𝑚_𝑖,𝑗=1𝜕𝐹𝜕𝑟_𝑖𝑗𝜑𝜕_{𝑢1𝑢1𝑢𝑖𝑢𝑗}𝜑. (6.38) Let us now calculate the quantity 𝜕_𝑢1𝑢1𝐹[𝜑] (at 𝑃₁). Since 𝜕_𝑢1𝑔^𝑠𝑞(𝑃₁)=0, we have 𝜕²𝐹𝜕𝑢¹𝜕𝑢¹𝑃₁,𝐷²𝜑𝑃₁=^𝑚_𝑠=1𝜎_{𝑘−1,𝑠}𝜆𝑃₁𝜎_𝑘𝜆𝑃₁×𝜕_𝑢1𝑢1𝑔^𝑠𝑠𝑃₁𝜕_𝑠𝑠𝜑𝑃₁. (6.39) But at 𝑃₁, 𝜕_𝑢1𝑢1𝑔^𝑠𝑠=−𝑔^𝑠𝑜𝑔^𝑞𝑠𝜕_𝑢1𝑢1𝑔_𝑞𝑜 and [𝑔^𝑖𝑗]_{1≤𝑖,𝑗≤𝑚}=2𝐼_𝑚, then 𝜕_𝑢1𝑢1𝑔^𝑠𝑠=−4𝜕_𝑢1𝑢1𝑔_𝑠𝑠 so that 𝜕_𝑢1𝑢1𝑔^𝑠𝑠=−𝜕_𝑢1𝑢1𝑔_𝑠𝑠−𝜕_𝑢1𝑢1𝑔_{(𝑠+𝑚)(𝑠+𝑚)}. Moreover Γ^𝑢𝑟_{𝑢𝑗𝑢𝑠}=(1/2)(𝜕_𝑢𝑗𝑔_𝑜𝑠+𝜕_𝑢𝑠𝑔_𝑜𝑗−𝜕_𝑢𝑜𝑔_𝑗𝑠)𝑔^𝑜𝑟, thus 𝜕_𝑢𝑖Γ^𝑢𝑟_{𝑢𝑗𝑢𝑠}=(1/2)(𝜕_{𝑢𝑖𝑢𝑗}𝑔_𝑟𝑠+𝜕_{𝑢𝑖𝑢𝑠}𝑔_𝑟𝑗−𝜕_{𝑢𝑖𝑢𝑟}𝑔_𝑗𝑠). Similarly, we have at 𝑃₁ : 𝜕_𝑢𝑖Γ^𝑢𝑠_{𝑢𝑗𝑢𝑟}=(1/2)(𝜕_{𝑢𝑖𝑢𝑗}𝑔_𝑟𝑠+𝜕_{𝑢𝑖𝑢𝑟}𝑔_𝑠𝑗−𝜕_{𝑢𝑖𝑢𝑠}𝑔_𝑗𝑟). Consequently, we deduce that 𝜕_{𝑢𝑖𝑢𝑗}𝑔_𝑟𝑠=𝜕_𝑢𝑖Γ^𝑢𝑟_{𝑢𝑗𝑢𝑠}+𝜕_𝑢𝑖Γ^𝑢𝑠_{𝑢𝑗𝑢𝑟}. Hence, we have at 𝑃₁: 𝜕_𝑢1𝑢1𝑔^𝑠𝑠=−2𝜕_𝑢1Γ^𝑢𝑠_{𝑢1𝑢𝑠}−2𝜕_𝑢1Γ^{𝑢𝑠+𝑚}_{𝑢1𝑢𝑠+𝑚}. Besides, 𝜕_𝑠𝑠𝜑=(1/4)(𝜕_{𝑢𝑠𝑢𝑠}𝜑+𝜕_{𝑢𝑠+𝑚𝑢𝑠+𝑚}𝜑), which infers that at 𝑃₁𝜕_𝑢1𝑢1𝐹𝜑=^𝑚_𝑠=1𝜎_{𝑘−1,𝑠}𝜆𝑃₁𝜎_𝑘𝜆𝑃₁×−2𝜕_𝑢1Γ^𝑢𝑠_{𝑢1𝑢𝑠}−2𝜕_𝑢1Γ^{𝑢𝑠+𝑚}_{𝑢1𝑢𝑠+𝑚}×14𝜕_{𝑢𝑠𝑢𝑠}𝜑+𝜕_{𝑢𝑠+𝑚𝑢𝑠+𝑚}𝜑. (6.40) Consequently, the inequality (6.38) becomes 𝜕_𝑢1𝑢1𝑣≤^2𝑚_𝑖,𝑗=1𝜕𝐹𝜕𝑟_𝑖𝑗𝜑𝜕_{𝑢1𝑢1𝑢𝑖𝑢𝑗}1𝜑−2^𝑚_𝑠=1𝜎_{𝑘−1,𝑠}𝜆𝑃₁𝜎_𝑘𝜆𝑃₁×𝜕_𝑢1Γ^𝑢𝑠_{𝑢1𝑢𝑠}+𝜕_𝑢1Γ^{𝑢𝑠+𝑚}_{𝑢1𝑢𝑠+𝑚}×𝜕_{𝑢𝑠𝑢𝑠}𝜑+𝜕_{𝑢𝑠+𝑚𝑢𝑠+𝑚}𝜑. (6.41)

6.5.6. Differentiation of the Functional Φ₁

We differentiate twice the functional Φ₁: Φ₁(𝑃)=∇²𝜑₁₁(𝑃)𝑔₁₁(𝑃)+12∇𝜑_𝑃²_𝑔,𝜕_𝑢𝑗Φ₁(𝑃)=𝜕_𝑢𝑗∇²𝜑₁₁𝑔₁₁(𝑃)−∇²𝜑₁₁𝜕_𝑢𝑗𝑔₁₁(𝑃)𝑔₁₁(𝑃)²+12𝜕_𝑢𝑗∇𝜑_𝑃²_𝑔,𝜕_{𝑢𝑖𝑢𝑗}Φ₁(𝑃)=𝜕_{𝑢𝑖𝑢𝑗}∇²𝜑₁₁𝑔₁₁(𝑃)−𝜕_𝑢𝑗∇²𝜑₁₁𝜕_𝑢𝑖𝑔₁₁(𝑃)𝑔₁₁(𝑃)²−𝜕_𝑢𝑖∇²𝜑₁₁𝜕_𝑢𝑗𝑔₁₁(𝑃)+∇²𝜑₁₁(𝑃)𝜕_{𝑢𝑖𝑢𝑗}𝑔₁₁(𝑃)𝑔₁₁(𝑃)²−∇²𝜑₁₁(𝑃)𝜕_𝑢𝑗𝑔₁₁(𝑃)𝜕_𝑢𝑖1𝑔₁₁(𝑃)²+12𝜕_{𝑢𝑖𝑢𝑗}∇𝜑_𝑃²_𝑔. (6.42) Hence, at 𝑃₁ in the chart 𝜓₁, we obtain 𝜕_{𝑢𝑖𝑢𝑗}Φ₁=𝜕_{𝑢𝑖𝑢𝑗}∇²𝜑₁₁−∇²𝜑₁₁𝑃₁𝜕_{𝑢𝑖𝑢𝑗}𝑔₁₁+12𝜕_{𝑢𝑖𝑢𝑗}∇𝜑_𝑃²_𝑔𝑃₁. (6.43) Let us now calculate the different terms of this formula (at 𝑃₁ in the chart 𝜓₁): 𝜕_{𝑢𝑖𝑢𝑗}∇²𝜑₁₁=𝜕_{𝑢𝑖𝑢𝑗}𝜕_𝑢1𝑢1𝜑−Γ^𝑢𝑠_𝑢1𝑢1𝜕_𝑢𝑠𝜑=𝜕_{𝑢𝑖𝑢𝑗𝑢1𝑢1}𝜑−𝜕_{𝑢𝑖𝑢𝑗}Γ^𝑢𝑠_𝑢1𝑢1𝜕_𝑢𝑠𝜑−𝜕_𝑢𝑗Γ^𝑢𝑠_𝑢1𝑢1𝜕_{𝑢𝑖𝑢𝑠}𝜑−𝜕_𝑢𝑖Γ^𝑢𝑠_𝑢1𝑢1𝜕_{𝑢𝑗𝑢𝑠}𝜑. (6.44) Besides, we have Γ^𝑢1_{𝑢𝑗𝑢1}=(1/2)(𝜕_𝑢𝑗𝑔_𝑠1+𝜕_𝑢1𝑔_𝑠𝑗−𝜕_𝑢𝑠𝑔_𝑗1)𝑔^𝑠1; therefore we deduce that 𝜕_𝑢𝑖Γ^𝑢1_{𝑢𝑗𝑢1}=(1/2)(𝜕_{𝑢𝑖𝑢𝑗}𝑔_𝑠1+𝜕_{𝑢𝑖𝑢1}𝑔_𝑠𝑗−𝜕_{𝑢𝑖𝑢𝑠}𝑔_𝑗1)𝑔^𝑠1+0=(1/2)𝜕_{𝑢𝑖𝑢𝑗}𝑔₁₁; namely, 𝜕_{𝑢𝑖𝑢𝑗}𝑔₁₁=2𝜕_𝑢𝑖Γ^𝑢1_{𝑢𝑗𝑢1}. Moreover, we have at 𝑃₁𝜕_{𝑢𝑖𝑢𝑗}∇𝜑_𝑃²_𝑔=𝜕_{𝑢𝑖𝑢𝑗}^2𝑚_𝑟,𝑠=1𝑔^𝑟𝑠𝜕_𝑢𝑟𝜑𝜕_𝑢𝑠𝜑=^2𝑚_𝑟,𝑠=1𝜕_{𝑢𝑖𝑢𝑗}𝑔^𝑟𝑠𝜕_𝑢𝑟𝜑𝜕_𝑢𝑠𝜑+𝑔^𝑟𝑠𝜕_{𝑢𝑖𝑢𝑗𝑢𝑟}𝜑𝜕_𝑢𝑠𝜑+𝑔^𝑟𝑠𝜕_{𝑢𝑗𝑢𝑟}𝜑𝜕_{𝑢𝑖𝑢𝑠}𝜑+𝑔^𝑟𝑠𝜕_{𝑢𝑖𝑢𝑟}𝜑𝜕_{𝑢𝑗𝑢𝑠}𝜑+𝑔^𝑟𝑠𝜕_𝑢𝑟𝜑𝜕_{𝑢𝑖𝑢𝑗𝑢𝑠}𝜑=^2𝑚_𝑟,𝑠=1𝜕_{𝑢𝑖𝑢𝑗}𝑔^𝑟𝑠𝜕_𝑢𝑟𝜑𝜕_𝑢𝑠𝜑+2^2𝑚_𝑠=1𝜕_{𝑢𝑖𝑢𝑗𝑢𝑠}𝜑𝜕_𝑢𝑠𝜑+2^2𝑚_𝑠=1𝜕_{𝑢𝑖𝑢𝑠}𝜑𝜕_{𝑢𝑗𝑢𝑠}𝜑. (6.45) But at 𝑃₁, 𝜕_{𝑢𝑖𝑢𝑗}𝑔^𝑟𝑠=−𝜕_{𝑢𝑖𝑢𝑗}𝑔_𝑟𝑠, in addition at this point 𝜕_{𝑢𝑖𝑢𝑗}𝑔_𝑟𝑠=𝜕_𝑢𝑖Γ^𝑢𝑟_{𝑢𝑗𝑢𝑠}+𝜕_𝑢𝑖Γ^𝑢𝑠_{𝑢𝑗𝑢𝑟}; therefore we obtain at 𝑃₁ in the chart 𝜓₁𝜕_{𝑢𝑖𝑢𝑗}∇𝜑_𝑃²_𝑔=−2^2𝑚_𝑟,𝑠=1𝜕_𝑢𝑖Γ^𝑢𝑟_{𝑢𝑗𝑢𝑠}𝜕_𝑢𝑟𝜑𝜕_𝑢𝑠𝜑+2^2𝑚_𝑠=1𝜕_{𝑢𝑖𝑢𝑗𝑢𝑠}𝜑𝜕_𝑢𝑠𝜑+2^2𝑚_𝑠=1𝜕_{𝑢𝑖𝑢𝑠}𝜑𝜕_{𝑢𝑗𝑢𝑠}𝜑. (6.46) Henceforth, and in order to lighten the notations, we use 𝜕_𝑖 instead of 𝜕_𝑢𝑖 and Γ^𝑠_𝑖𝑗 instead of Γ^𝑢𝑠_{𝑢𝑖𝑢𝑗}, so we have 𝜕_𝑖𝑗Φ₁=𝜕_𝑖𝑗11𝜑−𝜕_𝑖𝑗Γ^𝑠₁₁𝜕_𝑠𝜑−𝜕_𝑗Γ^𝑠₁₁𝜕_𝑖𝑠𝜑−𝜕_𝑖Γ^𝑠₁₁𝜕_𝑗𝑠𝜑−2𝜕_𝑖Γ¹_𝑗1∇²𝜑₁₁𝑃₁−^2𝑚_𝑟,𝑠=1𝜕_𝑖Γ^𝑟_𝑗𝑠𝜕_𝑟𝜑𝜕_𝑠𝜑+^2𝑚_𝑠=1𝜕_𝑖𝑗𝑠𝜑𝜕_𝑠𝜑+^2𝑚_𝑠=1𝜕_𝑖𝑠𝜑𝜕_𝑗𝑠𝜑. (6.47) Let us now consider the second order linear operator: 𝐿=^2𝑚_𝑖,𝑗=1𝜕𝐹𝜕𝑟_𝑖𝑗𝜑𝜕_𝑖𝑗. (6.48) 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 2𝐿Φ_1≤0−𝜕₁₁𝑣≥^2𝑚_𝑖,𝑗=1𝜕𝐹𝜕𝑟_𝑖𝑗𝜑𝜕_𝑖𝑗11𝜑−𝜕_𝑖𝑗Γ^𝑠₁₁𝜕_𝑠𝜑−𝜕_𝑗Γ^𝑖₁₁𝜕_𝑖𝑖𝜑−𝜕_𝑖Γ^𝑗₁₁𝜕_𝑗𝑗𝜑−2𝜕_𝑖Γ¹_𝑗1∇²𝜑₁₁𝑃₁−^2𝑚_𝑟,𝑠=1𝜕_𝑖Γ^𝑟_𝑗𝑠𝜕_𝑟𝜑𝜕_𝑠𝜑+^2𝑚_𝑠=1𝜕_𝑖𝑗𝑠𝜑𝜕_𝑠𝜑+𝛿^𝑗_𝑖𝜕_𝑖𝑖𝜑²−𝜕_11𝑖𝑗𝜑+12^𝑚_𝑠=1𝜎_{𝑘−1,𝑠}𝜆𝑃₁𝜎_𝑘𝜆𝑃₁𝜕₁Γ^𝑠_1𝑠+𝜕₁Γ₁₍₎^𝑠+𝑚_𝑠+𝑚𝜕_𝑠𝑠𝜑+𝜕_{(𝑠+𝑚)(𝑠+𝑚)}𝜑. (6.49) The fourth derivatives are simplified. Moreover, we have at 𝑃₁ : 𝜕_𝑠𝑣=(𝜕𝐹/𝜕𝑢¹)[𝜑]+∑^2𝑚_𝑖,𝑗=1(𝜕𝐹/𝜕𝑟_𝑖𝑗)[𝜑]𝜕_𝑠𝑖𝑗𝜑 with (𝜕𝐹/𝜕𝑢¹)(𝑃₁,𝐷²𝜑(𝑃₁))=0, consequently: 0≥𝜕₁₁𝑣+^2𝑚_𝑠=1𝜕_𝑠𝑣𝜕_𝑠𝜑+^2𝑚_𝑖,𝑗=1𝜕𝐹𝜕𝑟_𝑖𝑗𝜑−2𝜕_𝑖Γ¹_𝑗1∇²𝜑₁₁𝑃₁−𝜕_𝑖Γ^𝑗₁₁𝜕_𝑗𝑗𝜑−𝜕_𝑗Γ^𝑖₁₁𝜕_𝑖𝑖𝜑−^2𝑚_𝑠=1𝜕_𝑖𝑗Γ^𝑠₁₁𝜕_𝑠𝜑−^2𝑚_𝑟,𝑠=1𝜕_𝑖Γ^𝑟_𝑗𝑠𝜕_𝑟𝜑𝜕_𝑠𝜑+𝛿^𝑗_𝑖𝜕_𝑖𝑖𝜑²+12^𝑚_𝑠=1𝜎_{𝑘−1,𝑠}𝜆𝑃₁𝜎_𝑘𝜆𝑃₁𝜕₁Γ^𝑠_1𝑠+𝜕₁Γ₁₍₎^𝑠+𝑚_𝑠+𝑚𝜕_𝑠𝑠𝜑+𝜕_{(𝑠+𝑚)(𝑠+𝑚)}𝜑. (6.50) Let us now express the quantities 𝜕_𝑖Γ¹_𝑗1,𝜕_𝑖Γ^𝑗₁₁,𝜕_𝑗Γ^𝑖₁₁,𝜕_𝑖Γ^𝑟_𝑗𝑠 and 𝜕_𝑖𝑗Γ^𝑠₁₁ using the components of the Riemann curvature (at the point 𝑃₁ in the normal chart 𝜓₁): 𝜕_𝑖Γ¹_𝑗1=13𝑅_𝑗11𝑖+𝑅_𝑗𝑖11=0=13𝑅_𝑗11𝑖,𝜕_𝑖Γ^𝑗₁₁=13𝑅_1𝑗1𝑖+𝑅_1𝑖1𝑗=23𝑅_1𝑗1𝑖,𝜕_𝑗Γ^𝑖₁₁=23𝑅_1𝑖1𝑗,𝜕_𝑖Γ^𝑟_𝑗𝑠=13𝑅_{𝑗𝑟𝑠𝑖}+𝑅_{𝑗𝑖𝑠𝑟},𝜕₁Γ^𝑠_1𝑠=13𝑅_1𝑠𝑠1+𝑅_11𝑠𝑠=0=13𝑅_1𝑠𝑠1,𝜕₁Γ^𝑠+𝑚_1(𝑠+𝑚)=13𝑅_{1(𝑠+𝑚)(𝑠+𝑚)1},𝜕_𝑖𝑗Γ^𝑠₁₁=14∇_𝑖𝑅_1𝑗1𝑠+∇_𝑖𝑅_1𝑠1𝑗+∇_𝑗𝑅_1𝑠1𝑖+∇_𝑗𝑅_1𝑖1𝑠−1∇12^𝑠𝑅_1𝑖1𝑗+∇_𝑠𝑅_1𝑗1𝑖=12∇_𝑖𝑅_1𝑠1𝑗+∇_𝑗𝑅_1𝑠1𝑖−16∇_𝑠𝑅_1𝑖1𝑗. (6.51) We then obtain 0≥𝜕₁₁𝑣+^2𝑚_𝑠=1𝜕_𝑠𝑣𝜕_𝑠𝜑+^2𝑚_𝑖,𝑗=1𝜕𝐹𝜕𝑟_𝑖𝑗𝜑−23𝑅_𝑗11𝑖∇²𝜑₁₁𝑃₁−23𝑅_1𝑗1𝑖𝜕_𝑗𝑗2𝜑−3𝑅_1𝑖1𝑗𝜕_𝑖𝑖𝜑−^2𝑚_𝑠=112∇_𝑖𝑅_1𝑠1𝑗+12∇_𝑗𝑅_1𝑠1𝑖−16∇_𝑠𝑅_1𝑖1𝑗𝜕_𝑠𝜑−^2𝑚_𝑟,𝑠=113𝑅_{𝑗𝑟𝑠𝑖}+𝑅_{𝑗𝑖𝑠𝑟}𝜕_𝑟𝜑𝜕_𝑠𝜑+𝛿^𝑗_𝑖𝜕_𝑖𝑖𝜑²+12^𝑚_𝑠=1𝜎_{𝑘−1,𝑠}𝜆𝑃₁𝜎_𝑘𝜆𝑃₁13𝑅_1𝑠𝑠1+𝑅_{1(𝑠+𝑚)(𝑠+𝑚)1}𝜕_𝑠𝑠𝜑+𝜕_{(𝑠+𝑚)(𝑠+𝑚)}𝜑. (6.52)

6.5.7. The Uniform Upper Bound of 𝛽₁=(∇²𝜑)_𝑃1(𝜉₁,𝜉₁)

By the uniform estimate of the gradient we have |𝜕_𝑗𝜑_𝑡|≤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,…,𝛽_2𝑚). Consequently 0≥𝜕₁₁𝑣+^2𝑚_𝑠=1𝜕_𝑠𝑣𝜕_𝑠𝜑+^2𝑚_𝑖,𝑗=1𝜕𝐹𝜕𝑟_𝑖𝑗𝜑𝛿_𝑖𝑗𝛽_𝑖²−23𝑅_𝑗11𝑖𝛽₁−23𝑅_1𝑗1𝑖𝛽_𝑗−23𝑅_1𝑖1𝑗𝛽_𝑖−13^2𝑚_𝑟,𝑠=1𝑅_{𝑗𝑟𝑠𝑖}+𝑅_{𝑗𝑖𝑠𝑟}𝜕_𝑟𝜑𝜕_𝑠𝜑−12^2𝑚_𝑠=1∇_𝑖𝑅_1𝑠1𝑗+∇_𝑗𝑅_1𝑠1𝑖−13∇_𝑠𝑅_1𝑖1𝑗𝜕_𝑠𝜑+16^𝑚_𝑖=1𝜎_{𝑘−1,𝑖}𝜆𝑃₁𝜎_𝑘𝜆𝑃₁×𝑅_1𝑖𝑖1+𝑅_{1(𝑖+𝑚)(𝑖+𝑚)1}𝛽_𝑖+𝛽_𝑖+𝑚. (6.53) But for 𝐹[𝜑]=𝐹_𝑘([𝛿^𝑗_𝑖+𝑔^𝑗ℓ𝜕_𝑖ℓ𝜑]_{1≤𝑖,𝑗≤𝑚}) since 𝜕_𝑠𝑠𝜑=(1/4)(𝜕_{𝑢𝑠𝑢𝑠}+𝜕_{𝑢𝑠+𝑚𝑢𝑠+𝑚}), we obtain at 𝑃₁ in the chart 𝜓₁ that 𝜕𝐹𝜕𝑟_𝑖𝑗𝜑=^𝑚_𝑠=1𝜕𝐹_𝑘𝜕𝐵^𝑠_𝑠diag𝜆𝑃₁14𝜕𝑟_𝑠𝑠+𝑟_{(𝑠+𝑚)(𝑠+𝑚)}𝜕𝑟_𝑖𝑗. (6.54) Then ∀1≤𝑖≠𝑗≤2𝑚𝜕𝐹𝜕𝑟_𝑖𝑗𝜑=0,∀1≤𝑖≤𝑚𝜕𝐹𝜕𝑟_𝑖𝑖𝜑=𝜕𝐹𝜕𝑟_{(𝑖+𝑚)(𝑖+𝑚)}𝜑=14𝜕𝐹_𝑘𝜕𝐵^𝑖_𝑖diag𝜆𝑃₁=14𝜎_{𝑘−1,𝑖}𝜆𝑃₁𝜎_𝑘𝜆𝑃_{1>0since𝜆𝑃1∈Γ𝑘}. (6.55) Hence 0≥𝜕₁₁𝑣+^2𝑚_𝑠=1𝜕_𝑠𝑣𝜕_𝑠𝜑+^2𝑚_𝑖−1𝜕𝐹𝜕𝑟_𝑖𝑖𝜑×𝛽_𝑖²+23𝑅_1𝑖1𝑖𝛽₁−2𝛽_𝑖+13^2𝑚_𝑟,𝑠=1𝑅_{𝑖𝑟𝑖𝑠}𝜕_𝑟𝜑𝜕_𝑠𝜑−^2𝑚_𝑠=1∇_𝑖𝑅_1𝑠1𝑖−16∇_𝑠𝑅_1𝑖1𝑖𝜕_𝑠𝜑+16^𝑚_𝑖=1𝜎_{𝑘−1,𝑖}𝜆𝑃₁𝜎_𝑘𝜆𝑃₁𝑅_1𝑖𝑖1+𝑅_{1(𝑖+𝑚)(𝑖+𝑚)1}𝛽_𝑖+𝛽_𝑖+𝑚. (6.56) But at 𝑃₁ in the chart 𝜓₁, ‖𝑅‖²_𝑔=𝑔^𝑎𝑖𝑔^𝑏𝑗𝑔^𝑐𝑟𝑔^𝑑𝑠𝑅_{𝑎𝑏𝑐𝑑}𝑅_{𝑖𝑗𝑟𝑠}=∑^2𝑚_{𝑎,𝑏,𝑐,𝑑=1}(𝑅_{𝑎𝑏𝑐𝑑})²; then |𝑅_{𝑎𝑏𝑐𝑑}|≤‖𝑅‖_𝑔 for all 𝑎,𝑏,𝑐,𝑑∈{1,…,2𝑚}, consequently 13^2𝑚_𝑟,𝑠=1𝑅_{𝑖𝑟𝑖𝑠}𝜕_𝑟𝜑𝜕_𝑠𝜑≤13^2𝑚_𝑟,𝑠=1‖𝑅‖_𝑔𝐶₅²=13(2𝑚)²‖𝑅‖_𝑔𝐶₅²=43𝑚²𝐶₅²‖𝑅‖_𝑔. (6.57) Besides, at 𝑃₁ in the chart 𝜓₁, we have ‖∇𝑅‖²_𝑔=𝑔^𝑒𝑙𝑔^𝑎𝑖𝑔^𝑏𝑗𝑔^𝑐𝑟𝑔^𝑑𝑠∇_𝑒𝑅_{𝑎𝑏𝑐𝑑}∇_𝑙𝑅_{𝑖𝑗𝑟𝑠}

=∑^2𝑚_{𝑒,𝑎,𝑏,𝑐,𝑑=1}(∇_𝑒𝑅_{𝑎𝑏𝑐𝑑})², so |∇_𝑒𝑅_{𝑎𝑏𝑐𝑑}|≤‖∇𝑅‖_𝑔 for all 𝑒,𝑎,𝑏,𝑐,𝑑∈{1,…,2𝑚}, therefore −^2𝑚_𝑠=1∇_𝑖𝑅_1𝑠1𝑖−16∇_𝑠𝑅_1𝑖1𝑖𝜕_𝑠𝜑≤^2𝑚_𝑠=176‖∇𝑅‖_𝑔𝐶₅7=2𝑚6‖∇𝑅‖_𝑔𝐶₅=73𝑚𝐶₅‖∇𝑅‖_𝑔. (6.58) Hence at 𝑃₁ in the chart 𝜓₁, we obtain −𝑡𝜕₁₁𝑓−𝑡^2𝑚_𝑠=1𝜕_𝑠𝑓𝜕_𝑠𝜑≥^2𝑚_𝑖=1𝜕𝐹𝜕𝑟_𝑖𝑖𝜑𝛽_𝑖²+23𝑅_1𝑖1𝑖𝛽₁−2𝛽_𝑖+16^𝑚_𝑖=1𝜎_{𝑘−1,𝑖}𝜆𝑃₁𝜎_𝑘𝜆𝑃₁×𝑅_1𝑖𝑖1+𝑅_{1(𝑖+𝑚)(𝑖+𝑚)1}𝛽_𝑖+𝛽_𝑖+𝑚+^2𝑚_𝑖=1𝜕𝐹𝜕𝑟_𝑖𝑖𝜑×−43𝑚²𝐶₅²‖𝑅‖_𝑔−73𝑚𝐶₅‖∇𝑅‖_𝑔. (6.59) But |𝜕₁₁𝑓(𝑃₁)|≤‖𝑓‖_𝐶2(𝑀), |𝜕_𝑠𝑓(𝑃₁)|≤‖𝑓‖_𝐶2(𝑀) and |𝜕_𝑠𝜑|≤𝐶₅ for all 𝑠 then −𝑡𝜕₁₁𝑓−𝑡^2𝑚_𝑠=1𝜕_𝑠𝑓𝜕_𝑠𝜑≤𝑓_𝐶2(𝑀)1+2𝑚𝐶₅. (6.60) Besides ^2𝑚_𝑖=1𝜕𝐹𝜕𝑟_𝑖𝑖𝜑=^𝑚_𝑖=1𝜕𝐹𝜕𝑟_𝑖𝑖𝜑+𝜕𝐹𝜕𝑟_{(𝑖+𝑚)(𝑖+𝑚)}𝜑=12^𝑚_𝑖=1𝜎_{𝑘−1,𝑖}𝜆𝑃₁𝜎_𝑘𝜆𝑃₁. (6.61) Consequently, we obtain 𝑓_𝐶2(𝑀)1+2𝑚𝐶₅≥𝜕𝐹𝜕𝑟₁₁𝜑𝛽₁²+23^2𝑚_𝑖=1𝜕𝐹𝜕𝑟_𝑖𝑖𝜑𝑅_1𝑖1𝑖𝛽₁−2𝛽_𝑖+16^𝑚_𝑖=1𝜎_{𝑘−1,𝑖}𝜆𝑃₁𝜎_𝑘𝜆𝑃₁×𝑅_1𝑖𝑖1+𝑅_{1(𝑖+𝑚)(𝑖+𝑚)1}𝛽_𝑖+𝛽_𝑖+𝑚+12^𝑚_𝑖=1𝜎_{𝑘−1,𝑖}𝜆𝑃₁𝜎_𝑘𝜆𝑃₁×−43𝑚²𝐶₅²‖𝑅‖_𝑔−73𝑚𝐶₅‖∇𝑅‖_𝑔. (6.62) Let us now estimate |𝛽_𝑖| 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 (∇²𝜑_𝑡)_𝑃(𝜉,𝜉)≤𝛽₁+(1/2)(𝐶₅)²; then at 𝑃 in a holomorphic 𝑔-normal 𝑔-adapted chart 𝜓_𝑃, namely, a chart such that [𝑔_𝑖𝑗(𝑃)]_{1≤𝑖,𝑗≤𝑚}=𝐼_𝑚, 𝜕_ℓ𝑔_𝑖𝑗(𝑃)=0 and [𝑔_𝑖𝑗(𝑃)]_{1≤𝑖,𝑗≤𝑚}=diag(𝜆₁(𝑃),…,𝜆_𝑚(𝑃)), we deduce that for all 𝑗∈{1,…,𝑚}𝜕_{𝑥𝑗𝑥𝑗}𝜑_𝑡(𝑃)=2∇²𝜑_𝑡𝑃𝜕_𝑥𝑗√2,𝜕_𝑥𝑗√2≤2𝛽₁+𝐶₅²,𝜕_{𝑦𝑗𝑦𝑗}𝜑_𝑡(𝑃)≤2𝛽₁+𝐶₅². (6.63) Since 𝜆_𝑗(𝑃)≥−(𝑚−1)𝐶^′₂, we infer the following inequalities: ∀𝑗∈{1,…,𝑚}𝜕_{𝑥𝑗𝑥𝑗}𝜑_𝑡(𝑃)≥−4(𝑚−1)𝐶^′₂+1−2𝛽₁−𝐶₅²,𝜕_{𝑦𝑗𝑦𝑗}𝜑_𝑡(𝑃)≥−4(𝑚−1)𝐶^′₂+1−2𝛽₁−𝐶₅². (6.64) Consequently ∀1≤𝑖,𝑗≤2𝑚𝜕_{𝑢𝑖𝑢𝑗}𝜑_𝑡(𝑃)≤4𝛽₁+2𝐶₅²+4(𝑚−1)𝐶^′₂+1_=∶𝐶9, (6.65) in the chart 𝜓_𝑃.

Hence we infer that ∇²𝜑_𝑡𝑃²_𝑔=14^2𝑚_𝑖,𝑗=1𝜕_{𝑢𝑖𝑢𝑗}𝜑_𝑡(𝑃)²≤𝑚²4𝛽₁+2𝐶₅²+𝐶₉²∀𝑃. (6.66) But at 𝑃₁ in the chart 𝜓₁, |(∇²𝜑_𝑡)_𝑃1|²_𝑔=∑^2𝑚_𝑖=1(𝜕_{𝑢𝑖𝑢𝑖}𝜑_𝑡(𝑃₁))²=∑^2𝑚_𝑖=1(𝛽_𝑖)²; consequently we obtain ∀1≤𝑖≤2𝑚𝛽_𝑖≤𝑚4𝛽₁+2𝐶₅²+𝐶₉. (6.67) Thus 𝑅_1𝑖1𝑖𝛽₁−2𝛽_𝑖≤𝑅_1𝑖1𝑖𝛽₁+2𝛽_𝑖≤3𝑚‖𝑅‖_𝑔4𝛽₁+2𝐶₅²+𝐶₉. (6.68) Besides 𝑅_1𝑖𝑖1+𝑅_{1(𝑖+𝑚)(𝑖+𝑚)1}𝛽_𝑖+𝛽_𝑖+𝑚≤𝑅_1𝑖𝑖1+𝑅_{1(𝑖+𝑚)(𝑖+𝑚)1}𝛽_𝑖+𝛽_𝑖+𝑚≤4𝑚‖𝑅‖_𝑔4𝛽₁+2𝐶₅²+𝐶₉. (6.69) Hence 𝑓_𝐶2(𝑀)1+2𝑚𝐶₅≥14𝜎_𝑘−1,1𝜆𝑃₁𝜎_𝑘𝜆𝑃₁𝛽₁²+^𝑚_𝑖=1𝜎_{𝑘−1,𝑖}𝜆𝑃₁𝜎_𝑘𝜆𝑃₁(−𝑚)‖𝑅‖_𝑔4𝛽₁+2𝐶₅²+𝐶₉+^𝑚_𝑖=1𝜎_{𝑘−1,𝑖}𝜆𝑃₁𝜎_𝑘𝜆𝑃₁−23𝑚‖𝑅‖_𝑔4𝛽₁+2𝐶₅²+𝐶₉+^𝑚_𝑖=1𝜎_{𝑘−1,𝑖}𝜆𝑃₁𝜎_𝑘𝜆𝑃₁−76𝑚𝐶₅‖∇𝑅‖_𝑔−23𝑚²𝐶₅²‖𝑅‖_𝑔. (6.70) Then 𝑓_𝐶2(𝑀)1+2𝑚𝐶₅≥14𝜎_𝑘−1,1𝜆𝑃₁𝜎_𝑘𝜆𝑃₁𝛽₁²+^𝑚_𝑖=1𝜎_{𝑘−1,𝑖}𝜆𝑃₁𝜎_𝑘𝜆𝑃₁−53𝑚‖𝑅‖_𝑔4𝛽₁+2𝐶₅²+𝐶₉+^𝑚_𝑖=1𝜎_{𝑘−1,𝑖}𝜆𝑃₁𝜎_𝑘𝜆𝑃₁−76𝑚𝐶₅‖∇𝑅‖_𝑔−23𝑚²𝐶₅²‖𝑅‖_𝑔. (6.71) But using the uniform ellipticity and the inequalities 𝑒^{−2‖𝑓‖∞}(^𝑚_𝑘)≤𝜎_𝑘(𝜆(𝑃))≤𝑒^{2‖𝑓‖∞}(^𝑚_𝑘), we obtain ^𝑚_𝑖=1𝜎_{𝑘−1,𝑖}𝜆𝑃₁𝜎_𝑘𝜆𝑃₁≤𝑚𝑒^{2‖𝑓‖∞}𝐹₀(^𝑚_𝑘),𝜎_𝑘−1,1𝜆𝑃₁𝜎_𝑘𝜆𝑃₁≥𝑒^{−2‖𝑓‖∞}𝐸₀(^𝑚_𝑘). (6.72)(6.73) Then at 𝑃₁ in the chart 𝜓₁, we have 10≥4𝑒^{−2‖𝑓‖∞}𝐸₀(^𝑚_𝑘)𝛽₁²+𝑚𝑒^{2‖𝑓‖∞}𝐹₀(^𝑚_𝑘)−53𝑚‖𝑅‖_𝑔4𝛽₁+2𝐶₅²+𝐶₉−𝑚𝑒^{2‖𝑓‖∞}𝐹₀(^𝑚_𝑘)76𝑚𝐶₅‖∇𝑅‖_𝑔+23𝑚²𝐶₅²‖𝑅‖_𝑔−𝑓_𝐶2(𝑀)1+2𝑚𝐶₅. (6.74) The previous inequality means that some polynomial of second order in the variable 𝛽₁ is negative: 10≥4𝑒^{−2‖𝑓‖∞}𝐸₀(^𝑚_𝑘)𝛽₁²+𝑚𝑒^{2‖𝑓‖∞}𝐹₀(^𝑚_𝑘)−203𝑚‖𝑅‖_𝑔𝛽₁−𝑚𝑒^{2‖𝑓‖∞}𝐹₀(^𝑚_𝑘)76𝑚𝐶₅‖∇𝑅‖_𝑔+23𝑚²𝐶₅²‖𝑅‖_𝑔+53𝑚‖𝑅‖_𝑔2𝐶₅²+𝐶₉−𝑓_𝐶2(𝑀)1+2𝑚𝐶₅. (6.75) Set 𝐼∶=803𝑚²𝑒^{4‖𝑓‖∞}𝐹₀𝐸₀‖𝑅‖_𝑔>0,𝐽∶=4𝑚²𝑒^{4‖𝑓‖∞}𝐹₀𝐸₀76𝐶₅‖∇𝑅‖_𝑔+23𝑚𝐶₅²‖𝑅‖_𝑔+532𝐶₅²+𝐶₉‖𝑅‖_𝑔+4(^𝑚_𝑘)𝑒^{2‖𝑓‖∞}𝐸₀𝑓_𝐶2(𝑀)1+2𝑚𝐶₅>0. (6.76) The previous inequality writes then: 𝛽₁²−𝐼𝛽₁−𝐽≤0. (6.77) The discriminant of this polynomial of second order is equal to ▵=𝐼²+4𝐽>0, which gives an upper bound for 𝛽₁.

7. A 𝐶^2,𝛽 A Priori Estimate

We infer from the 𝐶² estimate a 𝐶^2,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. Let Ω be a bounded domain (i.e., an open connected set) of ℝ^𝑛, 𝑛≥2. Let one denote by ℝ^𝑛×𝑛 the set of real 𝑛×𝑛 symmetric matrices. 𝑢∈𝐶⁴(Ω,ℝ) is a solution of 𝐺[𝑢]=𝐺𝑥,𝐷²𝑢=0onΩ, 𝐸^′ where 𝐺∈𝐶²(Ω×ℝ^𝑛×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 ∀𝑥∈Ω,∀𝜉∈ℝ^𝑛,𝜆𝜉²≤𝐺_𝑖𝑗𝑥,𝐷²𝑢(𝑥)𝜉_𝑖𝜉_𝑗≤Λ𝜉². (7.1)(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 ∀𝑥∈Ω,∀𝜁∈ℝ^𝑛×𝑛,𝐺_{𝑖𝑗,𝑘𝑠}𝑥,𝐷²𝑢(𝑥)𝜁_𝑖𝑗𝜁_𝑘𝑠≤0. (7.2)Then for all Ω^′⊂⊂Ω, one has the following interior estimate: 𝐷²𝑢_𝛽;Ω′≤𝐶, (7.3) 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), 𝐺_𝑥, 𝐺_𝑟, 𝐺_𝑥𝑥 et 𝐺_𝑟𝑥. The notation 𝐺_𝑟𝑥 used here denotes the matrix 𝐺_𝑟𝑥=[𝐺_{𝑖𝑗,𝑥ℓ}]_{𝑖,𝑗,ℓ=1⋯𝑛} evaluated at (𝑥,𝐷²𝑢(𝑥)). It is the same for the notations 𝐺_𝑥, 𝐺_𝑟, and 𝐺_𝑥𝑥 [18, page 457].

7.1. The Evans-Trudinger Method

Let us suppose that there exists a constant 𝐶₁₁>0 such that for all 𝑖∈ℕ, we have ‖𝜑_𝑡𝑖‖_{𝐶2(𝑀,ℝ)}≤𝐶₁₁. In the following, we remove the index 𝑖 from 𝜑_𝑡𝑖 to lighten the notations. In order to construct a 𝐶^2,𝛽 estimate with 0<𝛽<1, we prepare the framework of application of Theorem 7.1.

Let ℛ=(𝑈_𝑗,𝜙_𝑗)_{1≤𝑗≤𝑁} be a finite covering of the compact manifold 𝑀 by charts, and let 𝒫=(𝜃_𝑗)_{1≤𝑗≤𝑁} be a partition of unity of class 𝐶^∞ subordinate to this covering. The family of continuity equations writes in the chart (𝑈_𝑠,𝜙_𝑠) where 1≤𝑠≤𝑁 is a fixed integer as follows: 𝐹_𝑘𝛿^𝑗_𝑖+𝑔^𝑗ℓ∘𝜙_𝑠⁻¹(𝑥)𝜕𝜑_𝑡∘𝜙_𝑠⁻¹𝜕𝑧_𝑖𝜕𝑧_ℓ(𝑥)_{1≤𝑖,𝑗≤𝑚}−𝑡𝑓∘𝜙_𝑠⁻¹(𝑥)−ln𝐴_𝑡=0𝑥∈𝜙_𝑠𝑈_𝑠⊂ℝ^2𝑚. 𝐸^′_𝑘,𝑡 Besides, we have 𝜕/𝜕𝑧_𝑎𝜕𝑧_𝑏=(1/4)(𝐷_𝑎𝑏+𝐷_{(𝑎+𝑚)(𝑏+𝑚)}+𝑖𝐷_{𝑎(𝑏+𝑚)}−𝑖𝐷_{(𝑎+𝑚)𝑏}) where the 𝐷_𝑎𝑏s denotes real derivatives; thus our equation writes: 𝐺𝑥,𝐷²𝜑_𝑡∘𝜙_𝑠⁻¹=0𝑥∈𝜙_𝑠𝑈_𝑠⊂ℝ^2𝑚with,𝐺(𝑥,𝑟)=𝐹_𝑘𝛿^𝑗_𝑖+14𝑔^𝑗ℓ𝜙_𝑠⁻¹(𝑥)𝑟_𝑖ℓ+𝑟_{(𝑖+𝑚)(ℓ+𝑚)}+𝑖𝑟_{𝑖(ℓ+𝑚)}−𝑖𝑟_{(𝑖+𝑚)ℓ1≤𝑖,𝑗≤𝑚}−𝑡𝑓∘𝜙_𝑠⁻¹(𝑥)−ln𝐴_𝑡. 𝐸^′′_𝑘,𝑡(7.4) 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 𝜙_𝑠(𝑈_𝑠), 𝐺(𝑥,⋅) is concave on 𝜌_𝜙⁻¹_{𝑠−1(𝑥)}(𝜆⁻¹(Γ_𝑘))⊂𝑆_2𝑚(ℝ)). For all 𝑠∈{1,…,𝑁}, let us consider Ω_𝑠 a bounded domain of ℝ^2𝑚 strictly included in 𝜙_𝑠(𝑈_𝑠): Ω_𝑠⊂⊂𝜙_𝑠𝑈_𝑠. (7.5) The notation 𝑆^′⊂⊂𝑆 means that 𝑆^′ is strictly included in 𝑆, namely, that 𝑆^′⊂𝑆. We will explain later how these domains Ω_𝑠 are chosen. The map 𝐺 is of class 𝐶² and the solution 𝜓^𝑠_𝑡∶=𝜑_𝑡∘𝜙_𝑠⁻¹∈𝐶⁴(Ω_𝑠,ℝ) since 𝜑_𝑡∈𝐶^ℓ,𝛼(𝑀) with ℓ≥5. The equation 𝐸^′′_𝑘,𝑡 on Ω_𝑠⊂𝜙_𝑠(𝑈_𝑠) is now written in the form corresponding to the Theorem 7.1; it remains to check the hypotheses of this theorem on Ω_𝑠, namely, that (1)𝐺 is uniformly elliptic with respect to 𝜓^𝑠_𝑡=𝜑_𝑡∘𝜙_𝑠⁻¹; that is, there exist two real numbers 𝜆_𝑠,Λ_𝑠>0 such that ∀𝑥∈Ω_𝑠,∀𝜉∈ℝ^2𝑚,𝜆_𝑠𝜉²≤𝐺_𝑖𝑗𝑥,𝐷²𝜓^𝑠_𝑡(𝑥)𝜉_𝑖𝜉_𝑗≤Λ_𝑠𝜉². (7.6) Moreover, we will impose ourselves to find real numbers 𝜆_𝑠,Λ_𝑠 independent of 𝑡. (2)𝐺 is concave with respect to 𝜓^𝑠_𝑡 in the variable 𝑟. Since 𝐺 is of class 𝐶², this concavity condition is equivalent to ∀𝑥∈Ω_𝑠,∀𝜁∈ℝ^2𝑚×2𝑚,𝐺_{𝑖𝑗,𝑘ℓ}𝑥,𝐷²𝜓^𝑠_𝑡(𝑥)𝜁_𝑖𝑗𝜁_𝑘ℓ≤0. (7.7) This has been checked before. (3)The derivatives 𝐺_𝑥, 𝐺_𝑟, 𝐺_𝑥𝑥, and 𝐺_𝑟𝑥 are controlled (these quantities are evaluated at (𝑥,𝐷²(𝜓^𝑠_𝑡)(𝑥))). Once these three points checked, and since we have a 𝐶² estimate of 𝜑_𝑡 by 𝐶₁₁, Theorem 7.1 allows us to deduce that for all open set Ω^′_𝑠⊂⊂Ω_𝑠 there exist two real numbers 𝛽_𝑠∈]0,1] and 𝐶𝑠𝑡𝑒_𝑠>0 depending only on 𝑚, 𝜆_𝑠, Λ_𝑠, dist(Ω^′_𝑠,𝜕Ω_𝑠), on the uniform estimate of |𝜓^𝑠_𝑡|_2;Ω′𝑠, and on the uniform estimates of the quantities 𝐺_𝑥, 𝐺_𝑟, 𝐺_𝑥𝑥, and 𝐺_𝑟𝑥, so in particular 𝛽_𝑠 and 𝐶𝑠𝑡𝑒_𝑠 are independent of 𝑡, such that 𝐷²𝜓^𝑠_{𝑡𝛽𝑠;Ω′𝑠}≤𝐶𝑠𝑡𝑒_𝑠. (7.8)

The Choice of Ω_𝑠 and Ω^′_𝑠
Let us denote by 𝐾_𝑠 the support of the function 𝜃_𝑠∘𝜙_𝑠⁻¹: 𝐾_𝑠∶=supp𝜃_𝑠∘𝜙_𝑠⁻¹=𝜙_𝑠supp𝜃_𝑠⊂𝜙_𝑠𝑈_𝑠. (7.9) The set 𝐾_𝑠 is compact, and it is included in the open set 𝜙_𝑠(𝑈_𝑠) of ℝ^2𝑚, and ℝ^2𝑚 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 Ω_𝑠 and Ω^′_𝑠 such that 𝐾_𝑠⊂Ω^′_𝑠⊂⊂Ω_𝑠⊂⊂𝜙_𝑠𝑈_𝑠. (7.10) The set Ω_𝑠 is required to be connected: for this, it suffices that 𝐾_𝑠 be connected even after restriction to a connected component in Ω_𝑠 of a point of 𝐾_𝑠; indeed, this connected component is an open set of Ω_𝑠 since Ω_𝑠 is locally connected (as an open set of ℝ^2𝑚); moreover it is bounded since Ω_𝑠 is bounded.

Application of the Theorem
Let 𝛽∶=min𝛽_𝑠; the norm ‖⋅‖_𝐶^2,𝛽 is submultiplicative; then 𝜑_𝑡𝐶^ℛ,𝒫_2,𝛽(𝑀)=^𝑁_𝑠=1𝜃_𝑠∘𝜙_𝑠⁻¹×𝜑_𝑠∘𝜙_𝑠⁻¹_{2,𝛽;Ω′𝑠}≤^𝑁_𝑠=1𝜃_𝑠∘𝜙_𝑠⁻¹_{2,𝛽;Ω′𝑠}×𝜓^𝑠_{𝑡2,𝛽;Ω′𝑠}. (7.11) But, by (7.8) we have |𝜓^𝑠_𝑡|_{2,𝛽𝑠;Ω′𝑠}=|𝜓^𝑠_𝑡|_2;Ω′𝑠+[𝐷²(𝜓^𝑠_𝑡)]_{𝛽𝑠;Ω′𝑠}≤|𝜓^𝑠_𝑡|_2;Ω′𝑠+𝐶𝑠𝑡𝑒_𝑠≤𝐶𝑠𝑡𝑒^′_𝑠 where 𝐶𝑠𝑡𝑒^′_𝑠 depends only on 𝑚, 𝜆_𝑠, Λ_𝑠, dist(Ω^′_𝑠,𝜕Ω_𝑠), 𝐶₁₁ (the constant of the 𝐶² estimate) and the uniform estimates of the quantities 𝐺_𝑥, 𝐺_𝑟, 𝐺_𝑥𝑥, and 𝐺_𝑟𝑥. We obtain consequently the needed 𝐶^2,𝛽 estimate: 𝜑_𝑡𝐶^ℛ,𝒫_2,𝛽(𝑀)≤^𝑁_𝑠=1𝜃_𝑠∘𝜙_𝑠⁻¹_{2,𝛽;Ω′𝑠}×𝐶𝑠𝑡𝑒^′_𝑠=∶𝐶₁₂. (7.12) Let us now check the hypotheses 1 and 3 above.

7.2. Uniform Ellipticity of 𝐺 on Ω_𝑠

Let 𝑥∈Ω_𝑠 and 𝜉∈ℝ^2𝑚: ^2𝑚_𝑖,𝑗=1𝐺_𝑖𝑗(𝑥,𝑟)𝜉_𝑖𝜉_𝑗=𝑑(𝐺(𝑥,⋅))_𝑟(𝑀)with𝑀=𝜉_𝑖𝜉_{𝑗1≤𝑖,𝑗≤𝑚}∈𝑆_2𝑚(ℝ)=𝑑𝐹_𝑘∘𝜌_𝜙𝑠⁻¹_(𝑥)𝑟(𝑀)=𝑑𝐹_{𝑘𝜌𝜙𝑠−1(𝑥)(𝑟)}⋅𝑑𝜌_𝜙𝑠⁻¹_(𝑥)𝑟(𝑀). (7.13) Let us recall that 𝜌_𝑃(𝑟)=[𝛿^𝑗_𝑖+(1/4)∑^𝑚_ℓ,𝑜=1(𝑔^−1/2(𝑃))_𝑖ℓ(𝑔^−1/2(𝑃))_𝑜𝑗(𝑟_ℓ𝑜+𝑟_{(ℓ+𝑚)(𝑜+𝑚)}+𝑖𝑟_{ℓ(𝑜+𝑚)}−𝑖𝑟_{(ℓ+𝑚)𝑜})]_{1≤𝑖,𝑗≤𝑚} (cf. (6.36)); we consequently obtain ^2𝑚_𝑖,𝑗=1𝐺_𝑖𝑗𝑥,𝐷²𝜓^𝑠_𝑡(𝑥)𝜉_𝑖𝜉_𝑗=𝑑𝐹_{𝑘𝜌𝜙𝑠−1(𝑥)(𝐷2(𝜓𝑠𝑡)(𝑥))}⋅14^𝑚_ℓ,𝑜=1(𝑔^−1/2(𝜙_𝑠⁻¹(𝑥)))_𝑖ℓ𝑔^−1/2𝜙_𝑠⁻¹(𝑥)_𝑜𝑗×𝑀_ℓ𝑜+𝑀_{(ℓ+𝑚)(𝑜+𝑚)}+𝑖𝑀_{ℓ(𝑜+𝑚)}−𝑖𝑀_{(ℓ+𝑚)𝑜1≤𝑖,𝑗≤𝑚}. (7.14) In the following, we denote 𝑀∶=[(1/4)(𝑀_ℓ𝑠+𝑀_{(ℓ+𝑚)(𝑠+𝑚)}+𝑖𝑀_{ℓ(𝑠+𝑚)}−𝑖𝑀_{(ℓ+𝑚)𝑠})]_{1≤ℓ,𝑠≤𝑚}. Thus 1𝑀=4𝜉_ℓ𝜉_𝑠+𝜉_ℓ+𝑚𝜉_𝑠+𝑚+𝑖𝜉_ℓ𝜉_𝑠+𝑚−𝑖𝜉_ℓ+𝑚𝜉_{𝑠1≤ℓ,𝑠≤𝑚}∈ℋ_𝑚(ℂ)=14𝜉_ℓ−𝑖𝜉_ℓ+𝑚𝜉_𝑠+𝑖𝜉_{𝑠+𝑚̃𝜉=∶𝑠1≤ℓ,𝑠≤𝑚}=14𝜉̃_ℓ𝜉̃_{𝑠1≤ℓ,𝑠≤𝑚}. (7.15) Besides, let us denote 𝑑_𝑖=𝜎_{𝑘−1,𝑖}[𝜆(𝑔⁻¹𝑔_𝜑𝑡(𝜙_𝑠⁻¹(𝑥)))]/𝜎_𝑘[𝜆(𝑔⁻¹𝑔_𝜑𝑡(𝜙_𝑠⁻¹(𝑥)))] and 𝑔^−1/2 instead of 𝑔^−1/2(𝜙_𝑠⁻¹(𝑥)) in order to lighten the formulas. We obtain by the invariance formula (2.7) that ^2𝑚_𝑖,𝑗=1𝐺_𝑖𝑗𝑥,𝐷²𝜓^𝑠_𝑡(𝑥)𝜉_𝑖𝜉_𝑗=𝑑𝐹_{𝑘[𝑔]−1/2𝑔𝜑𝑡[𝑔]−1/2}⋅𝑔^−1/2𝑀𝑔^−1/2=𝑑𝐹_{𝑘diag(𝜆1,…,𝜆𝑚)}⋅^𝑡𝑈𝑔^−1/2𝑀𝑔^−1/2𝑈where𝑈∈𝑈_𝑚(ℂ)with_𝑡𝑈𝑔^−1/2𝑔_𝜑𝑡𝑔^−1/2𝑈=diag𝜆₁,…,𝜆_𝑚weareatthepoint𝜙_𝑠⁻¹(𝑥)=^𝑚_𝑖=1𝑑_𝑖^𝑡𝑈𝑔^−1/2𝑀𝑔^−1/2𝑈_𝑖𝑖=^𝑚_𝑖=1𝑑_𝑖^𝑡𝑔^−1/2𝑈𝑀𝑔^−1/2𝑈_𝑖𝑖=^𝑚_{𝑖,ℓ,𝑗=1}𝑑_𝑖𝑔^−1/2𝑈_ℓ𝑖𝑀_ℓ𝑗𝑔^−1/2𝑈_𝑗𝑖=^𝑚_{𝑖,ℓ,𝑗=1}𝑑_𝑖𝑔^−1/2𝑈_ℓ𝑖14𝜉̃_ℓ𝜉̃_𝑗𝑔^−1/2𝑈_𝑗𝑖=14^𝑚_𝑖=1𝑑_𝑖^𝑚_𝑗=1𝜉̃_𝑗𝑔^−1/2𝑈_{𝑗𝑖=∶𝛼𝑖}^𝑚_ℓ=1𝜉̃_ℓ𝑔^−1/2𝑈_{ℓ𝑖=𝛼𝑖}=14^𝑚_𝑖=1𝑑_𝑖𝛼_𝑖². (7.16) But by Proposition 6.10 and the inequalities 𝑒^{−2‖𝑓‖∞}(^𝑚_𝑘)≤𝜎_𝑘(𝜆(𝑔⁻¹𝑔_𝜑𝑡(𝑃)))≤𝑒^{2‖𝑓‖∞}(^𝑚_𝑘), we have for (6.72) 𝑒^{−2‖𝑓‖∞}𝐸₀(^𝑚_𝑘)≤𝑑_𝑖≤𝑒^{2‖𝑓‖∞}𝐹₀(^𝑚_𝑘). (7.17) Combining (7.16) and (7.17), we obtain 14𝑒^{−2‖𝑓‖∞}𝐸₀(^𝑚_𝑘)^𝑚_𝑖=1𝛼_𝑖²≤^2𝑚_𝑖,𝑗=1𝐺_𝑖𝑗𝑥,𝐷²𝜓^𝑠_𝑡(𝑥)𝜉_𝑖𝜉_𝑗≤14𝑒^{2‖𝑓‖∞}𝐹₀(^𝑚_𝑘)^𝑚_𝑖=1𝛼_𝑖². (7.18) But ^𝑚_𝑖=1𝛼_𝑖²=^𝑚_𝑖=1^𝑚_𝑗=1𝜉̃_𝑗𝑔^−1/2𝑈_𝑗𝑖²=^𝑚_𝑖=1^𝑚_𝑗=1𝜉̃_𝑗𝑔^−1/2𝑈_𝑗𝑖^𝑚_ℓ=1𝜉̃_ℓ𝑔^−1/2𝑈_ℓ𝑖=^𝑚_𝑗,ℓ=1^𝑚_𝑖=1𝑔^−1/2𝑈_𝑗𝑖𝑔^−1/2𝑈_ℓ𝑖𝜉̃_𝑗𝜉̃_ℓ=^𝑚_𝑗,ℓ=1𝑔^−1/2𝑈×^𝑡[𝑔]^−1/2𝑈_𝑗ℓ𝜉̃_𝑗𝜉̃_ℓ. (7.19) And ([𝑔]^−1/2𝑈)×^𝑡([𝑔]^−1/2𝑈)=[𝑔]^−1/2𝑈^𝑡𝑈^𝑡[𝑔]^−1/2=[𝑔]^𝑡−1/2[𝑔]^−1/2=[𝑔]^−1/2[𝑔]^−1/2=[𝑔]⁻¹; then ^𝑚_𝑖=1𝛼_𝑖²=^𝑚_𝑗,ℓ=1𝑔⁻¹_𝑗ℓ𝜉̃_𝑗𝜉̃_ℓ=^𝑚_𝑗,ℓ=1𝑔^ℓ𝑗𝜙_𝑠⁻¹(𝑥)𝜉̃_𝑗𝜉̃_ℓ. (7.20) Consequently, and since |𝜉|²=|𝜉̃|², the checking of the hypothesis of uniform ellipticity of the Theorem 7.1 is reduced to find two real numbers 𝜆^𝑜_𝑠,Λ^𝑜_𝑠>0 such that ∀𝑥∈Ω_𝑠,∀𝜉̃∈ℂ^𝑚,𝜆^𝑜_𝑠𝜉̃²≤^𝑚_𝑗,ℓ=1𝑔^ℓ𝑗𝜙_𝑠⁻¹(𝑥)𝜉̃_ℓ𝜉̃_𝑗≤Λ^𝑜_𝑠𝜉̃². (7.21) By the min-max principle applied on ℂ^𝑚 to the Hermitian form ≺𝑋,𝑌≻_{𝑔(𝜙𝑠}⁻¹_(𝑥)) =𝑔^𝑎𝑏(𝜙_𝑠⁻¹(𝑥))𝑋_𝑎𝑌_𝑏 relatively to the canonical one, we have 𝜆_min𝑔^𝑎𝑏𝜙_𝑠⁻¹(𝑥)_{1≤𝑎,𝑏≤𝑚}𝜉̃²≤^𝑚_𝑎,𝑏=1𝑔^𝑎𝑏𝜙_𝑠⁻¹(𝑥)𝜉̃_𝑎𝜉̃_𝑏≤𝜆_max𝑔^𝑎𝑏𝜙_𝑠⁻¹(𝑥)_{1≤𝑎,𝑏≤𝑚}𝜉̃². (7.22) But the functions 𝑃↦𝜆_min[𝑔^𝑎𝑏(𝑃)]_{1≤𝑎,𝑏≤𝑚} and 𝑃↦𝜆_max[𝑔^𝑎𝑏(𝑃)]_{1≤𝑎,𝑏≤𝑚} are continuous on 𝜙_𝑠⁻¹(Ω_𝑠)⊂𝑈_𝑠 which is compact since it is a closed set of the compact manifold 𝑀 (cf. (7.5) for the choice of the domains Ω_𝑠), so these functions are bounded and reach their bounds; thus min_{𝑃∈𝜙𝑠−1Ω𝑠}𝜆_min𝑔^𝑎𝑏(𝑃)_{1≤𝑎,𝑏≤𝑚=∶𝜆𝑜𝑠}×𝜉̃²≤^𝑚_𝑎,𝑏=1𝑔^𝑎𝑏𝜙_𝑠⁻¹(𝑥)𝜉̃_𝑎𝜉̃_𝑏≤max_{𝑃∈𝜙𝑠−1Ω𝑠}𝜆_max𝑔^𝑎𝑏(𝑃)_{1≤𝑎,𝑏≤𝑚=∶Λ𝑜𝑠}×𝜉̃². (7.23) By the inequalities (7.18) and (7.23), we deduce that 𝜆_𝑠𝜉̃²≤^2𝑚_𝑖,𝑗=1𝐺_𝑖𝑗𝑥,𝐷²𝜓^𝑠_𝑡(𝑥)𝜉_𝑖𝜉_𝑗≤Λ_𝑠𝜉̃²with𝜆_𝑠1∶=4𝑒^{−2‖𝑓‖∞}𝐸₀(^𝑚_𝑘)𝜆^𝑜_𝑠,Λ_𝑠1∶=4𝑒^{2‖𝑓‖∞}𝐹₀(^𝑚_𝑘)Λ^𝑜_𝑠. (7.24) The real numbers 𝜆_𝑠 and Λ_𝑠 depend on 𝑘, 𝑚, ‖𝑓‖_∞, 𝐸₀, 𝐹₀, 𝑔, (𝑈_𝑠,𝜙_𝑠), and Ω_𝑠 and are independent of 𝑡, 𝑥 and 𝜉̃, which achieves the proof of the global uniform ellipticity.

7.3. Uniform Estimate of 𝐺_𝑥, 𝐺_𝑟, 𝐺_𝑥𝑥, and 𝐺_𝑟𝑥

In this subsection, we estimate uniformly the quantities 𝐺_𝑥, 𝐺_𝑟,𝐺_𝑥𝑥, and 𝐺_𝑟𝑥 (recall that these quantities are evaluated at (𝑥,𝐷²(𝜓^𝑠_𝑡)(𝑥))) by using the same technique as in the previous subsection for the proof of uniform ellipticity (7.24).

We have 𝐺_𝑥²=𝐺_{𝑥𝑖1≤𝑖≤2𝑚}²=^2𝑚_𝑖=1𝐺_𝑥𝑖²where𝐺_𝑥𝑖=𝜕𝐺𝜕𝑥_𝑖𝑥,𝐷²𝜓^𝑠_𝑡(𝑥). (7.25) For (7.14), we obtain 𝐺_𝑥𝑖=𝑑𝐹_{𝑘[𝑔−1𝑔𝜑𝑡(𝜙𝑠−1(𝑥))]}⋅^𝑚_ℓ=1𝜕𝑔^𝑞ℓ∘𝜙_𝑠⁻¹𝜕𝑥_𝑖(𝑥)𝜕_𝑜ℓ𝜑_𝑡𝜙_𝑠⁻¹(𝑥)_{1≤𝑜,𝑞≤𝑚=∶𝑀𝑜}𝜕−𝑡𝑓∘𝜙_𝑠⁻¹𝜕𝑥_𝑖(𝑥) (7.26) and for (7.16), we infer then by the invariance formula (2.7) that 𝐺_𝑥𝑖=^𝑚_𝑗=1𝑑_𝑗^𝑡𝑈𝑀^𝑜𝑈_𝑗𝑗−𝑡𝜕𝑓𝜕𝑥^𝑖𝜙_𝑠⁻¹(𝑥), (7.27) where 𝑈∈𝑈_𝑚(ℂ) such that (^𝑡𝑈[𝑔⁻¹𝑔_𝜑𝑡(𝜙_𝑠⁻¹(𝑥))]𝑈=diag(𝜆₁,…,𝜆_𝑚) and 𝑑_𝑖=𝜎_{𝑘−1,𝑖}[𝜆(𝑔⁻¹𝑔_𝜑𝑡(𝜙_𝑠⁻¹(𝑥)))]/𝜎_𝑘[𝜆(𝑔⁻¹𝑔_𝜑𝑡(𝜙_𝑠⁻¹(𝑥)))]. We can then write: 𝐺_𝑥𝑖=^𝑚_{𝑗,𝑝,𝑞=1}𝑑_𝑗𝑈_𝑝𝑗𝑈_𝑞𝑗𝑀^𝑜_𝑝𝑞−𝑡𝜕𝑓𝜕𝑥^𝑖𝜙_𝑠⁻¹(𝑥)=^𝑚_{𝑗,𝑝,𝑞=1}𝑑_𝑗𝑈_𝑝𝑗𝑈_𝑞𝑗^𝑚_ℓ=1𝜕𝑔^𝑞ℓ𝜕𝑥^𝑖𝜙_𝑠⁻¹(𝑥)𝜕_𝑝ℓ𝜑_𝑡𝜙_𝑠⁻¹(𝑥)−𝑡𝜕𝑓𝜕𝑥^𝑖𝜙_𝑠⁻¹(𝑥).Thus𝐺_𝑥𝑖≤^𝑚_{𝑗,𝑝,𝑞,ℓ=1}𝑒^{2‖𝑓‖∞}𝐹₀(^𝑚_𝑘)𝑈_𝑝𝑗𝑈_𝑞𝑗×max_{1≤𝑎,𝑏≤𝑚,1≤𝑖≤2𝑚}max_{𝑃∈𝜙𝑠−1(Ω𝑠)}𝜕𝑔^𝑎𝑏𝜕𝑥^𝑖(𝑃)_=∶Λ1𝑠𝜑_{𝑡𝐶2(𝑀,ℝ)}+𝑓_{𝐶1(𝑀,ℝ)}. (7.28) But 𝑈∈𝑈_𝑚(ℂ); then |𝑈_𝑞𝑗|≤1 for all 1≤𝑞, 𝑗≤𝑚, consequently 𝐺_𝑥𝑖≤𝑚⁴𝑒^{2‖𝑓‖∞}𝐹₀(^𝑚_𝑘)Λ¹_𝑠𝜑_{𝑡𝐶2(𝑀,ℝ)≤𝐶11𝐶2estimate}+𝑓_{𝐶1(𝑀,ℝ)}, (7.29) which gives the needed uniform estimate for 𝐺_𝑥: 𝐺_𝑥≤√2𝑚𝑚⁴𝑒^{2‖𝑓‖∞}𝐹₀(^𝑚_𝑘)Λ¹_𝑠𝐶₁₁+𝑓_{𝐶1(𝑀,ℝ)}. (7.30) Similarly 𝐺_𝑟²=𝐺_{𝑝𝑞1≤𝑝,𝑞≤2𝑚}²=^2𝑚_𝑝,𝑞=1𝐺_𝑝𝑞²,where𝐺_𝑝𝑞=𝜕𝐺𝜕𝑟_𝑝𝑞𝑥,𝐷²𝜓^𝑠_𝑡(𝑥). (7.31) And we have 𝐺_𝑝𝑞=𝑑𝐹_{𝑘[𝑔−1𝑔𝜑𝑡(𝜙𝑠−1(𝑥))]}⋅^𝑚_ℓ=1𝑔^𝑗ℓ𝜙_𝑠⁻¹(𝑥)𝐸_{𝑝𝑞𝑖ℓ1≤𝑖,𝑗≤𝑚=∶𝑀1}, (7.32) where 𝐸_𝑝𝑞 is the 𝑚×𝑚 matrix whose all coefficients are equal to zero except the coefficient 𝑝𝑞 which is equal to 1, and the matrix (𝐸_𝑝𝑞) is obtained from 𝐸_𝑝𝑞 by the formula 𝑀∶=[(1/4)(𝑀_ℓ𝑠+𝑀_{(ℓ+𝑚)(𝑠+𝑚)}+𝑖𝑀_{ℓ(𝑠+𝑚)}−𝑖𝑀_{(ℓ+𝑚)𝑠})]_{1≤ℓ,𝑠≤𝑚}, thus 𝐺_𝑝𝑞=^𝑚_𝑗=1𝑑_𝑗(^𝑡𝑈𝑀¹𝑈)_𝑗𝑗, (7.33) where  𝑈  and  𝑑_𝑖  are as before for  𝐺_𝑥.

Since |(𝐸_𝑝𝑞)_𝑖ℓ|≤1 for all 1≤𝑖,ℓ≤𝑚, we obtain for 𝐺_𝑥 that 𝐺_𝑝𝑞≤𝑚⁴𝑒^{2‖𝑓‖∞}𝐹₀(^𝑚_𝑘)Λ²_𝑠, (7.34) where Λ²_𝑠=max_{1≤𝑎,𝑏≤𝑚}max_{𝑃∈𝜙𝑠−1(Ω𝑠)}|𝑔^𝑎𝑏(𝑃)|, which gives the needed uniform estimate for 𝐺_𝑟: 𝐺_𝑟≤2𝑚⁵𝑒^{2‖𝑓‖∞}𝐹₀(^𝑚_𝑘)Λ²_𝑠. (7.35) Concerning 𝐺_𝑥𝑥, we have 𝐺_𝑥𝑥²=𝐺_{𝑥𝑝𝑥𝑞1≤𝑝,𝑞≤2𝑚}²=^2𝑚_𝑝,𝑞=1𝐺_{𝑥𝑝𝑥𝑞}²,where𝐺_{𝑥𝑝𝑥𝑞}=𝜕²𝐺𝜕𝑥_𝑝𝜕𝑥_𝑞𝑥,𝐷²𝜓^𝑠_𝑡(𝑥). (7.36) A calculation shows that 𝐺_{𝑥𝑝𝑥𝑞}𝜕=−𝑡²𝑓𝜕𝑥^𝑝𝜕𝑥^𝑞𝜙_𝑠⁻¹(𝑥)+^𝑚_{𝑖,𝑗,ℓ=1}𝜕𝐹_𝑘𝜕𝐵^𝑗_𝑖𝑔⁻¹𝑔_𝜑𝑡𝜙_𝑠⁻¹(𝑥)𝜕²𝑔^𝑗ℓ𝜕𝑥^𝑝𝜕𝑥^𝑞𝜙_𝑠⁻¹(𝑥)𝜕_𝑖ℓ𝜑_𝑡𝜙_𝑠⁻¹(𝑥)+^𝑚_{𝑖,𝑗,ℓ,𝜇,𝑜,𝜈=1}𝜕²𝐹_𝑘𝜕𝐵^𝑜_𝜇𝜕𝐵^𝑗_𝑖𝑔⁻¹𝑔_𝜑𝑡𝜙_𝑠⁻¹(𝑥)_=∶ℰ×𝜕𝑔^𝑜𝜈𝜕𝑥^𝑝𝜙_𝑠⁻¹(𝑥)𝜕𝑔^𝑗ℓ𝜕𝑥^𝑞𝜙_𝑠⁻¹(𝑥)𝜕_𝜇𝜈𝜑_𝑡𝜙_𝑠⁻¹(𝑥)𝜕_𝑖ℓ𝜑_𝑡𝜙_𝑠⁻¹(𝑥). (7.37) All the terms are uniformly bounded; it remains to justify that the term in second derivative ℰ is also uniformly bounded: ℰ=𝑑²𝐹_{𝑘[𝑔−1𝑔𝜑𝑡(𝜙𝑠−1(𝑥))]}⋅𝐸_𝜇𝑜,𝐸_𝑖𝑗()=thenbytheinvarianceformula2.7_{𝑚𝑎,𝑏,𝑐,𝑑=1}𝜕²𝐹_𝑘𝜕𝐵^𝑏_𝑎𝜕𝐵^𝑑_𝑐diag𝜆₁,…,𝜆_𝑚(^𝑡𝑈𝐸_𝜇𝑜𝑈)_𝑎𝑏(^𝑡𝑈𝐸_𝑖𝑗𝑈)_𝑐𝑑,where𝑈∈𝑈_𝑚(ℂ)islikebefore. (7.38) But we know the second derivatives of 𝐹_𝑘 at a diagonal matrix by (2.5). Besides, we have 0<𝜎_{𝑘−1,𝑖}(𝜆)/𝜎_𝑘(𝜆)=𝑑_𝑖≤𝑒^{2‖𝑓‖∞}𝐹₀/(^𝑚_𝑘) by (7.17), and since 𝑒^{−2‖𝑓‖∞}(^𝑚_𝑘)≤𝜎_𝑘(𝜆), it remains only to control the quantities |𝜎_{𝑘−2,𝑖𝑗}(𝜆)| with 𝑖≠𝑗 to prove that ℰ is uniformly bounded. But since 𝜆∈Γ_𝑘, we have 𝜎_{𝑘−2,𝑖𝑗}(𝜆)>0 [11]. Moreover, by the pinching of the eigenvalues, we deduce automatically that 𝜎_{𝑘−2,𝑖𝑗}(𝜆)≤𝐶𝑚−2𝑘−2^′2𝑘−1=∶𝐹₁, (7.39) which achieves the checking of the fact that 𝐺_𝑥𝑥 is uniformly bounded.

Similarly, we establish a uniform estimate of 𝐺_𝑥𝑟 using this calculation: 𝐺_{𝑥𝑜,𝑝𝑞}=𝜕²𝐺𝜕𝑥_𝑜𝜕𝑟_𝑝𝑞𝑥,𝐷²𝜓^𝑠_𝑡𝑥)=_{𝑚𝑖,𝑗,ℓ=1}(𝜕𝐹_𝑘𝜕𝐵^𝑗_𝑖𝑔⁻¹𝑔_𝜑𝑡𝜙_𝑠⁻¹(𝑥)𝜕𝑔^𝑗ℓ𝜕𝑥^𝑜𝜙_𝑠⁻¹(𝑥)𝐸_{𝑝𝑞𝑖ℓ}+^𝑚_{𝑖,𝑗,ℓ,𝜈,𝜇,𝛾=1}𝜕²𝐹_𝑘𝜕𝐵^𝜇_𝜈𝜕𝐵^𝑗_𝑖𝑔⁻¹𝑔_𝜑𝑡𝜙_𝑠⁻¹(𝑥)×𝜕𝑔^𝜇𝛾𝜕𝑥^𝑜𝜙_𝑠⁻¹(𝑥)𝜕_𝜈𝛾𝜑_𝑡𝜙_𝑠⁻¹(𝑥)𝑔^𝑗ℓ𝜙_𝑠⁻¹(𝑥)𝐸_{𝑝𝑞𝑖ℓ}, (7.40) which achieves the proof of the 𝐶^2,𝛽 estimate.

Acknowledgment

This work was accomplished when the author was a Ph.D. student at the Nice Sophia-Antipolis University in France.