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DEGENERATE COMPLEX MONGE-AMPE`RE FLOWS ON STRICTLY PSEUDOCONVEX DOMAINS DO HOANG SON Abstract. We study the equation u˙ = logdet(u ¯)+f(t,z,u) in domains of Cn. αβ This equation has a close connection with the Kähler-Ricci flow. In this paper, we consider the case of the boundary conditions are smooth and the initial conditions 5 are bounded. 1 0 2 r Contents p A Introduction 2 9 1. Strategy of the proof 3 2 2. Preliminaries 4 ] 3. Order 1 a priori estimates 7 V 4. Higher order estimates 11 C 5. C2,α estimate up to the boundary for the parabolic equation 19 . h 6. Proof of the main theorem 28 t a 7. Further directions 32 m References 33 [ 3 v 7 6 1 7 0 . 1 0 5 1 : v i X r a Date: April 30, 2015. 1 2 DO HOANGSON Introduction On Ka¨hler manifolds, a Ka¨hler-Ricci flow is an equation ∂ (1) ω = Ric(ω), ∂t − which starts from a Ka¨hler metric. Here, Ric(ω) is the form associated to the Ricci curvature of ω, i.e., if √ 1 ω = − g dzi dzj, 2π i¯j ∧ then √ 1 Ric(ω) = − (∂ ∂ logdetg)dzi dz¯j. − 2π i j ∧ Thisflowwasbecomeapowefultoolofgeometry. ThetheoryofKa¨hler-Ricciflowiswell developed in the case of compact Ka¨hler manifolds, see e.g. [Cao85], [PS05], [ST07], [Zha09], [Tos10], [GZ13], [BG13]. It can be seen as the parabolic problem associated to an “elliptic” problem which would be the complex Monge-Ampère equation. Monge-Ampère equations and their generalizations have long been studied in strictly pseudoconvex domainsofCn, seeforinstance[CKNS85]. Thisraisesanaturalquestion: what is the behavior of the corresponding parabolic equation in the case of Cn? Let Ω be a bounded smooth strictly pseudoconvex domain of Cn, i.e., there exists a smooth strictly plurisubharmonic function ρ defined on a bounded neighbourhood of Ω¯ such that Ω = ρ < 0 and dρ = 0. ∂Ω { } | 6 Let T (0, ]. We consider the equation ∈ ∞ u˙ = logdet(u )+f(t,z,u) on Ω (0,T), αβ¯ × (2) u = ϕ on ∂Ω [0,T),  ×  u = u on Ω¯ 0 , 0 ×{ } where u˙ = ∂u, u= ∂2u , u is a plurisubharmonic function in a neighbourhood of ∂t αβ¯ ∂zα∂z¯β 0 Ω and f is smooth in [0,T) Ω¯ R and non increasing in the last variable. × × This equation has a close connection with the Ka¨hler-Ricci flow. There are some previousresults. Ifu iscontinuous andϕdoesnotdependonthelastvariable, then(2) 0 admits a unique viscosity solution [EGZ14]. If u is a smooth strictly plurisubharmonic 0 function in Ω¯, ϕ is smooth in Ω¯ [0,T) and the compatibility conditions are satisfied, then (2) admits a unique solutio×n u C∞(Ω (0,T)) C2;1(Ω¯ [0,T)) [HL10]; we ∈ × ∩ × state their result in detail as Theorem 2.2 in Section 2. In this paper, we study the case where ϕ is smooth and u is merely bounded. The 0 main result is the following: Theorem 0.1. Let Ω be a bounded smooth strictly pseudoconvex domain of Cn and T (0, ]. Let u be a bounded plurisubharmonic function defined on a neighbourhood 0 ∈ ∞ Ω˜ of Ω. Assume that ϕ C∞(Ω¯ [0,T)) and f C∞([0,T) Ω¯ R) satisfying ∈ × ∈ × × (i) f 0. u ≤ (ii) ϕ(z,0) = u (z) for z ∂Ω. 0 ∈ Then there exists a unique function u C∞(Ω¯ (0,T)) such that ∈ × (3) u(.,t) is a strictly plurisubharmonic function on Ω for all t (0,T), ∈ PARABOLIC COMPLEX MONGE-AMPE`RE EQUATIONS 3 (4) u˙ = logdet(u )+f(t,z,u) on Ω (0,T), αβ¯ × (5) u = ϕ on ∂Ω (0,T), × (6) limu(z,t) = u (z) z Ω¯. 0 t→0 ∀ ∈ Moreover, u L∞(Ω¯ [0,T′)) for any 0 < T′ < T, and u(.,t) also converges to u in 0 ∈ × capacity when t 0. ˜ → ¯ If u C(Ω) then u C(Ω [0,T)). 0 ∈ ∈ × Here, we say that u(.,t) converges to u in capacity if the convergence is uniform 0 outside sets of arbitrarily small capacity. Thisimprovesthemainresultof[HL10]intwodirections: wedonotneedsmoothness of the initial data, and still have continuity when t 0; and we obtain the maximal → possible regularity when z tends to ∂Ω, for fixed t > 0. Some techniques used in this paper are from the corresponding result in the case of compact Ka¨hler manifolds. On a compact Ka¨hler manifold, results have been obtained in the more general case where u has zero or even positive Lelong numbers. We refer 0 the reader to [GZ13] and [DL14] for the details. Acknowledgements. I am deeply grateful to Pascal Thomas and Vincent Guedj for many inspiring discussions on the subject and encouragement me to write down this paper. It is improved significantly thanks to their thorough reading and editing. I also would like to thank Lu Hoang Chinh for very useful discussions about Proposition 3.3. 1. Strategy of the proof We fix some notation. We say that u C2;1(Ω¯ [0,T)) if u(.,t) C2(Ω¯) for any t [0,T), u(z,.) C1([0,T)) for any∈z Ω¯ a×nd u˙,u C(Ω¯ ∈ [0,T)) for ∈ ∈ ∈ sjsk ∈ × s ,s x ,y ,...x ,y . j k 1 1 n n ∈ { } In order to prove Theorem 0.1, we use an approximation process and we first will need to prove the following a priori estimates theorem: Theorem 1.1. Let Ω be a bounded smooth strictly pseudoconvex domain of Cn and T > 0. Let ϕ C∞(Ω¯ [0,T)) and f C∞([0,T) Ω¯ R) and let u C∞(Ω (0,T)) C2;1(Ω¯∈ [0,T))×, strictly plurisu∈bharmonic wi×th re×spect to z, be a∈solution o×f ∩ × the equation (7) u˙ = logdet(u )+f(t,z,u) on Ω (0,T). αβ¯ × Assume that (8) u = ϕ , ∂Ω×[0,T) ∂Ω×[0,T) | | (9) sup u(z,0) C , u | | ≤ (10) f (t,z,u) 0 (t,z,u) (0,T) Ω R, u ≤ ∀ ∈ × × (11) f C , C2((0,T)×Ω×R) f k k ≤ (12) ϕ C . C4(Ω×(0,T)) ϕ k k ≤ 4 DO HOANGSON Then there exists M = M (Ω,T,C ,C ,C ) and for any 0 < ǫ < T there exists 0 0 u ϕ f C = C(Ω,ǫ,T,C ,C ,C ) such that u ϕ f u M on Ω (0,T), 0 | | ≤ × u + u˙ +∆u C on Ω (ǫ,T). |∇ | | | ≤ × Remark 1.2. In the theorem above, we denote ϕ = sup DjDlϕ , k kCk(Ω×(0,T)) | s t | Ω×(0,T) |j|+2l≤k X f = sup Dj1Dj2Dj3f , k kCk((0,T)×Ω×R)) | t s u | j1+|jX2|+j3≤k where s = (s ,...,s ) = (x ,y ,...,x ,y ). 1 2n 1 1 n n For the proof of Theorem 0.1, the strategy is as follows. + Construct the solutions u C∞(Ω (0,T)) C2;1(Ω¯ [0,T)) of (4) such that m ∈ × ∩ × u and u converge pointwise, respectively, to u and ϕ . m|Ω¯×{0} m|∂Ω×(0,T) 0 |∂Ω×(0,T) We also ask that the u be uniformly bounded and u = ϕ m m|∂Ω×(ǫm,T) |∂Ω×(ǫm,T) for some ǫ 0. m ց + Use the a priori estimates to prove kumkC2(Ω¯×(ǫ,T′)) ≤ Cǫ,T′ for any 0 < ǫ < T′ < T, where Cǫ,T′ > 0 is independent of m. + Use C2,α estimates and to prove um Ck(Ω¯×(ǫ,T′)) Ck,ǫ,T′ k k ≤ for any 0 < ǫ < T′ < T and k > 0, where Ck,ǫ,T′ > 0 is independent on m. The C2,α estimates and the Ck,α regularity will be mentioned in section 5. + By Ascoli’s theorem, there exists a subsequence of u , denoted also by u , m m and u C∞(Ω¯ (0,T) such that { } { } ∈ × Ck(Ω¯×(ǫ,T′)) u u. m −→ Then, u satisfies (3), (4) and (5). + Use Comparison principle to prove (6). + Finally, we prove the uniqueness of u. We will study some important tools before we prove Theorem 0.1. In Section 2, we introduce some basic results about parabolic complex Monge-Ampère equations. In Sections 3 and 4, we prove the a priori estimates theorem (Theorem 1.1). In Section 5 we establish the C2,α estimate needed to solve our problem. Finally in Section 6 we prove Theorem 0.1. 2. Preliminaries 2.1. Hou-Li theorem. The Hou-Li theorem states that equation (2) has a unique solution when the conditions are good enough. We will use it in Section 6 to obtain smooth solutions to an approximating problem, to which we then will apply the a priori estimates from Theorem 1.1. PARABOLIC COMPLEX MONGE-AMPE`RE EQUATIONS 5 We first need the notion of subsolution. Definition 2.1. A function u C∞(Ω¯ [0,T)) is called a subsolution of the equation ∈ × (14) if and only if u(.,t)is a strictly plurisubharmonic function, u˙ logdet(u) +f(t,z,u), (13)  ≤ αβ¯ u|∂Ω×(0,T) = ϕ|∂Ω×(0,T), u(.,0) u . 0 ≤  Theorem 2.2. Let Ω Cn be a bounded domain with smooth boundary. Let T  ⊂ ∈ (0, ]. Assume that ∞ ϕ is a smooth function in Ω¯ [0,T). • f is a smooth function in [0,×T) Ω¯ R non increasing in the lastest variable. • × × u is a smooth strictly plurisubharmonic funtion in a neighborhood of Ω. 0 • u (z) = ϕ(z,0), z ∂Ω. 0 • ∀ ∈ The compatibility condition is satisfied, i.e. • ϕ˙ = logdet(u ) +f(t,z,u ), (z,t) ∂Ω 0 . 0 αβ¯ 0 ∀ ∈ ×{ } There exists a subsolution to the equation (14). • Then there exists a unique solution u C∞(Ω (0,T)) C2;1(Ω¯ [0,T)) of the equation ∈ × ∩ × u˙ = logdet(u )+f(t,z,u) on Ω (0,T), αβ¯ × (14) u = ϕ on ∂Ω [0,T),  ×  u = u on Ω¯ 0 . 0 ×{ } Remark 2.3. (i) There is a corresponding result in the case of a compact Kähler  manifold. On the compact Kähler manifold X, we must assume that 0 < T < T , where T depends on X. In the case of domain Ω Cn, we can max max assume that T = + if ϕ,u are defined on Ω¯ [0,+ ) and f⊂is defined on [0,+ ) Ω¯ R. ∞ × ∞ (ii) If Ω∞is a×bou×nded smooth strictly pseudoconvex domain of Cn then one can prove that a subsolution always exists, and so Theorem 2.2 does not need the additional assumpation of existence of a subsolution. 2.2. Maximum principle. The following maximum principle is a basic tool to establish upper and lower bounds in the sequel (see [BG13] and [IS13] for the proof). Theorem 2.4. Let Ω be a bounded domain of Cn and T > 0. Let ω be a t 0<t<T { } continuous family of continuous positive definite Hermitian forms on Ω. Denote by ∆ t the Laplacian with respect to ω : t nωn−1 ddcf ∆ f = t ∧ , f C∞(Ω). t ωn ∀ ∈ t Suppose that H C∞(Ω (0,T)) C(Ω¯ [0,T)) and satisfies ∈ × ∩ × 6 DO HOANGSON (ω +ddcH )n (∂ ∆ )H 0 or H˙ log t t . ∂t − t ≤ t ≤ ωn t Then sup H = sup H. Here we denote ∂ (Ω (0,T)) = ∂Ω (0,T) Ω¯ 0 . P × × ∪ ×{ } Ω¯×[0,T) ∂P(Ω×[0,T)) Corollary 2.5. (Comparison principle) Let Ω be a bounded domain of Cn and T (0, ]. Let u,v C∞(Ω (0,T)) C(Ω¯ [0,T)) satisfying ∈ ∞ ∈ × ∩ × u(.,t) and v(.,t) are strictly plurisubharmonic functions for any t [0,T), • ∈ u˙ logdet(u )+f(t,z,u), • ≤ αβ¯ v˙ logdet(v )+f(t,z,v), • ≥ αβ¯ where f C∞([0,T) Ω¯ R) is non increasing in the last variable. ∈ × × Then sup (u v) max 0, sup (u v) . − ≤ { − } Ω×(0,T) ∂P(Ω×(0,T)) Corollary 2.6. Let Ω be a bounded domain of Cn and T (0, ]. We denote by L a ∈ ∞ operator on C∞(Ω (0,T)) satisfying × ∂f ∂2f L(f) = a b.f, ∂t − αβ¯∂z ∂z¯ − α β X where a ,b C(Ω (0,T)), (a (z,t)) are positive definite Hermitian matrices and αβ¯ ∈ × αβ¯ b(z,t) < 0. Assume that φ C∞(Ω (0,T)) C(Ω¯ [0,T)) satisfies ∈ × ∩ × L(φ) 0. ≤ Then φ max(0, sup φ). ≤ ∂P(Ω×(0,T)) 2.3. The Laplacian inequalities. We shall need two standard auxiliary results (see [Yau78], [Siu87] for a proof). Theorem 2.7. Let ω ,ω be positive (1,1)-forms on a complex manifold X.Then 1 2 ωn 1/n ωn n 1 tr (ω ) n 1 (tr (ω ))n−1, ωn ≤ ω2 1 ≤ ωn ω1 2 (cid:18) 2(cid:19) (cid:18) 2(cid:19) nωn−1 ω where tr (ω ) = 1 ∧ 2. ω1 2 ωn 1 Theorem 2.8. Let ω, ω′ be two Kähler forms on a complex manifold X. If the holo- morphic bisectional curvature of ω is bounded below by a constant B R on X,then ∈ tr Ric(ω′) ∆ω′ logtrω(ω′) ≥ − tωr (ω′) +Btrω′(ω), ω where Ric(ω′) is the form associated to the Ricci curvature of ω′. Remark 2.9. Applying Theorem 2.8 for ω = ddc z 2 and ω′ = ddcu, we have | | ∆logdet(u ) uαβ¯(log∆u) αβ¯ . αβ¯ ≥ ∆u X PARABOLIC COMPLEX MONGE-AMPE`RE EQUATIONS 7 2.4. Construction of subsolutions. We give a first construction which will be used in the proof of Theorem 1.1. First we need a notion of subsolution weaker than the one in Definition 2.1. Definition 2.10. We say that a function u C∞(Ω¯ [0,T)) is a subsolution of the ∈ × equation (7) if u˙ logdet(u )+f(t,z,u). ≤ αβ¯ We will construct subsolutions of (7) inorder toprove some estimates onthe boundary. Let ρ SPSH(Ω¯) C∞(Ω¯) be a function which defines Ω. We also assume that infρ = ∈1. Let ζ C∞∩(R) such that 0 ζ 1, ζ = 1 and ζ = 0. [0,1] [2,∞) − ∈ ≤ ≤ | | Let ϕ and u be as in Theorem 1.1. For any m > 0, we denote the function ϕ 0 m C∞(Ω¯ [0,T)) by the formula ∈ × ϕ = ϕ Osc(u ) ζ(mt). m 0 − · Then there exists M > 0 depending on ρ,T,C ,C ,C such that the function u = m u ϕ f m ϕ +M ρ satisfies m m u˙ logdet(u ) +f(t,z,u ) on Ω (0,T), m ≤ m αβ¯ m × ddc(u ) ddc z 2 on Ω [0,T). m ≥ | | × Then u is a subsolution of (7). Moreover, m u u , m|∂P(Ω×(0,T)) ≤ |∂P(Ω×(0,T)) u = ϕ . m|∂Ω×(m2,T) |∂Ω×(m2,T) By the maximum principle, we have u u on Ω (0,T). m ≤ × In the next two sections, we will prove Theorem 1.1. For convenience, we define an operator L on C∞(Ω (0,T)) by the formula × (15) L(φ) = φ˙ uαβ¯φ f (t,z,u)φ, − αβ¯− u where u is the function in Theorem 1X.1 and (uαβ¯) is the transpose of inverse matrix of Hessian matrix (u ). αβ¯ 3. Order 1 a priori estimates In this section, we will estimate u, u˙ and u . |∇ | Clearly, u u sup ϕ on Ω (0,T). 1 ≤ ≤ × ∂Ω×(0,T) Then M 2sup ϕ C u(z,t) sup ϕ, 1 u − − | |− ≤ ≤ ∂Ω×(0,T) where M is the constant defined in 2.4. Let C = M +2C +C , we obtain 1 1 1 ϕ u (16) sup u C . 1 | | ≤ 8 DO HOANGSON 3.1. Bounds on u˙. Proposition 3.1. There exists C > 0 depending only on T,C ,C such that 2 f 1 t u˙ C on Ω (0,T). 2 | | ≤ × Proof. Take L as in (15), then L(tu˙ u) = tu¨ t uαβ¯u˙ +n (tu˙ u)f (t,z,u). − − αβ¯ − − u By equation (7), we have X tu¨ = t uαβ¯u˙ +t.f (t,z,u)+tu˙.f (t,z,u). αβ¯ t u X Then ′ ′ C L(tu˙ u) = n+t.f (t,z,u)+u.f (t,z,u) C , − 2 ≤ − t u ≤ 2 ′ where C = n+C (T +C ) > 0. 2 f 1 ′ ′ Since L(tu˙ u C t) 0 and L(tu˙ u + C t) 0, by the maximum principle, we − − 2 ≤ − 2 ≥ obtain ′ ′ ′ tu˙ u C t sup (tu˙ u C t) (C +C )T +C , − − 2 ≤ − − 2 ≤ ϕ 2 1 ∂P(Ω×(0,T)) ′ ′ ′ tu˙ u+C t inf (tu˙ u+C t) (C +C )T C . − 2 ≥ ∂P(Ω×(0,T)) − 2 ≥ − ϕ 2 − 1 Thus t u˙ C on Ω (0,T), where C = (C +2C′)T +2C . (cid:3) | | ≤ 2 × 2 ϕ 2 1 3.2. Gradient estimates. Proposition 3.2. Let m > 2. Then there exists C = C (Ω,M ,C ) > 0 such that T 3 3 m ϕ u C on ∂Ω (2,T). |∇ | ≤ 3 × m Proof. Let h C∞(Ω¯ [0,T)) be a spatial harmonic function (i.e. harmonic with ∈ × respect to z) satisfying h = ϕ on ∂Ω [0,T). × Then taking u as 2.4 , we have m u u h on Ω (2,T), m ≤ ≤ × m u = u = h = ϕ on ∂Ω (2,T). m × m Hence (u u ) (h u ) on ∂Ω (2,T). |∇ − m | ≤ |∇ − m | × m Thus u u + (h u ) C on ∂Ω (2,T), |∇ | ≤ |∇ m| |∇ − m | ≤ 3 × m where C > 0 depends only on Ω,C ,M . 3 ϕ m (cid:3) Proposition 3.3. Assume that m,C satisfy Proposition 3.2 and 2 < ǫ < T. Then 3 m there exists C = C (Ω,m,ǫ,T,C ,C ,C ,C ) > 0 such that 4 4 f 1 2 3 PARABOLIC COMPLEX MONGE-AMPE`RE EQUATIONS 9 u C on Ω (ǫ,T). 4 |∇ | ≤ × Proof. We will use the technique of Blocki as in [Blo08]. In this proof only, we denote 2 g(t) = nlog(t ), − m 1 1 γ(u) = Au Bu2 where A = ,B = , − 10C 20C2 1 1 1 η = , 4(diamΩ)2 φ = log u 2 +γ(u)+g(t)+η z 2, |∇ | | | and we assume that 0 Ω. ∈ Let ǫ < T′ < T, we will prove that sup φ C˜ , 4 ≤ Ω×(2,T′) m where C˜ depends on Ω,C ,C ,C ,m,T,C . 4 1 2 3 f Notice that the hypotheses and previous bounds on u imply that, for t (2,T′), | | ∈ m 2 (17) expφ(z,t) u(z,t) 2(t )nexp max γ(u)+ηmax z C u 2, ≤ |∇ | − m Ω×(2,T′) Ω | |! ≤ |∇ | m and in a similar way 2 u(z,t) 2 C(ǫ )−nexpφ(z,t) C expφ(z,t), t (ǫ,T′), ǫ |∇ | ≤ − m ≤ ∈ so the bound on φ yields a bound on u(z,t) . |∇ | Suppose that sup φ = φ(z ,t ). 0 0 Ω×(2,T′) m By anorthogonal change of coordinates, we can assume that (u (z ,t )) is diagonal. αβ¯ 0 0 For convenience, we denote u (z ,t ) = λ . αα¯ 0 0 α We also denote by L the operator ∂ ∂2 L = uαβ¯ . ∂t − ∂z ∂z¯ α β X If u 2(z ,t ) C, by (17), we are done. In particular, if z ∂Ω, we know that 0 0 0 |∇ | ≤ ∈ u(z,t) is bounded. So we may restrict attention to the case where u 2(z ,t ) > 1 0 0 |∇ | |∇ | 2 and (z ,t ) Ω ( ,T′]. Then L(φ) 0. 0 0 ∈ × m |(z0,t0) ≥ We compute L(φ) = L(log u 2)+γ′(u).u˙ +g′(t) γ′(u) uαβ¯u |∇ | − αβ¯ γ′′(u) uαβ¯u u η uαα¯ P − α β¯− = L(logP u 2)+γ′(u).(Pu˙ n)+g′(t) |∇ | − γ′′(u) uαβ¯u u η uαα¯. − α β¯− P P 10 DO HOANGSON When u = 0, we have |∇ | 6 u 2 u 2 u 2 (log u 2) = |∇ |αβ¯ |∇ |α|∇ |β¯ |∇ | αβ¯ u 2 − u 4 |∇ | |∇ | = h∇uαβ¯,∇ui + h∇u,∇uβα¯i + h∇uα,∇uβi u 2 u 2 u 2 u|∇, |u u|∇2 |u 2 |∇ | +h∇ β¯ ∇ α¯i |∇ |α|∇ |β¯. u 2 − u 4 |∇ | |∇ | L(log u 2) = h∇u˙,∇ui− huαβ¯∇uαβ¯,∇ui + h∇u,∇u˙i− h∇u,uβα¯∇uβα¯i |∇ | u 2 u 2 |P∇ | |P∇ | u , u + u , u ( u 2) ( u 2) uαβ¯h∇ α ∇ βi h∇ β¯ ∇ α¯i + uαβ¯ |∇ | α |∇ | β¯. − u 2 u 4 |∇ | |∇ | P P We have, by (7), u, f u 2 + u 2 L(log u 2) = 2Re h∇ ∇ i +2f (t,z,u) u 2 |∇ k| |∇ k¯| |∇ | |(z0,t0) u 2 u |∇ | − λ u 2 (cid:18) |∇ | (cid:19) k|∇ | + (|∇u|2)k(|∇u|2)k¯ P λ u 4 k |∇ | P2|∇f| + (|∇u|2)k(|∇u|2)k¯. ≤ u λ u 4 k |∇ | |∇ | P ′ ′ Hence, there exists C = C (m,C ,C ,C ) such that 4 4 1 2 f u 2 1 ( u 2) ( u 2) L(φ) C′ +g′(t) γ′′(u) | k| η + |∇ | k |∇ | k¯. |(z0,t0) ≤ 4 − λ − λ λ u 4 k k k |∇ | P P P By the condition ∂φ(z ,t ) = 0, we have ∂zk 0 0 ( u 2) ( u 2) η |∇ | k |∇ | k¯ = γ′(u)u +ηz¯ 2 2(γ′(u))2 u 2 +2η2 z 2 2(γ′(u))2 u 2 + , u 4 | k k| ≤ | k| | k| ≤ | k| 2 |∇ | where (z,t) = (z ,t ). 0 0 Then u 2 η 1 0 L(φ) C′ +g′(t)+(2(γ′(u))2 γ′′(u)) | k| ≤ |(z0,t0) ≤ 4 − λ − 2 λ k k C′ +g′(t) a( |uk|2 + 1 )P, P ≤ 4 − λ λ k k ηP P where a := min 2B (A+BC ), . Hence, at (z ,t ) 1 0 0 { − 2} u 2 1 1 (18) | k| + (C′ +g′(t)). λ λ ≤ a 4 k k X X ′′ ′′ Moreover, by Proposition 3.1 and by (16), there exists C = C (m,C ,C ) such that 4 4 1 2 (19) λ λ ...λ = det(u ) = eu˙−f(t,z,u) C′′. 1 2 n αβ¯ ≤ 4