Fu-Yau Hessian Equations
aa r X i v : . [ m a t h . DG ] J a n FU-YAU HESSIAN EQUATIONS
Duong H. Phong, Sebastien Picard, and Xiangwen Zhang
Abstract
We solve the Fu-Yau equation for arbitrary dimension and arbitrary slope α ′ .Actually we obtain at the same time a solution of the open case α ′ >
0, an improvedsolution of the known case α ′ <
0, and solutions for a family of Hessian equationswhich includes the Fu-Yau equation as a special case. The method is based on theintroduction of a more stringent ellipticity condition than the usual Γ k admissiblecone condition, and which can be shown to be preserved by precise estimates withscale. The main goal of this paper is to solve the following non-linear partial differential equationproposed in 2008 by J.X. Fu and S.T. Yau [10], i∂ ¯ ∂ ( e u ˆ ω − α ′ e − u ρ ) ∧ ˆ ω n − + α ′ i∂ ¯ ∂u ∧ i∂ ¯ ∂u ∧ ˆ ω n − + µ ˆ ω n = 0 . (1.1)Here the unknown is a scalar function u on a compact n -dimensional K¨ahler manifold( X, ˆ ω ), and the given data is a real (1 ,
1) form ρ , a function µ , and a number α ′ ∈ R calledthe slope. A key innovation in the solution is the introduction of an ellipticity conditionwhich is more restrictive than the usual cone conditions for fully non-linear second orderpartial differential equations, but which can be shown to be preserved by the continuitymethod using some precise estimates with scale. This innovation may be useful for otherequations as well, and we shall illustrate this by using it to solve a whole family of Hessianequations in which the equation (1.1) fits as only the simplest example.The equation (1.1) is a generalization of an equation in complex dimension 2, which wasshown in [10] to arise from the Hull-Strominger system [17, 18, 27]. The Hull-Stromingersystem is an extension of a proposal of Candelas, Horowitz, Strominger, and Witten [5]for supersymmetric compactifications of the heterotic string. It poses new geometric diffi-culties as it involves quadratic expressions in the curvature tensor, but it can potentiallylead to a new notion of canonical metric in non-K¨ahler geometry. From our point of view,the equation (1.1) is of particular interest as a model equation for an eventual extensionof the classical theory of Monge-Amp`ere equations of Yau [32] and Hessian equations ofCaffarelli, Nirenberg, and Spruck [4], to more general equations mixing the unknown, itsgradient, and several Hessians.When the dimension of X is n = 2, the equation (1.1) was solved by Fu and Yau in twoseparate papers, [10] for the case when α ′ >
0, and [11] for the case when α ′ < ′ = 0, the equation poses no difficulty as it reduces essentially to the Laplacian). As weshall discuss below, in the approach of [10, 11], the required estimates in the two cases α ′ > α ′ < n when α ′ <
0. However, the case α ′ > n remained open, as a key lower bound for the Hessian could not be established [19]. Inthis paper, we shall simultaneously solve the open case α ′ > n ,improve on the solution found in [21] for the case α ′ <
0, and do it actually for moregeneral equations where the factor ( i∂ ¯ ∂u ) in (1.1) is replaced by higher powers of i∂ ¯ ∂u .More precisely, let ( X, ˆ ω ), ρ , µ , α ′ be as above. For each fixed integer k , 1 ≤ k ≤ n − γ >
0, we consider the equation i∂ ¯ ∂ n e ku ˆ ω − α ′ e ( k − γ ) u ρ o ∧ ˆ ω n − + α ′ ( i∂ ¯ ∂u ) k +1 ∧ ˆ ω n − k − + µ ˆ ω n = 0 . (1.2)Clearly, when k = 1 and γ = 2, this equation reduces to the Fu-Yau equation (1.1). Weshall refer to (1.2) as Fu-Yau Hessian equations. Our main result is then the following: Theorem 1
Let α ′ ∈ R , ρ ∈ Ω , ( X, R ) , and µ : X → R be a smooth function such that R X µ ˆ ω n = 0 . Define the set Υ k by Υ k = n u ∈ C ( X, R ) : e − γu < δ, | α ′ || e − u i∂ ¯ ∂u | k ˆ ω < τ o , (1.3) where < δ, τ ≪ are explicit fixed constants depending only on ( X, ˆ ω ) , α ′ , ρ, µ, n, k, γ ,whose expressions are given in (2.6, 2.7) below. Then there exists M ≫ depending on ( X, ˆ ω ) , α ′ , n , k , γ , µ and ρ , such that for each M ≥ M , there exists a unique smoothfunction u ∈ Υ k with normalization R X e u ˆ ω n = M solving the Fu-Yau Hessian equation (1.2) . We outline now the key differences between the earlier approaches and the approachof the present paper.The earlier approaches [10, 11, 19, 20] were based on rewriting the equation (1.1) asˆ σ ( ω ′ ) = n ( n − e u − α ′ e u |∇ u | ) + ν (1.4)where ν is a linear combination of known functions, u and ∇ u , ω ′ is defined by ω ′ = e u ˆ ω + α ′ e − u ρ + 2 nα ′ i∂ ¯ ∂u , and ˆ σ k ( ω ′ ) is the k -th symmetric function of the eigenvaluesof ω ′ with respect to ˆ ω . We look then for solutions u satisfying the condition ω ′ ∈ Γ ,where Γ is defined by the conditions ˆ σ ( ω ′ ) > σ ( ω ′ ) >
0. The left hand side isthen >
0. When α ′ >
0, this implies immediately an upper bound on |∇ u | . However, thedifficulty is then to derive a positive lower bound for ˆ σ ( ω ′ ), and the arguments of [10]worked only when n = 2. On the other hand, when α ′ <
0, such a lower bound turns out tohold because there is no cancellation in the expression e u − α ′ e u |∇ u | . The estimate for2 ∇ u | and | ˆ σ ( ω ′ ) | can then be obtained respectively by applying the techniques of Dinew-Kolodziej [8], and Chou-X.J. Wang [6], Hou-Ma-Wu [16], Guan [14], and the authors [22].The approach in the present paper relies instead on a different strategy.First, the equation (1.1) corresponds to the case k = 1, γ = 2 of the Fu-Yau Hessianequations. As stated in Theorem 1, we look for solutions u ∈ Υ , which is a more stringentcondition than ω ′ ∈ Γ . The set Υ and its condition e − u | α ′ i∂ ¯ ∂u | ˆ ω < τ are inspired by thecondition | α ′ Rm ( ω ) | << . In the method of continuity, the given equation (1.1)is realized as the end point of a family of equations for each t ∈ [0 , u ∈ Υ implies that the diffusion operator F p ¯ q ∇ p ∇ ¯ q governing the evolution of | Du | and | α ′ i∂ ¯ ∂u | is a controllable perturbation of the Laplacian ∆ = g p ¯ q ∇ p ∇ ¯ q . The main problemis then to show that, if u ∈ Υ at time t = 0, it will stay in Υ at all times.This is accomplished by establishing a priori estimates, which we shall refer to as“estimates with scale”, which are more precise and delicate than the usual ones. Indeed,a priori estimates for | u | , | Du | , and | α ′ i∂ ¯ ∂u | are usually required only to be independentof z ∈ X and t ∈ [0 , Z X e u ˆ ω n = M (1.5)sets effectively a scale M , and the estimates with scale that we need are estimates for | u | , | Du | , and | α ′ i∂ ¯ ∂u | in terms of some specific powers of M . An example of such an estimateis the C estimate stated in Theorem 3 below, C − M ≤ e u ≤ C M , which is a version inthe present context of similar C estimates established earlier in [10, 11, 20]. The hardestpart of the paper resides in the proof of similar estimates with scale for | Du | and | i∂ ¯ ∂u | ,as stated in Theorems 4 and 5. Neither the set Υ nor the estimates with scale depend onthe sign of α ′ , which is why both cases α ′ > α ′ < u ∈ Υ , which is better than a solution in Γ . A vitalclue that a strategy based on Υ and estimates with scale could work was provided by theauthors’ earlier alternative proof [21, 22] by flow methods of the Fu-Yau theorem [10, 11]in dimension n = 2.The power of the new method is even more evident when it comes to the general Fu-YauHessian equation (1.2). For k ≥
2, it is no longer possible to express the equation (1.2) interms of a single Hessian ˆ σ k +1 ( ω ′ ) for some (1 , ω ′ as in (1.4). Rather, the equationleads to a combination of several Hessians, which makes it non-concave, and prevents aderivation of C and C ,α estimates by standard techniques of concave PDE’s. On theother hand, the method of an ellipticity condition Υ k preserved by estimates with scaleworks seamlessly in all cases of 1 ≤ k ≤ n −
1. In fact the C estimates that we obtain In these flows, a Hermitian metric ω evolves with time, and Rm ( ω ) is the curvature of the Chernunitary connection of ω . The condition | α ′ Rm ( ω ) | << C estimates established in the literature for any general class ofHessian equations besides the Laplacian and the Monge-Amp`ere equations. In our study of (1.2), we will assume that Vol( X, ˆ ω ) = 1, which can be acheived byscaling ˆ ω λ ˆ ω , α ′ λ k α ′ , ρ λ − k +1 ρ , µ λ − µ . Since the equation (1.2) reducesto the Laplace equation when α ′ = 0, we assume from now on that α ′ = 0. We will usethe notation C ℓn = n ! ℓ !( n − ℓ )! and ˆ σ ℓ ( i∂ ¯ ∂u ) ˆ ω n = C ℓn ( i∂ ¯ ∂u ) ℓ ∧ ˆ ω n − ℓ . Given ρ , we define thedifferential operator L ρ acting on functions by L ρ f ˆ ω n = ni∂ ¯ ∂ ( f ρ ) ∧ ˆ ω n − . (2.1)For each fixed k ∈ { , , , . . . , n − } and a real number γ >
0, the Fu-Yau Hessianequation (1.2) can be rewritten as1 k ∆ ˆ g e ku + α ′ (cid:26) L ρ e ( k − γ ) u + ˆ σ k +1 ( i∂ ¯ ∂u ) (cid:27) = µ. (2.2)We note that we adjusted our conventions compared to the introduction by redefining µ , ρ , and α ′ up to a constant. From this point on, we only work with the present conventions(2.2). The standard Fu-Yau equation can be recovered by letting k = 1, γ = 2. We remarkthat this equation is already of interest in the case when ρ ≡
0, in which case the term L ρ e ( k − γ ) u vanishes.We can also write L ρ as L ρ = a j ¯ k ∂ j ∂ ¯ k + b i ∂ i + b ¯ i ∂ ¯ i + c, (2.3)where a j ¯ k is a Hermitian section of ( T , X ) ∗ ⊗ ( T , X ) ∗ , b i is a section of ( T , X ) ∗ , and c is a real function. All these coefficients are characterized by the following equations ni∂ ¯ ∂f ∧ ρ ∧ ˆ ω n − = a j ¯ k ∂ j ∂ ¯ k f ˆ ω n , ni∂f ∧ ¯ ∂ρ ∧ ˆ ω n − = b i ∂ i f ˆ ω n , ni∂ ¯ ∂ρ ∧ ˆ ω n − = c ˆ ω n . (2.4)for an arbitrary function f , and can be expressed explicitly in terms of ρ and ˆ ω if desired.We will use the constant Λ depending on ρ defined by − Λˆ g j ¯ k ≤ a j ¯ k ≤ Λˆ g j ¯ k , ˆ ω = ˆ g ¯ kj idz j ∧ d ¯ z k , ˆ g j ¯ k = (ˆ g ¯ kj ) − . (2.5)We will look for solutions in the regionΥ k = n u ∈ C ( X, R ) : e − γu < δ, | α ′ || e − u i∂ ¯ ∂u | k ˆ ω < τ o , τ = 2 − C kn − , (2.6)4here 0 < δ ≪ X, ˆ ω ) , α ′ , ρ, µ, k, n, γ . Moreprecisely, it suffices for δ to satisfy the inequality δ ≤ min ( , − | α ′ | ( k + γ ) Λ , (cid:18) θ C X ( k µ k ∞ + k α ′ c k ∞ ) (cid:19) γ/γ ′ ) , (2.7)where θ = 12 C − , γ ′ = min { k, γ } , C = { C X + 1)( γ + k ) } n (cid:18) nn − (cid:19) n . (2.8)Here C X is the maximum of the constants appearing in the Poincar´e inequality and Sobolevinequality on ( X, ˆ ω ). The proof of Theorem 1 is based on the following a priori estimates: Theorem 2
Let u ∈ Υ k be a C ,β ( X ) function with normalization R X e u ˆ ω n = M solvingthe k -th Fu-Yau Hessian equation (2.2) . Then C − M ≤ e u ≤ CM, e − u | i∂ ¯ ∂u | ˆ ω ≤ CM − / , e − u |∇ ¯ ∇∇ u | ω ≤ C ′ , (2.9) where C > only depends on ( X, ˆ ω ) , α ′ , k , γ , n , ρ , and µ . Assuming Theorem 2, we can prove Theorem 1. Both the existence and uniquenessstatements will be proved by the continuity method. We begin with the existence. Fix α ′ ∈ R \{ } , γ >
0, 1 ≤ k ≤ ( n − ρ ∈ Ω , ( X, R ) and µ : X → R such that R X µ ˆ ω n = 0,and define the set Υ k as above. For a real parameter t , we consider the family of equations1 k ∆ ˆ g e ku t + α ′ n tL ρ e ( k − γ ) u t + ˆ σ k +1 ( i∂ ¯ ∂u t ) o = tµ. (2.10)As equations of differential forms, this family can be expressed as i∂ ¯ ∂ ( e ku k ˆ ω + α ′ te ( k − γ ) u ρ ) ∧ ˆ ω n − + α ′ C kn − k + 1 ( i∂ ¯ ∂u ) k +1 ∧ ˆ ω n − k − − t µn ˆ ω n = 0 . (2.11)We introduce the following spaces B M = { u ∈ C ,β ( X, R ) : Z X e u ˆ ω n = M } , (2.12) B = { ( t, u ) ∈ [0 , × B M : u ∈ Υ k } , (2.13) B = { ψ ∈ C ,β ( X, R ) : Z X ψ ˆ ω n = 0 } (2.14)and define the map Ψ : B → B byΨ( t, u ) = 1 k ∆ ˆ g e ku t + α ′ tL ρ e ( k − γ ) u t + α ′ ˆ σ k +1 ( i∂ ¯ ∂u t ) − tµ. (2.15)5e consider I = { t ∈ [0 ,
1] : there exists u ∈ B M such that ( t, u ) ∈ B and Ψ( t, u ) = 0 } . (2.16)First, 0 ∈ I : indeed the constant function u = log M − log R X ˆ ω n is in Υ k when M ≫
1, and u solves the equation at t = 0. In particular I is non-empty.Next, we show that I is open. Let ( t , u ) ∈ B , and let L = ( D u Ψ) ( t ,u ) be thelinearized operator at ( t , u ), L : (cid:26) h ∈ C ,β ( X, R ) : Z X he u ˆ ω n = 0 (cid:27) → (cid:26) ψ ∈ C ,β ( X, R ) : Z X ψ ˆ ω n = 0 (cid:27) , (2.17)defined by L ( h )ˆ ω n = i∂ ¯ ∂ { e ku h ˆ ω + α ′ ( k − γ ) t e ( k − γ ) u h ρ } ∧ ˆ ω n − + α ′ C kn − i∂ ¯ ∂h ∧ ( i∂ ¯ ∂u ) k ∧ ˆ ω n − k − . (2.18)The leading order terms are L ( h )ˆ ω n = e ku χ ( t ,u ) ∧ ˆ ω n − k − ∧ i∂ ¯ ∂h + · · · (2.19)where χ ( t,u ) = ˆ ω k + α ′ ( k − γ ) te − γu ρ ∧ ˆ ω k − + α ′ C kn − ( e − u i∂ ¯ ∂u ) k . (2.20)Since u ∈ Υ k , we see from the conditions (2.6) that χ ( t ,u ) > k, k ) form and hence L is elliptic. The L adjoint L ∗ is readily computed by integrating by parts: Z X ψL ( h ) ˆ ω n = Z X h e ku χ ( t ,u ) ∧ ˆ ω n − k − ∧ i∂ ¯ ∂ψ = Z X hL ∗ ( ψ ) ˆ ω n . (2.21)Since L ∗ is an elliptic operator with no zeroth order terms, by the strong maximum prin-ciple the kernel of L ∗ consists of constant functions. An index theory argument (see e.g.[21] or [10] for full details) shows that the kernel of L is spanned by a function of constantsign. It follows that L is an isomorphism. By the implicit function theorem, there existsa unique solution ( t, u t ) for t sufficiently close to t , with u t ∈ Υ k since Υ k is open. Weconclude that I is open.Finally, we apply Theorem 2 to show that I is closed. Consider a sequence t i ∈ I suchthat t i → t ∞ , and denote u t i ∈ Υ k ∩ B M the associated C ,β functions such that Ψ( t i , u t i ) =0. By differentating the equation e − ku ti Ψ( t i , u t i ) = 0 with the Chern connection ˆ ∇ of theK¨ahler metric ˆ ω , we obtain0 = χ ( t i ,u ti ) ∧ ˆ ω n − k − ∧ i∂ ¯ ∂ ( ∂ ℓ u t i )ˆ ω n + ˆ ∇ ℓ { α ′ t i e − γu ti (( k − γ ) a p ¯ q ∂ p u t i ∂ ¯ q u t i + ( k − γ ) b k ∂ k u t i + ( k − γ ) b ¯ k ∂ ¯ k u t i + c ) } + k ˆ ∇ ℓ | Du t i | g − α ′ ke − ku ti ˆ σ k +1 ( i∂ ¯ ∂u t i ) ∂ ℓ u t i − t i ∂ ℓ { e − ku ti µ } . (2.22)6ince the equations (2.10) are of the form (2.2) with uniformly bounded coefficients ρ and µ , Theorem 2 applies to give uniform control of | u t i | and | ∂ ¯ ∂∂u t i | ˆ ω along this sequence.Therefore ˆ∆ u t i is uniformly controlled in C β ( X ) for any 0 < β <
1. By Schauder estimates,we have k u t i k C ,β ≤ C .Thus the differentiated equation (2.22) is a linear elliptic equation for ∂ ℓ u t i with C β coefficients. This equation is uniformly elliptic along the sequence, since χ ( t i ,u ti ) ≥ ˆ ω k by (2.9) when M ≫
1. By Schauder estimates, we have uniform control of k Du t i k C ,β . Abootstrap argument shows that we have uniform control of k u t i k C ,β , hence we may extracta subsequence converging to u ∞ ∈ C ,β . Furthermore, for M ≥ M ≫ e − u ∞ ≪ , | e − u i∂ ¯ ∂u ∞ | ˆ ω ≪ , (2.23)hence u ∞ ∈ Υ k . Thus I is closed.Hence I = [0 ,
1] and consequently there exists a C ,β function u ∈ Υ k with normal-ization R X e u ˆ ω n = M solving the Fu-Yau equation (2.2). By applying Schauder estimatesand a bootstrap argument to the differentiated equation (2.22), we see that u is smooth.We complete now the proof of Theorem 1 with the proof of uniqueness.First, we show that the only solutions of the equation1 k i∂ ¯ ∂e ku ∧ ˆ ω n − + α ′ C kn − k + 1 ( i∂ ¯ ∂u ) k +1 ∧ ˆ ω n − k − = 0 (2.24)with | α ′ | C kn − | e − u i∂ ¯ ∂u | k ˆ ω < − are constant functions. Multiplying by u and integrating,we see that 0 = Z X i∂u ∧ ¯ ∂u ∧ (cid:26) e ku ˆ ω k + α ′ C kn − k + 1 ( i∂ ¯ ∂u ) k (cid:27) ∧ ˆ ω n − k − , (2.25)and hence u must be constant since e ku ˆ ω k + α ′ C kn − k +1 ( i∂ ¯ ∂u ) k > k, k ) form.Now suppose there are two distinct solutions u ∈ Υ k and v ∈ Υ k satisfying (2.2) underthe normalization R X e u ˆ ω n = R X e v ˆ ω n = M with M ≥ M . For t ∈ [0 , t, u ) = i∂ ¯ ∂ ( e ku k ˆ ω + α ′ (1 − t ) e ( k − γ ) u ρ ) ∧ ˆ ω n − + α ′ C kn − k + 1 ( i∂ ¯ ∂u ) k +1 ∧ ˆ ω n − k − − (1 − t ) µn ˆ ω n , (2.26)and consider the path t u t satisfying Φ( t, u t ) = 0, u t ∈ Υ k , R X e u t ˆ ω n = M with initialcondition u = u .The same argument which shows that I is open also shows that the path u t exists fora short-time: there exists ǫ > u t is defined on [0 , ǫ ). By our estimates (2.9),we may extend the path to be defined for t ∈ [0 , t = 1, we know that u = log M − log R X ˆ ω n . The same argument gives a path t v t t, v t ) = 0, v t ∈ Υ k , R X e v t ˆ ω n = M with v = v and v = log M − log R X ˆ ω n .But then at the first time 0 < t ≤ u t = v t , we contradict the local uniquenessof Φ( t, u t ) = 0 given by the implicit function theorem.It follows from our discussion that in order to prove Theorem 1, it remains to establishthe a priori estimates (2.9). Theorem 3
Suppose u ∈ Υ k solves (2.2) subject to the normalization R X e u ˆ ω n = M .Then C − M ≤ e u ≤ CM, (3.1) where C only depends on ( X, ˆ ω ) , k , and γ . We first note the following general identity which holds for any function u .0 = α ′ ( p − k ) Z X e ( p − k ) u i∂u ∧ ¯ ∂u ∧ ( i∂ ¯ ∂u ) k ∧ ˆ ω n − k − + α ′ Z X e ( p − k ) u ( i∂ ¯ ∂u ) k +1 ∧ ˆ ω n − k − . (3.2)Substituting the Fu-Yau Hessian equation (2.11) with t = 1, we obtain0 = α ′ C kn − k + 1 ( p − k ) Z X e ( p − k ) u i∂u ∧ ¯ ∂u ∧ ( i∂ ¯ ∂u ) k ∧ ˆ ω n − k − + Z X e ( p − k ) u µn − Z X e ( p − k ) u i∂ ¯ ∂ ( e ku k ˆ ω + α ′ e ( k − γ ) u ρ ) ∧ ˆ ω n − . (3.3)We integrate by parts to derive0 = α ′ C kn − k + 1 ( p − k ) Z X e ( p − k ) u i∂u ∧ ¯ ∂u ∧ ( i∂ ¯ ∂u ) k ∧ ˆ ω n − k − + Z X e ( p − k ) u µn + ( p − k ) Z X e pu i∂u ∧ i ¯ ∂u ∧ ˆ ω n − +( p − k ) α ′ Z X e ( p − k ) u i∂u ∧ i ¯ ∂ ( e ( k − γ ) u ρ ) ∧ ˆ ω n − . (3.4)Integrating by parts again gives( p − k ) Z X e pu i∂u ∧ ¯ ∂u ∧ ˆ ω n − k − ∧ χ = − Z X e ( p − k ) u µ ˆ ωn + p − kp − γ α ′ Z X e ( p − γ ) u ∧ i∂ ¯ ∂ρ ∧ ˆ ω n − , (3.5)where we now assume p > γ and we define χ = ˆ ω k + α ′ ( k − γ ) e − γu ρ ∧ ˆ ω k − + α ′ C kn − k + 1 ( e − u i∂ ¯ ∂u ) k . (3.6)8ext, we estimate i∂u ∧ ¯ ∂u ∧ ˆ ω n − k − ∧ χ = |∇ u | ω n ˆ ω n + α ′ ( k − γ ) e − γu a i ¯ j u i u ¯ j n ˆ ω n + α ′ C kn − k + 1 i∂u ∧ ¯ ∂u ∧ ( e − u i∂ ¯ ∂u ) k ∧ ˆ ω n − k − ≥ |∇ u | ω n ˆ ω n − | α ′ Λ( k − γ ) | δ |∇ u | ω n ˆ ω n −| α ′ | C kn − k + 1 | e − u i∂ ¯ ∂u | k ˆ ω |∇ u | ω n ˆ ω n . (3.7)Since u ∈ Υ k , by (2.6) and (2.7) the positive term dominates the expression and we canconclude i∂u ∧ ¯ ∂u ∧ ˆ ω n − k − ∧ χ ≥ |∇ u | ω n ˆ ω n . (3.8)The proof of Theorem 3 will be divided into three propositions. We note that in thefollowing arguments we will omit the background volume form ˆ ω n when integrating scalarfunctions. Proposition 1
Suppose u ∈ Υ k solves (2.2) subject to normalization R X e u = M . Thereexists C > such that e u ≤ C M, (3.9) where C only depends on ( X, ˆ ω ) , n , k and γ . In fact, C is given by (2.8). Combining (3.5) and (3.8) gives12 ( p − k ) Z X e pu |∇ u | ω ≤ − Z X e ( p − k ) u µ + p − kp − γ nα ′ Z X e ( p − γ ) u ∧ i∂ ¯ ∂ρ ∧ ˆ ω n − . (3.10)We estimate Z X |∇ e p u | ω ≤ p p − k ) (cid:26) k µ k L ∞ Z X e ( p − k ) u + p − kp − γ k α ′ c k L ∞ Z X e ( p − γ ) u (cid:27) . (3.11)For any p ≥ { γ, k } , there holds p p − k ) ≤ p and p − kp − γ ≤
2. Using e − γu ≤ δ ≤ Z X |∇ e p u | ω ≤ k µ k L ∞ + k α ′ c k L ∞ ) δ min { k,γ } γ p Z X e pu ≤ θC X p Z X e pu ≤ pC X Z X e pu , (3.12)9or any p ≥ γ + k ). Let β = nn − . The Sobolev inequality gives us (cid:18)Z X e βpu (cid:19) /β ≤ C X (cid:18) Z X |∇ e p u | ω + Z X e pu (cid:19) . (3.13)Therefore for all p ≥ γ + k ), k e u k L pβ ≤ ( C X + 1) /p p /p k e u k L p . (3.14)Iterating this inequality gives k e u k L pβ ( k +1) ≤ { ( C X + 1) p } p P ki =0 1 βi · β p P ki =1 iβi k e u k L p . (3.15)Letting k → ∞ , we obtainsup X e u ≤ C ′ k e u k L γ + k ) , C ′ = { C X + 1)( γ + k ) } γ + k ) P ∞ i =0 1 βi · β γ + k ) P ∞ i =1 iβi . (3.16)It follows that sup X e u ≤ C ′ (sup X e u ) − (2( γ + k )) − (cid:18)Z X e u (cid:19) / γ + k ) , (3.17)and we conclude that sup X e u ≤ C Z X e u , C = ( C ′ ) γ + k ) . (3.18)This proves the estimate. As it will be needed in the future, we note that the precise formof C agrees with the definition given in (2.8). Proposition 2
Suppose u ∈ Υ k solves (2.2) subject to normalization R X e u = M . Thereexists a constant C only depending on ( X, ˆ ω ) , n , k and γ such that Z X e − u ≤ CM − . (3.19)Setting p = − k + 1) Z X e − u i∂u ∧ ¯ ∂u ∧ ˆ ω n − k − ∧ χ (3.20)= Z X e − (1+ k ) u µ ˆ ω n n − k γ Z X e − (1 − γ ) u i∂ ¯ ∂ρ ∧ ˆ ω n − ≤ n k µ k L ∞ Z X e − (1+ k ) u + 1 + k (1 + γ ) n k α ′ c k L ∞ Z X e − (1+ γ ) u . (3.21)Since u ∈ Υ k , we may use (3.8) and e − γu ≤ δ ≤ Z X e − u |∇ u | ω ≤ δ min { k,γ } γ ( k µ k L ∞ + k α ′ c k L ∞ ) Z X e − u . (3.22)10y the Poincar´e inequality Z X e − u − (cid:18)Z X e − u/ (cid:19) ≤ C X Z X |∇ e − u/ | ω . (3.23)After using the definition of δ (2.7), it follows that Z X e − u ≤ − θ (cid:18)Z X e − u/ (cid:19) . (3.24)Let U = { x ∈ X : e u ≥ M } . From Proposition 1, and using Vol( X, ˆ ω ) = 1, M = Z X e u ≤ C M | U | + (1 − | U | ) M . (3.25)Hence | U | ≥ θ >
0, where we recall that θ was defined in (2.7). Using | U | ≥ θ and (3.24),it was shown in [21] that the estimate Z X e − u ≤ − θ (cid:18) θ (cid:19) (cid:18) θ (cid:19) M − (3.26)follows. Proposition 3
Suppose u ∈ Υ k solves (2.2) subject to the normalization R X e u = M .There exists C such that sup X e − u ≤ CM − , (3.27) where C only depends on ( X, ˆ ω ) , n , k and γ . Exchanging p for − p in (3.5) and using (3.8) gives( p + k ) Z X e − pu i∂u ∧ ¯ ∂u ∧ ˆ ω n − (3.28) ≤ Z X e − ( p + k ) u µ ˆ ω n n − α ′ p + kp + γ Z X e − ( p + γ ) u i∂ ¯ ∂ρ ∧ ˆ ω n − . By using e γu ≤ δ ≤
1, we obtain Z X |∇ e − p u | ω ≤ p p + k ) δ min { k,γ } γ ( k µ k L ∞ + p + kp + γ k α ′ c k L ∞ ) Z X e − pu . (3.29)We may use (2.7) to obtain a constant C depending on ( X, ˆ ω ), n , k , and γ such that Z X |∇ e − p u | ω ≤ Cp Z X e − pu . (3.30)for any p ≥
1. Using the Sobolev inequality and iterating in a similar way to Proposition1, we obtain sup X e − u ≤ C k e − u k L . (3.31)Applying Proposition 2 gives the desired estimate.11 Setup and Notation
We come now to the key steps of establishing the gradient and the C estimates. It turnsout that, for these steps, it is more natural to view the equation (2.2) as an equation forthe unknown, non-K¨ahler, Hermitian form ω = e u ˆ ω (4.1)and to carry out calculations with respect to the Chern unitary connection ∇ of ω . Asusual, we identify the metrics ˆ g and g via ˆ ω = ˆ g ¯ kj idz j ∧ d ¯ z k and ω = g ¯ kj idz j ∧ d ¯ z k , anddenote ˆ g j ¯ k , g j ¯ k to be the inverse matrix of ˆ g ¯ kj , g ¯ kj . Then g ¯ kj = e u ˆ g ¯ kj , g j ¯ k = e − u ˆ g j ¯ k . Recallthat the Chern unitary connection ∇ is defined by ∇ ¯ k V j = ∂ ¯ k V j , ∇ k V j = g j ¯ m ∂ k ( g ¯ mp V p ) (4.2)and its torsion and curvature by[ ∇ α , ∇ β ] V γ = R βαγ δ V δ + T δβα ∇ δ V γ . (4.3)Explicitly, R ¯ kqjp = − ∂ ¯ k ( g j ¯ m ∂ j g ¯ mq ) , T j pq = g j ¯ m ( ∂ p g ¯ mq − ∂ q g ¯ mp ) . (4.4)The curvatures and torsions of the metrics g ¯ kj and ˆ g ¯ kj are then related by R ¯ kj pi = ˆ R ¯ kjpi − u ¯ kj δ pi , T λkj = u k δ λj − u j δ λk . (4.5)The following commutation formulas with either 3 or 4 covariant derivatives will be par-ticularly useful, ∇ j ∇ p ∇ ¯ q u = ∇ p ∇ ¯ q ∇ j u + T mpj ∇ m ∇ ¯ q u (4.6)and ∇ ¯ k ∇ j ∇ p ∇ ¯ q u = ∇ p ∇ ¯ q ∇ j ∇ ¯ k u − R ¯ qp ¯ k ¯ m ∇ ¯ m ∇ j u + R ¯ kj mp ∇ m ∇ ¯ q u + T ¯ m ¯ q ¯ k ∇ p ∇ ¯ m ∇ j u + T mpj ∇ ¯ k ∇ m ∇ ¯ q u. (4.7)They reduce in our case to ∇ j ∇ p ∇ ¯ q u = ∇ p ∇ ¯ q ∇ j u + u p u ¯ qj − u j u ¯ qp . (4.8)and to ∇ ¯ k ∇ j ∇ p ∇ ¯ q u = ∇ p ∇ ¯ q ∇ j ∇ ¯ k u + u p ∇ ¯ k ∇ j ∇ ¯ q u − u j ∇ ¯ k ∇ p ∇ ¯ q u + u ¯ q ∇ p ∇ ¯ k ∇ j u − u ¯ k ∇ p ∇ ¯ q ∇ j u + ˆ R ¯ kjλp u ¯ qλ − ˆ R ¯ qp ¯ k ¯ λ u ¯ λj . (4.9)12t will also be convenient to use the symmetric functions of the eigenvalues of i∂ ¯ ∂u with respect to ω rather than with respect to ˆ ω . Thus we define σ ℓ ( i∂ ¯ ∂u ) to be the ℓ -thelementary symmetric polynomial of the eigenvalues of the endomorphism h ij = g i ¯ k u ¯ kj .Explicitly, if λ i are the eigenvalues of the endomorphism h ij = g i ¯ k u ¯ kj , then σ ℓ ( i∂ ¯ ∂u ) = P i < ···
First, at a point z where g p ¯ q = δ pq and u ¯ qp is diagonal, the above lemma implies | α ′ σ p ¯ pk +1 | = | α ′ σ k ( λ | p ) | ≤ | α ′ | C kn − ( n − k/ |∇ ¯ ∇ u | kg . (4.22)The condition u ∈ Υ k gives | α ′ σ p ¯ pk +1 ( z ) | ≤ − . This argument shows that α ′ σ p ¯ qk +1 is on theorder of 2 − g p ¯ q in arbitrary coordinates.Next, u ∈ Υ k also implies that | α ′ ( k − γ ) e − γu Λ | ≤ − . Since − Λˆ g p ¯ q ≤ a p ¯ q ≤ Λˆ g p ¯ q , wecan put everything together and obtain the estimate (4.21). Q.E.D. Theorem 4
Let u ∈ Υ k be a C ( X, R ) function solving the Fu-Yau Hessian equation(2.2). Then |∇ u | g ≤ C, (5.1) where C only depends on ( X, ˆ ω ) , α ′ , k , γ , k ρ k C ( X, ˆ ω ) and k µ k C ( X ) . In view of Theorem 3, this estimate is equivalent to |∇ u | g ≤ CM − , (5.2)where C only depends on ( X, ˆ ω ), α ′ , k , γ , k ρ k C ( X, ˆ ω ) and k µ k C ( X ) . We will prove thisestimate by applying the maximum principle to the following test function G = log |∇ u | g + (1 + σ ) u, (5.3)for a parameter 0 < σ <
1. Though there is a range of values of σ which makes theargument work, to be concrete we will take σ = 2 − .14 .1 Estimating the leading terms Suppose G attains a maximum at p ∈ X . Then0 = ∇|∇ u | g |∇ u | g + (1 + σ ) ∇ u. (5.4)We will compute the operator F p ¯ q ∇ p ∇ ¯ q acting on G at p . F p ¯ q ∇ p ∇ ¯ q G = 1 |∇ u | g F p ¯ q ∇ p ∇ ¯ q |∇ u | g − |∇ u | g F p ¯ q ∇ p |∇ u | g ∇ ¯ q |∇ u | g + (1 + σ ) F p ¯ q u ¯ qp . (5.5)By direct computation F p ¯ q ∇ p ∇ ¯ q |∇ u | g = F p ¯ q g j ¯ i ∇ p ∇ ¯ q ∇ j u ∇ ¯ i u + F p ¯ q g j ¯ i ∇ j u ∇ p ∇ ¯ q ∇ ¯ i u + |∇ ¯ ∇ u | F g + |∇∇ u | F g . (5.6)where |∇∇ u | F g = F p ¯ q g j ¯ i ∇ p ∇ j u ∇ ¯ q ∇ ¯ i u and |∇ ¯ ∇ u | F g = F p ¯ q g j ¯ i u ¯ qj u ¯ ip . Commuting deriva-tives according to the relation[ ∇ j , ∇ ¯ ℓ ] u ¯ i = R ¯ ℓj ¯ i ¯ p u ¯ p = ˆ R ¯ ℓj ¯ i ¯ p u ¯ p − u ¯ ℓj u ¯ i , (5.7)we obtain F p ¯ q g j ¯ i ∇ j u ∇ p ∇ ¯ q ∇ ¯ i u = F p ¯ q g j ¯ i ∇ p ∇ ¯ q ∇ j u ∇ ¯ i u + F p ¯ q g j ¯ i u j ˆ R ¯ qp ¯ i ¯ λ u ¯ λ − F p ¯ q g j ¯ i u j u ¯ qp u ¯ i . (5.8)Thus F p ¯ q ∇ p ∇ ¯ q |∇ u | g = 2Re { F p ¯ q g j ¯ i ∇ p ∇ ¯ q ∇ j u ∇ ¯ i u } + F p ¯ q g j ¯ i u j ˆ R ¯ qp ¯ i ¯ λ u ¯ λ − F p ¯ q g j ¯ i u j u ¯ qp u ¯ i + |∇ ¯ ∇ u | F g + |∇∇ u | F g . (5.9)Next, we use the equation. Expanding L ρ = a p ¯ q ∂ p ∂ ¯ q + b i ∂ i + ¯ b i ∂ ¯ i + c , equation (4.10)becomes 0 = ∆ g u + α ′ n ( k − γ ) e − (1+ γ ) u a p ¯ q u ¯ qp + σ k +1 ( i∂ ¯ ∂u ) o + k |∇ u | g + α ′ ( k − γ ) e − (1+ γ ) u a p ¯ q u p u ¯ q + 2 α ′ ( k − γ ) e − (1+ γ ) u Re { b i u i } + α ′ e − (1+ γ ) u c − e − ( k +1) u µ. (5.10)We covariantly differentiate equation (5.10), using (4.12) to differentiate σ k +1 and usingthe notation F p ¯ q introduced in (4.18). This leads to0 = F p ¯ q ∇ j ∇ p ∇ ¯ q u + k ∇ j |∇ u | g + E j , (5.11)15here E j = α ′ ( k − γ ) e − (1+ γ ) u (cid:26) − γa p ¯ q u ¯ qp u j + ˆ ∇ j a p ¯ q u ¯ qp (cid:27) + α ′ ( k − γ ) e − (1+ γ ) u (cid:26) − γa p ¯ q u p u ¯ q u j + ˆ ∇ j a p ¯ q u p u ¯ q + a p ¯ q ∇ j ∇ p uu ¯ q + a p ¯ q u p u ¯ qj (cid:27) + α ′ ( k − γ ) e − (1+ γ ) u (cid:26) − γ )Re { b i u i } u j + ˆ ∇ j b i u i + u j b i u i + ∂ j ¯ b i u ¯ i + b i ∇ j ∇ i u + ¯ b i u ¯ ij (cid:27) − (1 + γ ) α ′ e − (1+ γ ) u cu j + α ′ e − (1+ γ ) u ∂ j c +( k + 1) e − ( k +1) u µu j − e − ( k +1) u ∂ j µ. (5.12)We used ∇ i W j = ˆ ∇ i W j + u i W j to replace ∇ by ˆ ∇ in the above calculation. We willeventually see that the terms E j play a minor role when u ∈ Υ k , and will only perturb thecoefficients of the leading terms. Commuting covariant derivatives using (4.8), we obtain F p ¯ q ∇ p ∇ ¯ q ∇ j u = − F p ¯ q u p u ¯ qj + F p ¯ q u j u ¯ qp − k ∇ j |∇ u | g − E j . (5.13)Substituting (5.13) into (5.9), an important partial cancellation occurs, and we obtain F p ¯ q ∇ p ∇ ¯ q |∇ u | g = − { F p ¯ q g j ¯ i u ¯ i u p u ¯ qj } + |∇ u | g F p ¯ q u ¯ qp − k Re { g j ¯ i ∇ ¯ i u ∇ j |∇ u | g }− { g j ¯ i E j u ¯ i } + F p ¯ q g j ¯ i u j ˆ R ¯ qp ¯ i ¯ λ u ¯ λ + |∇ ¯ ∇ u | F g + |∇∇ u | F g . (5.14)We note the identity F p ¯ q u ¯ qp = ∆ g u + α ′ ( k − γ ) e − (1+ γ ) u a p ¯ q u ¯ qp + ( k + 1) α ′ σ k +1 ( i∂ ¯ ∂u ) . (5.15)Substituting the equation (5.10) into the identity (5.15), we obtain F p ¯ q u ¯ qp = − k |∇ u | g + ˜ E , (5.16)where ˜ E = kα ′ σ k +1 ( i∂ ¯ ∂u ) − α ′ ( k − γ ) e − (1+ γ ) u a p ¯ q u p u ¯ q − α ′ ( k − γ ) e − (1+ γ ) u Re { b i u i } − α ′ e − (1+ γ ) u c + e − ( k +1) u µ. (5.17)will turn out to be another perturbative term. Substituting (5.14) and (5.16) into (5.5) F p ¯ q ∇ p ∇ ¯ q G = 1 |∇ u | g |∇ ¯ ∇ u | F g + 1 |∇ u | g |∇∇ u | F g − |∇ u | g Re { F p ¯ q g j ¯ i u ¯ i u p u ¯ qj }− |∇ u | g F p ¯ q ∇ p |∇ u | g ∇ ¯ q |∇ u | g − k |∇ u | g Re { g j ¯ i u ¯ i ∇ j |∇ u | g }− (2 + σ ) k |∇ u | g + 1 |∇ u | g F p ¯ q g j ¯ i u j ˆ R ¯ qp ¯ i ¯ λ u ¯ λ − |∇ u | g Re { g j ¯ i E j u ¯ i } + (2 + σ ) ˜ E . (5.18)16sing the critical equation (5.4), − |∇ u | g F p ¯ q ∇ p |∇ u | g ∇ ¯ q |∇ u | g − k |∇ u | g Re { g j ¯ i u ¯ i ∇ j |∇ u | g } = − (1 + σ ) |∇ u | F + 2(1 + σ ) k |∇ u | g . (5.19)Here we introduced the notation |∇ f | F = F p ¯ q f p f ¯ q for a real-valued function f . The criticalequation (5.4) can also be written as g j ¯ i ∇ p u j u ¯ i |∇ u | g = − g j ¯ i u j u ¯ ip |∇ u | g − (1 + σ ) u p . (5.20)We now combine this identity with the Cauchy-Schwarz inequality, which will lead to apartial cancellation of terms. This idea is also used to derive a C estimate for the complexMonge-Amp`ere equation, [2, 14, 24, 25, 33].1 |∇ u | g |∇∇ u | F g ≥ (cid:12)(cid:12)(cid:12)(cid:12) g j ¯ i ∇ u j u ¯ i |∇ u | g (cid:12)(cid:12)(cid:12)(cid:12) F (5.21)= 1 |∇ u | g | g j ¯ i u j ∇ u ¯ i | F + (1 + σ ) |∇ u | F + 2(1 + σ ) |∇ u | g Re { F p ¯ q g j ¯ i u j u ¯ ip u ¯ q } . Let ε >
0. Combining (5.19) and (5.21) and dropping a nonnegative term, − |∇ u | g F p ¯ q ∇ p |∇ u | g ∇ ¯ q |∇ u | g − k |∇ u | g Re { g j ¯ i u ¯ i ∇ j |∇ u | g } + (1 − ε ) 1 |∇ u | g |∇∇ u | F g ≥ − (1 + σ ) ε |∇ u | F + 2(1 + σ ) k |∇ u | g + 2(1 + σ )(1 − ε ) |∇ u | g Re { F p ¯ q g j ¯ i u j u ¯ ip u ¯ q } . (5.22)Substituting this inequality into (5.18), partial cancellation occurs and we are left with F p ¯ q ∇ p ∇ ¯ q G ≥ |∇ u | g |∇ ¯ ∇ u | F g + ε |∇ u | g |∇∇ u | F g + { σ − ε (1 + σ ) } |∇ u | g Re { F p ¯ q g j ¯ i u ¯ i u p u ¯ qj } + σk |∇ u | g − (1 + σ ) ε |∇ u | F + 1 |∇ u | g F p ¯ q g j ¯ i u j ˆ R ¯ qp ¯ i ¯ λ u ¯ λ − |∇ u | g Re { g j ¯ i E j u ¯ i } + (2 + σ ) ˜ E . (5.23)Since u ∈ Υ k , we now use (4.21) in Lemma 2 to pass the norms with respect to F p ¯ q to g p ¯ q up to an error of order 2 − . We choose ε = (1 + σ ) − (1 + 2 − ) − σ . (5.24)17hen (1 + σ ) ε |∇ u | F ≤ σ |∇ u | g , (5.25)and ε |∇ u | g |∇∇ u | F g ≥ σ σ ) − − − |∇ u | g |∇∇ u | g . (5.26)Since σ = 2 − , we have the inequality of numbers
12 1 − − (1+ σ ) (1+2 − ) ≥ . Thus ε |∇ u | g |∇∇ u | F g ≥ σ |∇ u | g |∇∇ u | g . (5.27)We also note the inequalities1 |∇ u | g |∇ ¯ ∇ u | F g ≥ (1 − − ) 1 |∇ u | g |∇ ¯ ∇ u | g , (5.28)and { σ − ε (1 + σ ) } |∇ u | g Re { F p ¯ q g j ¯ i u ¯ i u p u ¯ qj }≥ −{ − (1 + σ ) − (1 + 2 − ) − } σ (1 + 2 − ) |∇ ¯ ∇ u | g ≥ − σ (1 + 2 − ) |∇ ¯ ∇ u | g . (5.29)The main inequality (5.23) becomes F p ¯ q ∇ p ∇ ¯ q G ≥ (1 − − ) 1 |∇ u | g |∇ ¯ ∇ u | g + σ |∇∇ u | g |∇ u | g − σ (1 + 2 − ) |∇ ¯ ∇ u | g + σ |∇ u | g + 1 |∇ u | g F p ¯ q g j ¯ i u j ˆ R ¯ qp ¯ i ¯ λ u ¯ λ − |∇ u | g Re { g j ¯ i E j u ¯ i } + (2 + σ ) ˜ E . (5.30) E j terms Recall the constant Λ is such that − Λˆ g j ¯ i ≤ a j ¯ i ≤ Λˆ g j ¯ i . We will go through each term inthe definition of E j (5 .
12) and estimate the terms appearing in |∇ u | g Re { g j ¯ i E j u ¯ i } by groups.In the following, we will use C to denote constants possibly depending on α ′ , k , γ , a p ¯ q , b i , c , µ , and their derivatives.First, using 2 ab ≤ a + b , we estimate the terms involving ∇ ¯ ∇ u | α ′ ( k − γ ) ||∇ u | g e − (1+ γ ) u | g j ¯ i u ¯ i ( − γa p ¯ q u ¯ qp u j + ˆ ∇ j a p ¯ q u ¯ qp + ( k − γ ) a p ¯ q u p u ¯ qj + ¯ b q u ¯ qj ) | | α ′ Λ( k − γ )( k + 2 γ ) | e − γ |∇ ¯ ∇ u | g + Ce − γu e − u/ |∇ ¯ ∇ u | g |∇ u | g ≤ (cid:26) | α ′ Λ | / ( k − γ ) | δ / |∇ u | g (cid:27)(cid:26) δ / ( k + 2 γ ) | Λ α ′ | / |∇ ¯ ∇ u | g |∇ u | g (cid:27) + Ce − u/ |∇ ¯ ∇ u | g |∇ u | g ≤ | α ′ | Λ( k − γ ) δ |∇ u | g + 4 | Λ α ′ | ( k + γ ) δ |∇ ¯ ∇ u | g |∇ u | g + σ |∇ ¯ ∇ u | g |∇ u | g + C ( σ ) e − u . (5.31)Second, we estimate the terms involving ∇∇ u | α ′ ( k − γ ) ||∇ u | g e − (1+ γ ) u | g j ¯ i u ¯ i { ( k − γ ) a p ¯ q ∇ j ∇ p u u ¯ q + b p ∇ j ∇ p u }|≤ | α ′ | ( k − γ ) Λ e − γu |∇∇ u | g + 2 (cid:26) C | α ′ Λ | / e − (1+ γ ) u/ (cid:27)(cid:26) | α ′ Λ | / | k − γ | e − γu/ |∇∇ u | g |∇ u | g (cid:27) ≤ | α ′ | ( k − γ ) Λ δ (cid:26) |∇∇ u | g |∇ u | g + |∇ u | g (cid:27) + | α ′ Λ | ( k − γ ) e − γu |∇∇ u | g |∇ u | g + C | α ′ Λ | e − (1+ γ ) u ≤ | α ′ | Λ( k − γ ) δ |∇∇ u | g |∇ u | g + δ | α ′ | ( k − γ ) Λ |∇ u | g + Ce − u . (5.32)Third, we estimate the terms involving ∇ u quadratically2 | α ′ ( k − γ ) ||∇ u | g e − (1+ γ ) u | g j ¯ i u ¯ i { ( k − γ ) ˆ ∇ j a p ¯ q u p u ¯ q − γ )Re { b p u p } u j + u j b i u i }|≤ Ce − γu e − u/ |∇ u | g ≤ σ |∇ u | g + C ( σ ) e − (1+2 γ ) u ≤ σ |∇ u | g + Ce − u . (5.33)Finally, for all the other terms in E j , we can estimate2 | α ′ ( k − γ ) ||∇ u | g e − (1+ γ ) u | g j ¯ i u ¯ i {− γ ( k − γ ) a p ¯ q u p u ¯ q u j + ˆ ∇ j b p u p + ∂ j ¯ b q u ¯ q }| + 2 |∇ u | g | g j ¯ i u ¯ i {− (1 + γ ) α ′ ce − (1+ γ ) u u j + α ′ e − (1+ γ ) u ∂ j c + ( k + 1) e − ( k +1) u µu j − e − ( k +1) u ∂ j µ }|≤ | α ′ | Λ( k − γ ) γe − γu |∇ u | g + Ce − (1+ γ ) u + Ce − (1+ γ ) u e − u/ |∇ u | g + Ce − ( k +1) u + Ce − ( k +1) u e − u/ |∇ u | g ≤ | α ′ | Λ( k − γ ) γδ |∇ u | g + Ce − u + Ce − u e − u/ |∇ u | g . (5.34)Putting everything together, we obtain the following estimate for the terms comingfrom E j .2 |∇ u | g | g j ¯ i E j u ¯ i | ≤ (cid:26) | α ′ | Λ( k − γ ) (1 + γ ) δ + σ (cid:27) |∇ u | g + Ce − u + Ce − u e − u/ |∇ u | g + { | α ′ | Λ( k + γ ) δ + σ } |∇ ¯ ∇ u | g |∇ u | g + 2 | α ′ | Λ( k − γ ) δ |∇∇ u | g |∇ u | g (5.35)19 .2.2 The ˜ E terms Next, estimating ˜ E defined in (5.17) gives(2 + σ ) | ˜ E | ≤ k (2 + σ ) | α ′ || σ k +1 ( i∂ ¯ ∂u ) | + (2 + σ ) | α ′ Λ | ( k − γ ) e − γu |∇ u | g +2 k α ′ ( k − γ ) b i k ∞ e − γu e − u/ |∇ u | g + Ce − (1+ γ ) u + Ce − ( k +1) u . (5.36)Using e − u ≤ δ ≤ k α ′ ( k − γ ) b k ∞ e − γu e − u/ |∇ u | g ≤ σ |∇ u | g + C ( σ ) e − u e − γu , (5.37)we obtain(2 + σ ) | ˜ E | ≤ k (2 + σ ) | α ′ || σ k +1 ( i∂ ¯ ∂u ) | + (2 + σ ) | α ′ Λ | ( k − γ ) δ |∇ u | g + σ |∇ u | g + Ce − u . By Lemma 1, we have k | α ′ || σ k +1 ( i∂ ¯ ∂u ) | ≤ k | α ′ | C k +1 n n / n k/ |∇ ¯ ∇ u | kg |∇ ¯ ∇ u | g ≤ {| α ′ | C kn − |∇ ¯ ∇ u | kg }|∇ ¯ ∇ u | g . (5.38)Since u ∈ Υ k , we have | α ′ | C kn − |∇ ¯ ∇ u | kg ≤ − . Thus(2 + σ ) | ˜ E | ≤ (cid:26) (2 + σ ) | α ′ Λ | ( k − γ ) δ + σ (cid:27) |∇ u | g + 2 − (2 + σ ) |∇ ¯ ∇ u | g + Ce − u . (5.39) Combining (5.35) and (5.39),2 |∇ u | g | g j ¯ i E j u ¯ i | + (2 + σ ) | ˜ E | ≤ (cid:26) | α ′ | Λ( k − γ ) (1 + γ ) δ + σ (cid:27) |∇ u | g +2 | α ′ Λ | ( k − γ ) δ |∇∇ u | g |∇ u | g + { | α ′ | Λ( k + γ ) δ + σ } |∇ ¯ ∇ u | g |∇ u | g +2 − (2 + σ ) |∇ ¯ ∇ u | g + Ce − u + Ce − u e − u/ |∇ u | g . (5.40)Recall σ = 2 − and using ( k − γ ) (1 + γ ) ≤ ( k + γ ) , the definition (2.7) of δ implies5 | α ′ | Λ( k − γ ) (1 + γ ) δ ≤ σ | α ′ Λ | ( k + γ ) δ ≤ − . Then, we have2 |∇ u | g | g j ¯ i E j u ¯ i | + (2 + σ ) | ˜ E | ≤ σ |∇ u | g + σ |∇∇ u | g |∇ u | g + 2 − |∇ ¯ ∇ u | g |∇ u | g + 2 − (2 + σ ) |∇ ¯ ∇ u | g + Ce − u + Ce − u e − u/ |∇ u | g . (5.41)20sing (5.41), the main inequality (5.30) becomes F p ¯ q ∇ p ∇ ¯ q G ≥ (1 − − ) 1 |∇ u | g |∇ ¯ ∇ u | g − n σ (1 + 2 − ) + 2 − (2 + σ ) o |∇ ¯ ∇ u | g + σ |∇ u | g + 1 |∇ u | g F p ¯ q g j ¯ i u j ˆ R ¯ qp ¯ i ¯ λ u ¯ λ − Ce − u − Ce − u e − u/ |∇ u | g . (5.42)By our choice σ = 2 − , we have the inequality of numbers n σ (1 + 2 − ) + 2 − (2 + σ ) o − − ≤ σ . (5.43)Thus n σ (1 + 2 − ) + 2 − (2 + σ ) o |∇ ¯ ∇ u | g ≤ (1 − − ) 1 |∇ u | g |∇ ¯ ∇ u | g + 14 n σ (1 + 2 − ) + 2 − (2 + σ ) o − − |∇ u | g ≤ (1 − − ) 1 |∇ u | g |∇ ¯ ∇ u | g + σ |∇ u | g . (5.44)We may also estimate 1 |∇ u | g F p ¯ q g j ¯ i u j ˆ R ¯ qp ¯ i ¯ λ u ¯ λ ≥ − Ce − u . (5.45)Putting everything together, at p there holds0 ≥ F p ¯ q ∇ p ∇ ¯ q G ≥ σ |∇ u | g − Ce − u e − u/ |∇ u | g − Ce − u . (5.46)From this inequality, we can conclude |∇ u | g ( p ) ≤ Ce − u ( p ) . (5.47)By definition G ( x ) ≤ G ( p ), and we have |∇ u | g ≤ Ce − u ( p ) e (1+ σ )( u ( p ) − u ) ≤ CM − , (5.48)since e u ( p ) e − u ≤ C and e − u ≤ CM − by Theorem 3. This completes the proof of Theorem4. Theorem 5
Let u ∈ Υ k be a C ( X ) function with normalization R X e u ˆ ω n = M solvingthe Fu-Yau equation (2.2). Then |∇ ¯ ∇ u | g ≤ CM − . (6.1) where C only depends on ( X, ˆ ω ) , α ′ , k , γ , k ρ k C ( X, ˆ ω ) and k µ k C ( X ) .
21e begin by noting the following elementary estimate.
Lemma 3
Let ℓ ∈ { , , . . . , n } . The following estimate holds: | g j ¯ i σ p ¯ q,r ¯ sℓ ∇ j u ¯ qp ∇ ¯ i u ¯ sr | ≤ C ℓ − n − |∇ ¯ ∇ u | ℓ − g |∇ ¯ ∇∇ u | g . (6.2) Proof:
Since the inequality is invariant, we may work at a point p ∈ X where g is theidentity and u ¯ qp is diagonal. At p , we can use (4.17) and conclude | g j ¯ i σ p ¯ q,r ¯ sℓ ∇ j u ¯ qp ∇ ¯ i u ¯ sr | ≤ X i X p,q | σ ℓ − ( λ | pq ) ||∇ i u ¯ qp | . (6.3)By Lemma 1, | σ ℓ − ( λ | pq ) | ≤ C ℓ − n − ( n − ( ℓ − / |∇ ¯ ∇ u | ℓ − g . (6.4)This inequality proves the Lemma. Q.E.D. Lemma 4
Let u ∈ Υ k be a C ( X ) function solving (2.2) with normalization R X e u = M .There exists a constant C > depending only on ( X, ˆ ω ) , α ′ , k , γ , k ρ k C ( X, ˆ ω ) and k µ k C ( X ) such that F p ¯ q ∇ p ∇ ¯ q |∇ ¯ ∇ u | g ≥ − − ) |∇ ¯ ∇∇ u | g − (1 + 2 k ) | α ′ | − /k τ /k |∇∇ u | g − (1 + 2 k ) | α ′ | − /k τ /k |∇ ¯ ∇ u | g − |∇ ¯ ∇ u | g |∇ ¯ ∇∇ u | g − CM − / |∇ ¯ ∇∇ u | g − CM − |∇∇ u | g − CM − . (6.5)We start by differentiating (5.11) and using the definition of F p ¯ q to obtain0 = α ′ ∇ ¯ i σ p ¯ qk +1 ∇ j ∇ p ∇ ¯ q u + F p ¯ q ∇ ¯ i ∇ j ∇ p ∇ ¯ q u + k ∇ ¯ i ∇ j |∇ u | g − α ′ ( k − γ )(1 + γ ) e − (1+ γ ) u a p ¯ q u ¯ i ∇ j ∇ p ∇ ¯ q u + α ′ ( k − γ ) e − (1+ γ ) u ∇ ¯ i a p ¯ q ∇ j ∇ p ∇ ¯ q u + ∇ ¯ i E j . (6.6)Next, we use (4.13) and (4.9) to conclude F p ¯ q ∇ p ∇ ¯ q u ¯ ij = − α ′ σ p ¯ q,r ¯ sk +1 ∇ j ∇ p ∇ ¯ q u ∇ ¯ i ∇ r ∇ ¯ s u − F p ¯ q [ u p ∇ ¯ i ∇ j ∇ ¯ q u − u j ∇ ¯ i ∇ p ∇ ¯ q u + u ¯ q ∇ p ∇ ¯ i ∇ j u − u ¯ i ∇ p ∇ ¯ q ∇ j u ] − F p ¯ q ˆ R ¯ ijλp u ¯ qλ + F p ¯ q ˆ R ¯ qp ¯ i ¯ λ u ¯ λj − k ∇ ¯ i ∇ j |∇ u | g + α ′ ( k − γ )(1 + γ ) e − (1+ γ ) u a p ¯ q u ¯ i ∇ j ∇ p ∇ ¯ q u − α ′ ( k − γ ) e − (1+ γ ) u ∇ ¯ i a p ¯ q ∇ j ∇ p ∇ ¯ q u − ∇ ¯ i E j . (6.7)22irect computation gives F p ¯ q ∇ p ∇ ¯ q |∇ ¯ ∇ u | g = 2 g s ¯ i g j ¯ r F p ¯ q ∇ p ∇ ¯ q u ¯ ij u ¯ rs + 2 |∇ ¯ ∇∇ u | F gg . (6.8)Recall (4.21) that we can pass from F p ¯ q to the metric g p ¯ q up to an error of order 2 − .Substituting (6.7) into (6.8) and estimating terms gives F p ¯ q ∇ p ∇ ¯ q |∇ ¯ ∇ u | g ≥ (cid:26) (1 − − ) |∇ ¯ ∇∇ u | g − | α ′ g m ¯ i g j ¯ n σ i ¯ j,r ¯ sk +1 ∇ j u ¯ qp ∇ ¯ i u ¯ sr u ¯ nm | (cid:27) − C |∇ ¯ ∇ u | g |∇ ¯ ∇∇ u | g (cid:26) | Du | g + e − γu |∇ u | g + e − γu e − u (cid:27) − C |∇ ¯ ∇ u | g (cid:26) e − u |∇ ¯ ∇ u | g (cid:27) − k (cid:12)(cid:12)(cid:12)(cid:12) g s ¯ i g j ¯ r ∇ ¯ i ∇ j |∇ u | g u ¯ rs (cid:12)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12) g s ¯ i g j ¯ r ∇ ¯ i E j u ¯ rs (cid:12)(cid:12)(cid:12)(cid:12) . (6.9)The condition u ∈ Υ k (2.6) together with k ≤ ( n −
1) gives C k − n − | α ′ ||∇ ¯ ∇ u | kg ≤ | α ′ | C kn − |∇ ¯ ∇ u | kg ≤ − . (6.10)Therefore by (6.2) | α ′ g m ¯ i g j ¯ n σ p ¯ q,r ¯ sk +1 ∇ j u ¯ qp ∇ ¯ k u ¯ sr u ¯ nm | ≤ − |∇ ¯ ∇∇ u | g . (6.11)In the coming estimates, we will often use the C and C estimates, and the condition u ∈ Υ k (2.6), which we record here for future reference. e − u ≤ CM − , |∇ u | g ≤ CM − , |∇ ¯ ∇ u | g ≤ | α ′ | − /k τ /k , (6.12)where τ = ( C kn − ) − − . Since u ∈ Υ k , we have M = R X e u ˆ ω n ≥
1, and so we will oftenonly keep the leading power of M since M ≥
1. Applying all this to (6.9), we have F p ¯ q ∇ p ∇ ¯ q |∇ ¯ ∇ u | g ≥ − − ) |∇ ¯ ∇∇ u | g − CM − / |∇ ¯ ∇ u | g |∇ ¯ ∇∇ u | g − CM − |∇ ¯ ∇ u | g |∇ ¯ ∇ u | g − k (cid:12)(cid:12)(cid:12)(cid:12) g s ¯ i g j ¯ r ∇ ¯ i ∇ j |∇ u | g u ¯ rs (cid:12)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12) g s ¯ i g j ¯ r ∇ ¯ i E j u ¯ rs (cid:12)(cid:12)(cid:12)(cid:12) . (6.13)We will now estimate the two last terms. We compute the first of these directly, using(4.5) to commute derivatives.2 kg s ¯ i g j ¯ r ∇ ¯ i ∇ j |∇ u | g u ¯ rs = 2 kg s ¯ i g j ¯ r (cid:26) g p ¯ q u ¯ q ∇ j ∇ ¯ i ∇ p u + g p ¯ q u p ∇ ¯ i ∇ j ∇ ¯ q u + g p ¯ q ∇ j ∇ p u ∇ ¯ i ∇ ¯ q u + g p ¯ q u ¯ ip u ¯ qj + g p ¯ q u ¯ q ˆ R ¯ ij ℓp u ℓ − g p ¯ q u ¯ q u ¯ ij u p (cid:27) u ¯ rs . (6.14)23e estimate (cid:12)(cid:12)(cid:12)(cid:12) kg s ¯ i g j ¯ r ∇ ¯ i ∇ j |∇ u | g u ¯ rs (cid:12)(cid:12)(cid:12)(cid:12) ≤ k (cid:26) |∇ ¯ ∇∇ u | g |∇ u | g + 2 |∇ ¯ ∇ u | g + 2 |∇∇ u | g + Ce − u |∇ u | g + 2 |∇ u | g |∇ ¯ ∇ u | g (cid:27) |∇ ¯ ∇ u | g . (6.15)We will use (6.12). Then (cid:12)(cid:12)(cid:12)(cid:12) kg s ¯ i g j ¯ r ∇ ¯ i ∇ j |∇ u | g u ¯ rs (cid:12)(cid:12)(cid:12)(cid:12) ≤ k | α ′ | − /k τ /k |∇ ¯ ∇ u | g + 2 k | α ′ | − /k τ /k |∇∇ u | g (6.16)+ CM − / |∇ ¯ ∇∇ u | g + CM − + CM − . Next, using the definition (5.12) of E j , we keep track of the order of each term and obtainthe estimate | g s ¯ i g j ¯ r ∇ ¯ i E j u ¯ rs | ≤ C ( a, b, c, α ′ ) |∇ ¯ ∇ u | g |∇ ¯ ∇∇ u | g (cid:26) e − γu e − u/ + e − γu |∇ u | g (cid:27) + C ( a, b, c ) |∇ ¯ ∇ u | g (cid:26) e − γu |∇ u | g + e − γu e − u/ |∇ u | g + e − (1+ γ ) u (cid:27) + C ( a, b, c, α ′ ) |∇ ¯ ∇ u | g |∇∇ u | g (cid:26) e − γu |∇ u | g + e − γu e − u/ |∇ u | g + e − (1+ γ ) u (cid:27) + C ( a, b, c, α ′ ) |∇ ¯ ∇ u | g (cid:26) e − (2+ γ ) u + e − (1+ γ ) u e − u/ |∇ u | g + e − (1+ γ ) u |∇ u | g + e − (1+ γ ) u e − u/ |∇ u | g + e − (1+ γ ) u |∇ u | g (cid:27) + C ( µ ) |∇ ¯ ∇ u | g (cid:26) e − ( k +1) u |∇ ¯ ∇ u | g + e − ( k +1) u |∇ u | g + e − ( k +1) u e − u/ |∇ u | g + e − ( k +2) u (cid:27) +( k − γ ) g s ¯ k g j ¯ r | ( α ′ e − (1+ γ ) u a p ¯ q ∇ ¯ k ∇ j ∇ p uu ¯ q ) u ¯ rs | + | k − γ | g s ¯ k g j ¯ r | ( α ′ e − (1+ γ ) u b i ∇ ¯ k ∇ j ∇ i u ) u ¯ rs | + | k − γ | g s ¯ k g j ¯ r | ( α ′ e − (1+ γ ) u γa p ¯ q u ¯ qp u ¯ kj u ¯ rs | +( k − γ ) g s ¯ k g j ¯ r | ( α ′ e − (1+ γ ) u a p ¯ q u ¯ kp u ¯ qj ) u ¯ rs | +( k − γ ) g s ¯ k g j ¯ r | ( α ′ e − (1+ γ ) u a p ¯ q ∇ j ∇ p u ∇ ¯ k ∇ ¯ q u ) u ¯ rs | . (6.17)We will use our estimates (6.12). We also recall the notation − Λˆ g p ¯ q ≤ a p ¯ q ≤ Λˆ g p ¯ q . We usethese estimates and commute covariant derivatives to obtain | g s ¯ k g j ¯ r ∇ ¯ k E j u ¯ rs | ≤ CM − / |∇ ¯ ∇∇ u | g + CM − |∇∇ u | g + CM − + CM − + CM − ( k +1) + CM − ( k +2) +( k − γ ) e − (1+ γ ) u g s ¯ k g j ¯ r | ( α ′ a p ¯ q ∇ j ∇ ¯ k ∇ p uu ¯ q + α ′ a p ¯ q R ¯ kjλp u λ u ¯ q ) u ¯ rs | + | k − γ | e − (1+ γ ) u g s ¯ k g j ¯ r | ( α ′ b i ∇ j ∇ ¯ k ∇ i u + α ′ b i R ¯ kj λi u λ ) u ¯ rs | +2 e − γu | α ′ | Λ( k + γ ) |∇ ¯ ∇ u | g |∇ ¯ ∇ u | g + e − γu | α ′ | Λ( k + γ ) |∇ ¯ ∇ u | g |∇∇ u | g . (6.18)24ince u ∈ Υ k , we have 2 | α ′ | Λ( k + γ ) e − γu ≤ | g s ¯ k g j ¯ r ∇ ¯ k E j u ¯ rs | ≤ | α ′ | − /k τ /k |∇∇ u | g + | α ′ | − /k τ /k |∇ ¯ ∇ u | g + CM − / |∇ ¯ ∇∇ u | g + CM − |∇∇ u | g + CM − . (6.19)Substituting (6.16) and (6.19) into (6.13) and keeping the leading orders of M , we arriveat (6.5). Let G = |∇ ¯ ∇ u | g + Θ |∇ u | g . (6.20)where Θ ≫ n, k, α ′ . To be precise, we letΘ = (1 − − ) − { (1 + 2 k ) | α ′ | − /k τ /k + 1 } . (6.21)By (5.9), F p ¯ q ∇ p ∇ ¯ q |∇ u | g ≥ |∇ ¯ ∇ u | F g + |∇∇ u | F g − |∇ u | g |∇ ¯ ∇∇ u | g −|∇ u | g |∇ ¯ ∇ u | g − Ce − u |∇ u | g . (6.22)Applying (6.12) and converting F p ¯ q to g p ¯ q yields F p ¯ q ∇ p ∇ ¯ q |∇ u | g ≥ (1 − − ) |∇ ¯ ∇ u | g + (1 − − ) |∇∇ u | g − CM − / |∇ ¯ ∇∇ u | g − CM − |∇ ¯ ∇ u | g − CM − . (6.23)Combining (6.5) and (6.23), we have F p ¯ q ∇ p ∇ ¯ q G ≥ − − ) |∇ ¯ ∇∇ u | g + |∇ ¯ ∇ u | g + |∇∇ u | g −|∇ ¯ ∇ u | g |∇ ¯ ∇∇ u | g − CM − / |∇ ¯ ∇∇ u | g − CM − |∇∇ u | g − CM − . (6.24)We will split the linear terms into quadratic terms by applying CM − / |∇ ¯ ∇∇ u | g ≤ |∇ ¯ ∇∇ u | g + C M − , (6.25) |∇ ¯ ∇ u | g |∇ ¯ ∇∇ u | g ≤ |∇ ¯ ∇∇ u | g + 12 |∇ ¯ ∇ u | g . (6.26) CM − |∇∇ u | g ≤ C M − + |∇∇ u | g . (6.27)25pplying these estimates, we may discard the remaining quadratic positive terms and(6.24) becomes F p ¯ q ∇ p ∇ ¯ q G ≥ |∇ ¯ ∇ u | g − CM − , (6.28)Let p ∈ X be a point where G attains its maximum. From the maximum principle, |∇ ¯ ∇ u | g ( p ) ≤ CM − . We conclude from G ≤ G ( p ) that |∇ ¯ ∇ u | g ≤ CM − . (6.29)establishing Theorem 5.We note that many equations involving the derivative of the unknown and/or severalHessians have been studied recently in the literature (see e.g. [3, 4, 7, 13, 15, 29, 26, 28]and references therein). It would be very interesting to determine when estimates withscale hold. Theorem 6
Let u ∈ A k be a C ( X ) function solving equation (2.2). Then |∇ ¯ ∇∇ u | g ≤ C. (7.1) where C only depends on ( X, ˆ ω ) , α ′ , k , γ , k ρ k C ( X, ˆ ω ) and k µ k C ( X ) . To prove this estimate, we will apply the maximum principle to the test function G = ( |∇ ¯ ∇ u | g + η ) |∇ ¯ ∇∇ u | g + B ( |∇ u | g + A ) |∇∇ u | g , (7.2)where A, B ≫ η = mτ /k | α ′ | − /k . We willspecify m ≫ τ = ( C kn − ) − − . The condition (2.6) u ∈ Γ implies | α ′ | /k |∇ ¯ ∇ u | g ≤ τ /k . (7.3)Our choice of constants ensures that η and |∇ ¯ ∇ u | g are of the same α ′ scale. By ourprevious work, we may estimate by C any term involving u , ∇ u , ∇ ¯ ∇ u , or the curvatureor torsion of g = e u ˆ g . Lemma 5
Let u ∈ A k be a C ( X ) function solving equation (2.2). Then for all A ≫ larger than a fixed constant only depending on |∇ u | g and for all B > , F p ¯ q ∇ p ∇ ¯ q n ( |∇ u | g + A ) |∇∇ u | g o ≥ A |∇∇∇ u | g + (1 − − ) |∇∇ u | g − B |∇ ¯ ∇∇ u | g − C ( A, B ) . (7.4) where C ( A, B ) only depends on A , B , ( X, ˆ ω ) , α ′ , k , γ , k ρ k C ( X, ˆ ω ) and k µ k C ( X ) . F p ¯ q ∇ ℓ ∇ j ∇ p ∇ ¯ q u = − α ′ ( k − γ ) ∇ ℓ ( e − (1+ γ ) u a p ¯ q ) ∇ j u ¯ qp − α ′ ( ∇ ℓ σ p ¯ qk +1 ) ∇ j u ¯ qp − k ∇ ℓ ∇ j |∇ u | g − ∇ ℓ E j , (7.5)Commuting derivatives F p ¯ q ∇ p ∇ ¯ q ∇ ℓ ∇ j u = F p ¯ q ∇ ℓ ∇ j ∇ p ∇ ¯ q u + F p ¯ q ∇ p ( ˆ R ¯ qℓλj ∇ λ u − u ¯ qℓ u j ) − F p ¯ q T λpℓ ∇ λ ∇ j ∇ ¯ q u − F p ¯ q ∇ ℓ ( u p ∇ j ∇ ¯ q u − u j ∇ p ∇ ¯ q u ) . (7.6)We compute directly and commute derivatives to derive F p ¯ q ∇ p ∇ ¯ q |∇∇ u | g = 2Re { g ℓ ¯ b g j ¯ d F p ¯ q ∇ p ∇ ¯ q ∇ ℓ ∇ j u ∇ ¯ b ∇ ¯ d u } (7.7)+ g ℓ ¯ b g j ¯ d ∇ ℓ ∇ j uF p ¯ q R ¯ qp ¯ b ¯ λ ∇ ¯ λ ∇ ¯ d u + g ℓ ¯ b g j ¯ d ∇ ℓ ∇ j uF p ¯ q R ¯ qp ¯ d ¯ λ ∇ ¯ b ∇ ¯ λ u + F p ¯ q g ℓ ¯ b g j ¯ d ∇ p ∇ ℓ ∇ j u ∇ ¯ q ∇ ¯ b ∇ ¯ d u + F p ¯ q g ℓ ¯ b g j ¯ d ∇ ¯ q ∇ ℓ ∇ j u ∇ p ∇ ¯ b ∇ ¯ d u. Combining (7.5), (7.6), (7.7) and converting F p ¯ q to g p ¯ q using Lemma 2, we estimate F p ¯ q ∇ p ∇ ¯ q |∇∇ u | g ≥ (1 − − ) |∇∇∇ u | g + (1 − − ) | ¯ ∇∇∇ u | g − α ′ Re { g ℓ ¯ b g j ¯ d σ p ¯ q,r ¯ sk +1 ∇ ℓ u ¯ sr ∇ j u ¯ qp ∇ ¯ b ∇ ¯ d u } − { g ℓ ¯ b g j ¯ d ∇ ℓ E j ∇ ¯ b ∇ ¯ d u }− C |∇∇ u | g ( |∇∇∇ u | g + |∇ ¯ ∇∇ u | g + |∇∇ u | g + 1) (7.8)We used the identity (4.8) to estimate |∇∇ ¯ ∇ u | by |∇ ¯ ∇∇ u | and lower order terms.Next, using Lemma 6.2 we estimate − { α ′ g ℓ ¯ b g j ¯ d σ p ¯ q,r ¯ sk +1 ∇ ℓ u ¯ sr ∇ j u ¯ qp ∇ ¯ b ∇ ¯ d u } ≥ − C k − n − | α ′ ||∇ ¯ ∇ u | k − g |∇∇ u | g |∇ ¯ ∇∇ u | g ≥ − C k − n − τ − (1 /k ) | α ′ | /k |∇∇ u | g |∇ ¯ ∇∇ u | g (7.9)and | g ℓ ¯ b g j ¯ d ∇ ℓ E j ∇ ¯ b ∇ ¯ d u | ≤ C |∇∇ u | g { |∇∇ u | g + |∇ ¯ ∇∇ u | g + |∇∇∇ u | g } . (7.10)Thus F p ¯ q ∇ p ∇ ¯ q |∇∇ u | g ≥ (1 − − ) |∇∇∇ u | g + (1 − − ) | ¯ ∇∇∇ u | g (7.11) − C |∇∇ u | g {|∇ ¯ ∇∇ u | g + |∇∇∇ u | g + |∇ ¯ ∇∇ u | g + |∇∇ u | g + 1 } By (5.14), F p ¯ q ∇ p ∇ ¯ q |∇ u | g ≥ (1 − − ) |∇ ¯ ∇ u | g + (1 − − ) |∇∇ u | g − C |∇∇ u | g − C. (7.12)Direct computation gives F p ¯ q ∇ p ∇ ¯ q n ( |∇ u | g + A ) |∇∇ u | g o = ( |∇ u | g + A ) F p ¯ q ∇ p ∇ ¯ q |∇∇ u | g + |∇∇ u | g F p ¯ q ∇ p ∇ ¯ q |∇ u | g +2Re { F p ¯ q ∇ p |∇ u | g ∇ ¯ q |∇∇ u | g } . (7.13)27e estimate2 (cid:12)(cid:12)(cid:12) F p ¯ q ∇ p |∇ u | g ∇ ¯ q |∇∇ u | g (cid:12)(cid:12)(cid:12) ≤ − ) |∇∇ u | g |∇ u | g | ¯ ∇∇∇ u | g +2(1 + 2 − ) |∇∇ u | g |∇ u | g |∇∇∇ u | g + C | ¯ ∇∇∇ u | g |∇∇ u | g + C |∇∇∇ u | g |∇∇ u | g . (7.14)Substituting (7.11), (7.12), (7.14) into (7.13), F p ¯ q ∇ p ∇ ¯ q { ( |∇ u | g + A ) |∇∇ u | g } ≥ A (1 − − ) n |∇∇∇ u | g + | ¯ ∇∇∇ u | g o + (1 − − ) |∇∇ u | g − |∇∇ u | g |∇ u | g n | ¯ ∇∇∇ u | g + |∇∇∇ u | g o − C ( A ) |∇∇ u | g (cid:26) |∇ ¯ ∇∇ u | g + |∇∇∇ u | g + |∇ ¯ ∇∇ u | g + |∇∇ u | g + |∇∇ u | g + 1 (cid:27) . (7.15)Using 2 ab ≤ a + b ,3 |∇∇ u | g |∇ u | g | ¯ ∇∇∇ u | ≤ − |∇∇ u | g + 2 |∇ u | g | ¯ ∇∇∇ u | g , (7.16)3 |∇∇ u | g |∇ u | g |∇∇∇ u | ≤ − |∇∇ u | g + 2 |∇ u | g |∇∇∇ u | g , (7.17) C ( A ) |∇∇∇ u | g |∇∇ u | g ≤ |∇∇∇ u | g + C ( A ) |∇∇ u | g (7.18) C ( A ) |∇ ¯ ∇∇ u | g |∇∇ u | g ≤ B |∇ ¯ ∇∇ u | g + 2 C ( A ) B |∇∇ u | g (7.19)for a constant B ≫ F p ¯ q ∇ p ∇ ¯ q n ( |∇ u | g + A ) |∇∇ u | g o ≥ n A (1 − − ) − |∇ u | g − o |∇∇∇ u | g + n A (1 − − ) − |∇ u | g − o | ¯ ∇∇∇ u | g +(1 − − ) |∇∇ u | g − B |∇ ¯ ∇∇ u | g (7.20) − C ( A, B ) (cid:26) |∇∇ u | g + |∇∇ u | g + |∇∇ u | g (cid:27) . The terms |∇∇ u | g + |∇∇ u | g + |∇∇ u | g can be absorbed into |∇∇ u | g by Young’s inequality.For A ≫
1, obtain (7.4).
Lemma 6
Let u ∈ A k be a C ( X ) function solving equation (2.2). Then F p ¯ q ∇ p ∇ ¯ q n ( |∇ ¯ ∇ u | g + η ) |∇ ¯ ∇∇ u | g o ≥ |∇ ¯ ∇∇ u | g C |∇∇∇ u | g (cid:26) |∇ ¯ ∇∇ u | g |∇∇ u | g + |∇ ¯ ∇∇ u | g + |∇∇ u | g (cid:27) − C (cid:26) |∇ ¯ ∇∇ u | g |∇∇ u | g + |∇ ¯ ∇∇ u | g |∇∇ u | g + |∇ ¯ ∇∇ u | g |∇∇ u | g + |∇ ¯ ∇∇ u | g |∇∇ u | g + 1 (cid:27) . (7.21) where C only depends on ( X, ˆ ω ) , α ′ , k , γ , k ρ k C ( X, ˆ ω ) and k µ k C ( X ) . To start this computation, we differentiate (6.7). F p ¯ q ∇ i ∇ p ∇ ¯ q u ¯ ℓj = − α ′ ∇ i ( σ p ¯ q,r ¯ sk +1 ) ∇ j u ¯ qp ∇ ¯ ℓ u ¯ sr − α ′ σ p ¯ q,r ¯ sk +1 ∇ i ∇ j u ¯ qp ∇ ¯ ℓ u ¯ sr − α ′ σ p ¯ q,r ¯ sk +1 ∇ j u ¯ qp ∇ i ∇ ¯ ℓ u ¯ sr + ∇ i [ − F p ¯ q u p ∇ ¯ ℓ u ¯ qj + F p ¯ q u j ∇ ¯ ℓ u ¯ qp ]+ ∇ i h − F p ¯ q u ¯ q ∇ p u ¯ ℓj + F p ¯ q u ¯ ℓ ∇ p u ¯ qj i + ∇ i [ F p ¯ q ˆ R ¯ qp ¯ ℓ ¯ λ u ¯ λj − F p ¯ q ˆ R ¯ ℓj λp u ¯ qλ ] − k ∇ i (cid:20) g p ¯ q u ¯ q ∇ j u ¯ ℓp + g p ¯ q u p ∇ ¯ ℓ u ¯ qj + g p ¯ q ∇ j ∇ p u ∇ ¯ ℓ ∇ ¯ q u + g p ¯ q u ¯ ℓp u ¯ qj + g p ¯ q u ¯ q ˆ R ¯ ℓj λp u λ − g p ¯ q u ¯ q u ¯ ℓj u p (cid:21) + ∇ i [ α ′ ( k − γ )(1 + γ ) e − (1+ γ ) u a p ¯ q u ¯ ℓ ∇ j u ¯ qp ] −∇ i [ α ′ ( k − γ ) e − (1+ γ ) u ∇ ¯ ℓ a p ¯ q ∇ j u ¯ qp ] − ∇ i ∇ ¯ ℓ E j . (7.22)Our conventions (4.3) imply the following commutator identities for any tensor W ¯ kj . ∇ p ∇ ¯ q W ¯ kj = ∇ ¯ q ∇ p W ¯ kj + R ¯ qp ¯ k ¯ λ W ¯ λj − R ¯ qpλj W ¯ kλ , (7.23) ∇ p ∇ ¯ q ∇ i W ¯ kj = ∇ i ∇ p ∇ ¯ q W ¯ kj + T λip ∇ λ W ¯ kj − ∇ p [ R ¯ qi ¯ k ¯ λ W ¯ λj − R ¯ qiλj W ¯ kλ ] . (7.24)Thus commuting derivatives gives F p ¯ q ∇ p ∇ ¯ q ∇ i u ¯ kj = F p ¯ q ∇ i ∇ p ∇ ¯ q u ¯ kj + F p ¯ q u i ∇ p ∇ ¯ q u ¯ kj − F p ¯ q u p ∇ i ∇ ¯ q u ¯ kj + F p ¯ q ∇ p [ ˆ R ¯ qiλj u ¯ kλ − ˆ R ¯ qi ¯ k ¯ λ u ¯ λj ] (7.25)We compute the expression for F p ¯ q ∇ p ∇ ¯ q acting on |∇ ¯ ∇∇ u | g , and exchange covariantderivatives to obtain F p ¯ q ∇ p ∇ ¯ q |∇ ¯ ∇∇ u | g = 2Re { g i ¯ d g a ¯ k g j ¯ b F p ¯ q ∇ p ∇ ¯ q ∇ i u ¯ kj ∇ ¯ d u ¯ ba } + F p ¯ q g a ¯ d g e ¯ b g c ¯ f ∇ p ∇ a u ¯ bc ∇ ¯ q ∇ ¯ d u ¯ fe + F p ¯ q g a ¯ d g e ¯ b g c ¯ f ∇ a ∇ ¯ q u ¯ bc ∇ ¯ d ∇ p u ¯ fe + F p ¯ q g a ¯ d g e ¯ b g c ¯ f ∇ a ∇ ¯ q u ¯ bc R ¯ dp ¯ f ¯ λ u ¯ λe − F p ¯ q g a ¯ d g e ¯ b g c ¯ f ∇ a ∇ ¯ q u ¯ bc R ¯ dpλe u ¯ fλ − F p ¯ q g a ¯ d g e ¯ b g c ¯ f R ¯ qa ¯ b ¯ λ u ¯ λc ∇ p ∇ ¯ d u ¯ fe + F p ¯ q g a ¯ d g e ¯ b g c ¯ f R ¯ qaλc u ¯ bλ ∇ p ∇ ¯ d u ¯ fe + g a ¯ d g e ¯ b g c ¯ f ∇ a u ¯ bc F p ¯ q R ¯ qp ¯ d ¯ λ ∇ ¯ λ u ¯ fe + g a ¯ d g e ¯ b g c ¯ f ∇ a u ¯ bc F p ¯ q R ¯ qp ¯ f ¯ λ ∇ ¯ d u ¯ λe − g a ¯ d g e ¯ b g c ¯ f ∇ a u ¯ bc F p ¯ q R ¯ qpλe ∇ ¯ d u ¯ fλ . (7.26)29ubstituting (7.22) and (7.25) into (7.26), and using Lemma 2, we have F p ¯ q ∇ p ∇ ¯ q |∇ ¯ ∇∇ u | g ≥ (1 − − ) |∇∇ ¯ ∇∇ u | g + (1 − − ) |∇ ¯ ∇∇ ¯ ∇ u | g − α ′ Re { g i ¯ d g a ¯ k g j ¯ b ∇ i ( σ p ¯ q,r ¯ sk +1 ) ∇ j u ¯ qp ∇ ¯ k u ¯ sr ∇ ¯ d u ¯ ba }− α ′ Re { g i ¯ d g a ¯ k g j ¯ b σ p ¯ q,r ¯ sk +1 ∇ i ∇ j u ¯ qp ∇ ¯ k u ¯ sr ∇ ¯ d u ¯ ba }− α ′ Re { g i ¯ d g a ¯ k g j ¯ b σ p ¯ q,r ¯ sk +1 ∇ j u ¯ qp ∇ i ∇ ¯ k u ¯ sr ∇ ¯ d u ¯ ba }− C (cid:26) ( |∇ ¯ ∇∇ ¯ ∇ u | g + |∇∇ ¯ ∇∇ u | g ) |∇ ¯ ∇∇ u | g + |∇ ¯ ∇∇ ¯ ∇ u | g +( |∇∇∇ u | g + | ¯ ∇∇∇ u | g + 1) |∇∇ u | g |∇ ¯ ∇∇ u | g + |∇ ¯ ∇∇ u | g + |∇ ¯ ∇∇ u | g + |∇ ¯ ∇∇ u | g (cid:27) − { g i ¯ d g a ¯ k g j ¯ b ∇ i ∇ ¯ k E j ∇ ¯ d u ¯ ba } (7.27)For the following steps, we will use that | α ′ | /k |∇ ¯ ∇ u | g ≤ τ /k for any u ∈ A k , where τ = ( C kn − ) − − . We also recall that we use the notation C ℓm = m ! ℓ !( m − ℓ )! . If k >
1, we canestimate2 | α ′ g i ¯ d g a ¯ ℓ g j ¯ b ∇ i ( σ p ¯ q,r ¯ sk +1 ) ∇ j u ¯ qp ∇ ¯ ℓ u ¯ sr ∇ ¯ d u ¯ ba | ≤ | α ′ | C k − n − |∇ ¯ ∇ u | k − |∇ ¯ ∇∇ u | g ≤ (2 C kn − τ ) | α ′ | /k τ − /k |∇ ¯ ∇∇ u | g = 2 − | α ′ | /k τ − /k |∇ ¯ ∇∇ u | g . (7.28)We used C k − n − ≤ C kn − . If k = 1, the term on the left-hand side vanishes and the inequalitystill holds. Using the same ideas, we can also estimate − α ′ Re { g i ¯ d g a ¯ ℓ g j ¯ b σ p ¯ q,r ¯ sk +1 ∇ i ∇ j u ¯ qp ∇ ¯ ℓ u ¯ sr ∇ ¯ d u ¯ ba } − α ′ Re { g i ¯ d g a ¯ ℓ g j ¯ b σ p ¯ q,r ¯ sk +1 ∇ j u ¯ qp ∇ i ∇ ¯ ℓ u ¯ sr ∇ ¯ d u ¯ ba }≥ − | α ′ | C k − n − |∇ ¯ ∇ u | k − g |∇ ¯ ∇∇ u | g (cid:26) |∇∇ ¯ ∇∇ u | g + |∇ ¯ ∇∇ ¯ ∇ u | g (cid:27) ≥ − (2 C kn − τ ) | α ′ | /k τ − /k |∇ ¯ ∇∇ u | g (cid:26) |∇∇ ¯ ∇∇ u | g + |∇ ¯ ∇∇ ¯ ∇ u | g (cid:27) = − − | α ′ | /k τ − /k |∇ ¯ ∇∇ u | g (cid:26) |∇∇ ¯ ∇∇ u | g + |∇ ¯ ∇∇ ¯ ∇ u | g (cid:27) . (7.29)The perturbative terms can be estimated roughly by using the definition (5.12) of E j andkeeping track of the orders of terms that we do not yet control. − { g i ¯ d g a ¯ k g j ¯ b ∇ i ∇ ¯ k E j ∇ ¯ d u ¯ ba } ≥ − C |∇ ¯ ∇∇ u | g (cid:26) |∇ ¯ ∇∇ ¯ ∇ u | g + |∇ ¯ ∇∇∇ u | g +( |∇ ¯ ∇∇ u | g + |∇∇∇ u | g ) |∇∇ u | g + |∇ ¯ ∇∇ u | g + |∇∇∇ u | g + |∇∇ u | g + |∇∇ u | g + 1 (cid:27) . (7.30)Applying these estimates leads to F p ¯ q ∇ p ∇ ¯ q |∇ ¯ ∇∇ u | g ≥ (1 − − ) h |∇∇ ¯ ∇∇ u | g + |∇ ¯ ∇∇ ¯ ∇ u | g i − − | α ′ | /k τ − /k |∇ ¯ ∇∇ u | g − | α ′ | /k τ − /k |∇ ¯ ∇∇ u | g h |∇∇ ¯ ∇∇ u | g + |∇ ¯ ∇∇ ¯ ∇ u | g i − C P (7.31)where P = |∇ ¯ ∇∇ ¯ ∇ u | g |∇ ¯ ∇∇ u | g + |∇∇ ¯ ∇∇ u | g |∇ ¯ ∇∇ u | g + |∇ ¯ ∇∇ ¯ ∇ u | g + |∇∇∇ u | g |∇ ¯ ∇∇ u | g |∇∇ u | g + |∇∇∇ u | g |∇ ¯ ∇∇ u | g + |∇ ¯ ∇∇ u | g |∇∇ u | g + |∇ ¯ ∇∇ u | g |∇∇ u | g + |∇ ¯ ∇∇ u | g |∇∇ u | g + |∇∇∇ u | g |∇∇ u | g + |∇ ¯ ∇∇ u | g + |∇ ¯ ∇∇ u | g + |∇ ¯ ∇∇ u | g . (7.32)We used the fact that the difference between |∇ ¯ ∇∇∇ u | g and |∇∇ ¯ ∇∇ u | g is a lower orderterm according to the commutation formula (7.23).Next, we apply (6.5) to obtain F p ¯ q ∇ p ∇ ¯ q |∇ ¯ ∇ u | g ≥ |∇ ¯ ∇∇ u | g − C |∇ ¯ ∇∇ u | g − C |∇∇ u | g − C |∇∇ u | g − C. (7.33)We directly compute F p ¯ q ∇ p ∇ ¯ q n ( |∇ ¯ ∇ u | g + η ) |∇ ¯ ∇∇ u | g o = |∇ ¯ ∇∇ u | g F p ¯ q ∇ p ∇ ¯ q |∇ ¯ ∇ u | g +( |∇ ¯ ∇ u | g + η ) F p ¯ q ∇ p ∇ ¯ q |∇ ¯ ∇∇ u | g +2Re { F p ¯ q ∇ p |∇ ¯ ∇ u | g ∇ ¯ q |∇ ¯ ∇∇ u | g } . (7.34)We can estimate2Re { F p ¯ q ∇ p |∇ ¯ ∇ u | g ∇ ¯ q |∇ ¯ ∇∇ u | g } ≥ − − ) |∇ ¯ ∇ u | g |∇ ¯ ∇∇ u | g |∇ ¯ ∇∇ ¯ ∇ u | g (7.35) − − ) |∇ ¯ ∇ u | g |∇ ¯ ∇∇ u | g |∇∇ ¯ ∇∇ u | g ≥ − − ) | α ′ | − /k τ /k |∇ ¯ ∇∇ u | g |∇ ¯ ∇∇ ¯ ∇ u | g − − ) | α ′ | − /k τ /k |∇ ¯ ∇∇ u | g |∇∇ ¯ ∇∇ u | g . Combining (7.31), (7.33), (7.35) with (7.34), setting η = m | α ′ | − /k τ /k and using |∇ ¯ ∇ u | g ≤| α ′ | − /k τ /k leads to F p ¯ q ∇ p ∇ ¯ q n ( |∇ ¯ ∇ u | g + η ) |∇ ¯ ∇∇ u | g o ≥ m (1 − − ) | α ′ | − /k τ /k (cid:26) |∇∇ ¯ ∇∇ u | g + |∇ ¯ ∇∇ ¯ ∇ u | g (cid:27) − − ) | α ′ | − /k τ /k |∇ ¯ ∇∇ u | g (cid:26) |∇∇ ¯ ∇∇ u | g + |∇ ¯ ∇∇ ¯ ∇ u | g (cid:27) − − ( m + 1) | α ′ | − /k τ /k |∇ ¯ ∇∇ u | g (cid:26) |∇∇ ¯ ∇∇ u | g + |∇ ¯ ∇∇ ¯ ∇ u | g (cid:27) + (cid:26) − − ( m + 1) (cid:27) |∇ ¯ ∇∇ u | g − C |∇ ¯ ∇∇ u | g |∇∇ u | g − C P . (7.36)31sing 2 ab ≤ a + b , we estimate4(1 + 2 − ) | α ′ | − /k τ /k |∇ ¯ ∇∇ u | g {|∇ ¯ ∇∇ ¯ ∇ u | g + |∇∇ ¯ ∇∇ u | g }≤ − ) | α ′ | − /k τ /k {|∇ ¯ ∇∇ ¯ ∇ u | g + |∇∇ ¯ ∇∇ u | g } + 12 |∇ ¯ ∇∇ u | g , (7.37)and 2 − ( m + 1) | α ′ | − /k τ /k |∇ ¯ ∇∇ u | g {|∇∇ ¯ ∇∇ u | g + |∇ ¯ ∇∇ ¯ ∇ u | g }≤ | α ′ | − /k τ /k {|∇∇ ¯ ∇∇ u | g + |∇ ¯ ∇∇ ¯ ∇ u | g } + 2 − ( m + 1) |∇ ¯ ∇∇ u | g . (7.38)The main inequality becomes F p ¯ q ∇ p ∇ ¯ q n ( |∇ ¯ ∇ u | g + η ) |∇ ¯ ∇∇ u | g o ≥ { m (1 − − ) − − ) − }| α ′ | − /k τ /k (cid:26) |∇∇ ¯ ∇∇ u | g + |∇ ¯ ∇∇ ¯ ∇ u | g (cid:27) + (cid:26) − − ( m + 1) − − ( m + 1) (cid:27) |∇ ¯ ∇∇ u | g − C |∇ ¯ ∇∇ u | g |∇∇ u | g − C P . (7.39)Next, we estimate terms on the first line in the definition (7.32) of P C {|∇ ¯ ∇∇ ¯ ∇ u | g + |∇∇ ¯ ∇∇ u | g }|∇ ¯ ∇∇ u | g ≤ | α ′ | − /k τ /k {|∇ ¯ ∇∇ ¯ ∇ u | g + |∇∇ ¯ ∇∇ u | g } + 8 C | α ′ | /k τ − /k |∇ ¯ ∇∇ u | g (7.40)and C |∇ ¯ ∇∇ ¯ ∇ u | g ≤ | α ′ | − /k τ /k |∇ ¯ ∇∇ ¯ ∇ u | g + 4 C | α ′ | /k τ − /k (7.41)and absorb |∇ ¯ ∇∇ u | g + |∇ ¯ ∇∇ u | g + |∇ ¯ ∇∇ u | g into 2 − |∇ ¯ ∇∇ u | g plus a large constant. Wecan now let m = 18 and drop the positive fourth order terms. We are left with F p ¯ q ∇ p ∇ ¯ q n ( |∇ ¯ ∇ u | g + η ) |∇ ¯ ∇∇ u | g o ≥ (cid:26) − − ( m + 1) − − ( m + 1) − − (cid:27) |∇ ¯ ∇∇ u | g − C |∇∇∇ u | g (cid:26) |∇ ¯ ∇∇ u | g |∇∇ u | g + |∇ ¯ ∇∇ u | g + |∇∇ u | g (cid:27) − C (cid:26) |∇ ¯ ∇∇ u | g |∇∇ u | g + |∇ ¯ ∇∇ u | g |∇∇ u | g + |∇ ¯ ∇∇ u | g |∇∇ u | g + |∇ ¯ ∇∇ u | g |∇∇ u | g + 1 (cid:27) . (7.42)Since m = 18, 12 − − ( m + 1) − − ( m + 1) − − ≥ − , (7.43)and we obtain (7.21). 32 .3 Using the test function We have computed F p ¯ q ∇ p ∇ ¯ q acting on the two terms of the test function G defined in(7.2). Combining (7.4) and (7.21) F p ¯ q ∇ p ∇ ¯ q G ≥ |∇ ¯ ∇∇ u | g + AB |∇∇∇ u | g + (1 − − ) B |∇∇ u | g − C (cid:26) |∇∇∇ u | g |∇ ¯ ∇∇ u | g |∇∇ u | g + |∇∇∇ u | g |∇ ¯ ∇∇ u | g + |∇∇∇ u | g |∇∇ u | g + |∇ ¯ ∇∇ u | g |∇∇ u | g + |∇ ¯ ∇∇ u | g |∇∇ u | g + |∇ ¯ ∇∇ u | g |∇∇ u | g + |∇ ¯ ∇∇ u | g |∇∇ u | g (cid:27) − C ( A, B ) . The negative terms are readily split and absorbed into the positive terms on the first line.For example, C |∇∇∇ u | g |∇ ¯ ∇∇ u | g |∇∇ u | g ≤ |∇∇∇ u | g + C |∇ ¯ ∇∇ u | g |∇∇ u | g , (7.44) C |∇ ¯ ∇∇ u | g |∇∇ u | g ≤ − |∇ ¯ ∇∇ u | g + 2 C |∇∇ u | g (7.45) C |∇ ¯ ∇∇ u | g |∇∇ u | g ≤ − |∇ ¯ ∇∇ u | g + 2 C |∇∇ u | g . (7.46) C |∇ ¯ ∇∇ u | g |∇∇ u | g ≤ − |∇ ¯ ∇∇ u | g + 2 C |∇∇ u | g . (7.47)This leads to F p ¯ q ∇ p ∇ ¯ q G ≥ − |∇ ¯ ∇∇ u | g + { AB − }|∇∇∇ u | g + { B − C }|∇∇ u | g − C ( A, B ) . (7.48)By choosing A, B ≫ p where G attains a maximum, we have |∇ ¯ ∇∇ u | g ( p ) ≤ C, |∇∇ u | g ( p ) ≤ C. (7.49)Therefore |∇ ¯ ∇∇ u | g and |∇∇ u | g are both uniformly bounded. k = 1 In the case of the standard Fu-Yau equation ( k = 1), to prove Theorem 1 we can insteadappeal to a general theorem of concave elliptic PDE and obtain H¨older estimates for thesecond order derivatives of the solution. To exploit the concave structure, we must rewritethe Fu-Yau equation into the standard form of complex Hessian equation.33ecall that ˆ σ ( χ ) ˆ ω n = nχ ∧ ˆ ω n − , ˆ σ ( χ ) ˆ ω n = n ( n − χ ∧ ˆ ω n − . A direct computationwith equation (1.1) givesˆ σ ( e u ˆ ω + α ′ e − u ρ + 2 α ′ i∂ ¯ ∂u ) = n ( n − e u − n − α ′ e u | Du | ω − n − α ′ µ (7.50)+2( n − α ′ ) e − u ( a j ¯ k u j u ¯ k − b i u i − b ¯ i u ¯ i )+2( n − α ′ ) e − u c + ( n − e − u ˆ σ ( α ′ ρ ) + e − u ˆ σ ( α ′ ρ )We note that the right hand side of the equation involves the given data α ′ , ρ , µ , u and Du . Since u ∈ Υ , the (1 , ω ′ = e u ˆ ω + α ′ e − u ρ + 2 α ′ i∂ ¯ ∂u is positive definite, andthus both sides of the above equation have a positive lower bound. Moreover, our previousestimates imply that we have uniform a priori estimates on k u k C ,β ( X ) for any 0 < β < C β ( X ). 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