In this paper, we consider the parabolic Hessian quotient equation on compact Hermitian manifolds. By setting up a priori estimates of the admissible solutions, we prove the long-time existence of the solution to the parabolic Hessian quotient equation and its convergence. As an application, we show the solvability of a class of complex Hessian quotient equations, which generalizes the relevant results.
Citation: Jundong Zhou, Yawei Chu. The complex Hessian quotient flow on compact Hermitian manifolds[J]. AIMS Mathematics, 2022, 7(5): 7441-7461. doi: 10.3934/math.2022416
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In this paper, we consider the parabolic Hessian quotient equation on compact Hermitian manifolds. By setting up a priori estimates of the admissible solutions, we prove the long-time existence of the solution to the parabolic Hessian quotient equation and its convergence. As an application, we show the solvability of a class of complex Hessian quotient equations, which generalizes the relevant results.
Let (M,ω) be a complex n-dimensional compact Hermitian manifold and χ be a smooth real (1, 1)-form on (M,ω). Γωk is the set of all real (1, 1)-forms whose eigenvalues belong to the k-positive cone Γk. For any u∈C2(M), we can get a new (1, 1)-form
χu:=χ+√−1∂¯∂u. |
In any local coordinate chart, χu can be expressed as
χu=√−1(χi¯j+ui¯j)dzi∧d¯zj. |
In this article, we study the following form of parabolic Hessian quotient equations
{∂u(x,t)∂t=logCknχku∧ωn−kClnχlu∧ωn−l−logϕ(x,u), (x,t)∈M×[0,T),u(x,0)=u0(x), x∈M, | (1.1) |
where 0≤l<k≤n, [0,T) is the maximum time interval in which the solution exists and ϕ(x,z)∈C∞(M×R) is a given strictly positive function.
The study of the parabolic flows is motivated by complex equations
χku∧ωn−k=ClnCknϕ(x,u)χlu∧ωn−l, χu∈Γωk. | (1.2) |
Equation (1.2) include some important geometry equations, for example, complex Monge-Ampère equation and Donaldson equation [6], which have attracted extensive attention in mathematics and physics since Yau's breakthrough in the Calabi conjecture [28]. Since Eq (1.2) are fully nonlinear elliptic, a classical way to solve them is the continuity method. Using this method, the complex Monge-Ampère equation
χnu=ϕ(x)ωn, χu∈Γωn |
was solved by Yau [28]. Donaldson equation
χnu=∫Mχn∫Mχ∧ωn−1χu∧ωn−1, χu∈Γωn |
was independently solved by Li-Shi-Yao [11], Collins-Szèkelyhidi [3] and Sun [17]. Equation (1.2) also include the complex k-Hessian equation and complex Hessian quotient equation, which, respectively, correspond to
Cknχku∧ωn−k=ϕ(x)ωn, χu∈Γωk, |
χku∧ωn−k=ClnCknϕ(x)χlu∧ωn−l, χu∈Γωk. |
Dinew and Kolodziej [7] proved a Liouville type theorem for m-subharmonic functions in Cn, and combining with the estimate of Hou-Ma-Wu [10], solved the complex k-Hessian equation by using the continuity method. Under the cone condition, Sun [16] solved the complex Hessian quotient equation by using the continuity method. There have been many extensive studies for complex Monge-Ampère equation, Donaldson equation, the complex k-Hessian equation and the complex Hessian quotient equation on closed complex manifolds, see, e.g., [4,12,20,22,23,29,30]. When the right hand side function ϕ in Eq (1.2) depends on u, that is ϕ=ϕ(x,u), it is interesting to ask whether we can solve them. We intend to solve (1.2) by the parabolic flow method.
Equation (1.1) covers some of the important geometric flows in complex geometry. If k=n and l=0, (1.1) is known as the complex Monge-Ampère flow
∂u(x,t)∂t=logχnuωn−logϕ(x), (x,t)∈M×[0,T), |
which is equivalent to the Kähler-Ricci flow. The result of Yau [28] was reproduced by Cao [2] through Kähler-Ricci flow. Using the complex Monge-Ampère flow, similar results on a compact Hermitian manifold and a compact almost Hermitian manifold were proved by Gill [9] and Chu [5], respectively. To study the normalized twisted Chern-Ricci flow
∂ωt∂t=−Ric(ωt)−ωt+η, |
which is equivalent to the following Mong-Ampère flow
∂φ∂t=log(θt+ddcφ)nΩ−φ, |
Tô [25,26] considered the following complex Monge-Ampère flow
∂φ∂t=log(θt+ddcφ)nΩ−F(t,x,φ), |
where Ω is a smooth volume form on M. From this, we can see that the given function ϕ depends on u in some geometric flows. If l=0, (1.1) is called as the complex k-Hessian flow
∂u(x,t)∂t=logCknχku∧ωn−kωn−logϕ(x,u), (x,t)∈M×[0,T). |
The solvability of complex k-Hessian flow was showed by Sheng-Wang [21].
In this paper, our research can be viewed as a generalization of Tô's work in [26] and Sheng-Wang's work in [21]. To solve the complex Hessian quotient flow, the condition of the parabolic C-subsolution is needed. According to Phong and Tô [14], we can give the definition of the parabolic C-subsolution to Eq (1.1).
Definition 1.1. Let u_(x,t)∈C2,1(M×[0,T)) and χu_∈Γωk, if there exist constants δ, R>0, such that for any (x,t)∈M×[0,T),
logσk(λ)σl(λ)−∂tu_≤logϕ(x,u_), λ−λ(u_)+δI∈Γn, |
implies that
|λ|<R, |
then u_ is said to be a parabolic C-subsolution of (1.1), where λ(u_) denotes eigenvalue set of χu_.
Obviously, we can give the equivalent definition of parabolic C-subsolution of (1.1).
Definition 1.2. Let u_(x,t)∈C2,1(M×[0,T)) and χu_∈Γωk, if there exist constant ˜δ>0, for any (x,t)∈M×[0,T), such that
limμ→∞logσk(λ(u_)+μei)σl(λ(u_)+μei)>∂u_∂t+˜δ+logϕ(x,u_), 1≤i≤n, | (1.3) |
then u_ is said to be a parabolic C-subsolution of (1.1).
Our main result is
Theorem 1.3. Let (M,g) a compact Hermitian manifold and χ be a smooth real (1,1)-form on M. Assume there exists a parabolic C-subsolutionu_ for Eq (1.1) and
∂tu_≥max{supM(logσk(λ(u0))σl(λ(u0))−logϕ(x,u0)),0}, | (1.4) |
ϕz(x,z)ϕ>cϕ>0, | (1.5) |
where cϕ is a constant.Then there exits a unique smooth solution u(x,t) to (1.1) all timewith
supx∈M(u0(x)−u_(x,0))=0. | (1.6) |
Moreover, u(x,t) isC∞ convergent to a smooth function u∞, which solves Eq (1.2).
The rest of this paper is organized as follows. In Section 2, we give some important lemmas and estimate on |ut(x,t)|. In Section 3, we prove C0 estimates of Eq (1.1) by the parabolic C-subsolution condition and the Alexandroff-Bakelman-Pucci maximum principle. In Section 4, using the parabolic C-subsolution condition, we establish the C2 estimate for Eq (1.1) by the method of Hou-Ma-Wu [10]. In Section 5, we adapt the blowup method of Dinew and Kolodziej [7] to obtain the gradient estimate. In Section 6, we give the proof of the long-time existence of the solution to the parabolic equation and its convergence, that is Theorem 1.3.
In this section, we give some notations and lemmas. In holomorphic coordinates, we can set
ω=√−1gi¯jdzi∧d¯zj=√−1δijdzi∧d¯zj, χ=√−1χi¯jdzi∧d¯zj, |
χu=√−1(χi¯j+ui¯j)dzi∧d¯zj=√−1Xi¯jdzi∧d¯zj, |
χu_=√−1(χi¯j+u_i¯j)dzi∧d¯zj=√−1X_i¯jdzi∧d¯zj. |
λ(u) and λ(u_) denote the eigenvalue set of {Xi¯j} and {X_i¯j} with respect to {gi¯j}, respectively. In local coordinates, (1.1) can be written as
∂tu=logσk(λ(u))σl(λ(u))−logϕ(x,u). | (2.1) |
For simplicity, set
F(λ(u))=logσk(λ(u))σl(λ(u)), |
then (2.1) is abbreviated as
∂tu=F(λ(u))−logϕ(x,u). | (2.2) |
We use the following notation
Fi¯j=∂F∂Xi¯j, F=∑iFi¯i, Fi¯j,p¯q=∂2F∂Xi¯j∂Xp¯q. |
For any x0∈M, we can choose a local holomorphic coordinates such that the matrix {Xi¯j} is diagonal and X1¯1≥⋯≥Xn¯n, then we have, at x0∈M,
λ(u)=(λ1,⋯,λn)=(X1¯1,⋯,Xn¯n), |
Fi¯j=Fi¯iδij=(σk−1(λ|i)σk−σl−1(λ|i)σl)δij, F1¯1≤⋯≤Fn¯n. |
To prove a priori C0-estimate for solution to Eq (1.1), we need the following variant of the Alexandroff-Bakelman-Pucci maximum principle, which is Proposition 10 in [20].
Lemma 2.1. [20] Let v:B(1)→R be a smooth function, which meets the conditionv(0)+ϵ≤inf∂B(1)v, whereB(1) denotes the unit ball in Rn.Define the set
Ω={x∈B(1):|Dv(x)|<ϵ2, and v(y)≥v(x)+Dv(x)⋅(y−x),∀y∈B(1)}. |
Then there exists a costant c0>0 such that
c0ϵn≤∫Ωdet(D2v). |
Next, we give an estimate on |ut(x,t)|.
Lemma 2.2. Under the assumption of Theorem 1.3, let u(x,t) be a solution to (1.1). Then for any (x,t)∈M×[0,T), we have
min{infMut(x,0),0}≤ut(x,t)≤max{supMut(x,0),0}. | (2.3) |
Furthermore, there is a constant C>0 such that
supM×[0,T)|∂tu(x,t)|≤supM|∂tu(x,0)|≤C, |
where C depends on H=|u0|C2(M) and |ϕ|C0(M×[−H,H]).
Proof. Differentiating (2.2) on both sides simultaneously at t, we obtain
(ut)t=Fi¯jXi¯jt−ϕzϕut=Fi¯j(ut)i¯j−ϕzϕut. | (2.4) |
Set uεt=ut−εt, ε>0. For any T′∈(0,T), suppose uεt achieves its maximum Mt at (x0,t0)∈M×[0,T′]. Without loss of generality, we may suppose Mt≥0. If t0>0, From the parabolic maximum principle and (2.4), we get
0≤(uεt)t−Fi¯j(uεt)i¯j+ϕzϕuεt≤(ut)t−ε−Fi¯j(ut)i¯j+ϕzϕut−εϕzϕt0≤−ε−εϕzϕt0. |
This is obviously a contradiction, so t0=0 and
supM×[0,T′]uεt(x,t)=supMut(x,0), |
that is
supM×[0,T′]ut(x,t)=supM×[0,T′](uεt(x,t)+εt)≤supMut(x,0)+εT′. |
Letting ε→0, we obtain
supM×[0,T′]ut(x,t)≤supMut(x,0). |
Since T′∈(0,T) is arbitrary, we have
supM×[0,T)ut(x,t)≤supMut(x,0). | (2.5) |
Similarly, setting uεt=ut+εt, ε>0, we obtain
infM×[0,T)ut(x,t)≥infMut(x,0). | (2.6) |
(2.1) yields
|ut(x,0)|=|logσk(λ(u0))σl(λ(u0))−logϕ(x,u0)|≤C. | (2.7) |
Combining (2.5)–(2.7), we complete the proof of Proposition 2.2.
From the concavity of F(λ(u)) and the condition of the parabolic C-subsolution, we give the following lemma, which plays an important role in the estimation of C2.
Lemma 2.3. Under the assumption of Theorem 1.3 and assuming that X1¯1≥⋯≥Xn¯n, there exists two positive constantsN and θ such that we have either
Fi¯i(u_i¯i−ui¯i)−∂t(u_−u)≥θ(1+F) | (2.8) |
or
F1¯1≥θN(1+F). | (2.9) |
Proof. Since u_ is a parabolic C-subsolution to Eq (1.1), from Definition 1.2, there are uniform constants ˜δ>0 and N>0, such that
logσk(λ(u_)+Ne1)σl(λ(u_)+Nue1)>∂u_∂t+˜δ+logϕ(x,u_). | (2.10) |
If ϵ>0 is sufficiently small, it can be obtained from (2.10)
logσk(λ(u_)−ϵI+Ne1)σl(λ(u_)−ϵI+Nue1)≥∂u_∂t+˜δ+logϕ(x,u_). |
Set λ′=λ(u_)−ϵI+Ne1, then
F(λ′)≥∂u_∂t+˜δ+logϕ(x,u_). | (2.11) |
Using the concavity of F(λ(u)) gives
Fi¯i(u_i¯i−ui¯i)=Fi¯i({X_i¯i−Xi¯i)=Fi¯i({X_i¯i−ϵδii+Nδi1−Xi¯i)+ϵF−NF1¯1≥F(λ′)−F(λ(u))+ϵF−NF1¯1. | (2.12) |
From Lemma 2.2 and (1.4), we obtain
u_t(x,t)≥ut(x,t), ∀(x,t)∈M×[0,T). | (2.13) |
In addition, it can be obtained from the condition (1.6)
u_(x,0)≥u(x,0), ∀x∈M×[0,T). | (2.14) |
(2.13) and (2.14) deduce that
u_(x,t)≥u(x,t), ∀(x,t)∈M×[0,T). | (2.15) |
It follows from this that
ϕ(x,u_)≥ϕ(x,u). | (2.16) |
Combining (2.2), (2.11) and (2.16) gives that
F(λ′)−F(λ(u))≥u_t(x,t)−ut(x,t)+˜δ+logϕ(x,u_)−logϕ(x,u)≥u_t(x,t)−ut(x,t)+˜δ. | (2.17) |
Put (2.17) into (2.12)
Fi¯i(u_i¯i−ui¯i)≥u_t(x,t)−ut(x,t)+˜δ+ϵF−NF1¯1≥˜δ+ϵF−NF1¯1. |
Let
θ=min{˜δ2, ϵ2}. |
If F1¯1N≤θ(1+F), Inequality (2.8) is obtained, otherwise Inequality (2.9) must be true.
In this section, we prove the C0 estimates by the existence of the parabolic C-subsolution and the Alexandroff-Bakelman-Pucci maximum principle.
Proposition 3.1. Under the assumption of Theorem1.3, let u(x,t) be a solution to (1.1). Then there exists a constant C>0 such that
|u(x,t)|C0(M×[0,T))≤C, |
where C depends on |u0|C2(M) and |u_|C2(M×[0,T)).
Proof. Combining (2.13), (2.14) and ∂ϕ(x,z)∂z≥0 yields
u_t(x,t)+logϕ(x,u_)≥ut(x,t)+logϕ(x,u). | (3.1) |
Let's rewrite Eq (2.2) as
F(λ(u))=∂tu+logϕ(x,u). | (3.2) |
when fix t∈[0,T), Eq (3.2) is elliptic. From (3.1), we see that the parabolic C-subsolution u_(x,t) is a C-subsolution to Eq (3.2) in the elliptic sense. From (2.15), we have
supM×[0,T)(u−u_)=0. |
Our goal is to obtain a lower bound for L=infM×t(u−u_). Note that λ(u)∈Γk, which implies that λ(u)∈Γ1, then Δ(u−u_)≥−˜C, where Δ is the complex Laplacian with respect to ω. According to Tosatti-Weinkove's method [22], we can prove that ‖u−u_‖L1(M) is bounded uniformly. Let G:M×M→R be the associated Green's function, then, by Yau [28], there is a uniform constant K such that
G(x,y)+K≥0, ∀(x,y)∈M×M, and ∫y∈MG(x,y)ωn(y)=0. |
Since
supM×[0,T)(u−u_)=0, |
then for fixed t∈[0,T) there exists a point x0∈M such that (u−u_)(x0,t)=0. Thus
(u−u_)(x0,t)=∫M(u−u_)dμ−∫y∈MG(x0,y)Δ(u−u_)(y)ωn(y)=∫M(u−u_)dμ−∫y∈M(G(x0,y)+K)Δ(u−u_)(y)ωn(y)≤∫M(u−u_)dμ+˜CK∫Mωn, |
that is
∫M(u_−u)dμ=∫M|(u−u_)|dμ≤˜CK∫Mωn. |
Let us work in local coordinates, for which the infimum L is achieved at the origin, that is L=u(0,t)−u_(0,t). We write B(1)={z:|z|<1}. Let v=u−u_+ϵ|z|2, for a small ϵ>0. We have infv=L=v(0), and v(z)≥L+ϵ for z∈∂B(1). From Lemma 2.1, we get
c0ϵ2n≤∫Ωdet(D2v). | (3.3) |
At the same time, if x∈Ω, then D2v(x)≥0 implies that
ui¯j(x)−u_i¯j(x)+ϵδij≥0. |
If ϵ is sufficiently small, then
λ(u)∈λ(u_)−δI+Γn. |
Set μ=λ(u)−λ(u_). Since λ(u) satisfies Eq (3.2), then
F(λ(u_)+μ)=∂tu+logϕ(x,u), μ+δI∈Γn. | (3.4) |
u_ is a C-subsolution to Eq (3.2) in the elliptic sense, so there is a uniform constant R>0, such that
|μ|≤R, |
which means |vi¯j|≤C, for any x∈Ω. As in Blocki [1], for x∈Ω, we have D2v(x)≥0 and so
D2v(x)≤22ndet(vi¯j)2≤C′. |
From this and (3.3), we obtain
c0ϵ2n≤∫Ωdet(D2v)≤C′⋅vol(Ω). | (3.5) |
On the other hand, by the definition of Ω in Lemma 2.1, for x∈Ω, we get
v(0)≥v(x)−Dv(x)⋅x>v(x)−ϵ2, |
and so
|v(x)|>|L+ϵ2|. |
It follows that
∫M|v(x)|≥∫Ω|v(x)|≥|L+ϵ2|⋅vol(Ω). | (3.6) |
Since ‖u−u_‖L1(M) is bounded uniformly, ∫M|v(x)| is also bounded uniformly. If L is very large, Inequality (3.6) contradicts (3.5), which means that L has a lower bound. For any t∈[0,T), Inequality (3.1) holds, thus
|u(x,t)|C0(M×[0,T))≤|L|+supM×[0,T)|u_|≤C. |
In this section, we prove that the second-order estimates are controlled by the square of the gradient estimate linearly. Our calculation is a parabolic version of that in Hou-Ma-Wu [10].
Proposition 4.1. Under the assumption of Theorem1.3, let u(x,t) be a solution to (1.1). Then there exists a constant ˜C such that
supM×[0,T)|√−1∂¯∂u|≤˜C(supM×[0,T)|∇u|2+1), |
where ˜C depends χ, ω, |ϕ|C2(M×[−C,C]), |u_|C2(M×[0,T)), |∂tu_|C0(M×[0,T)) and |u0|C2(M).
Proof. Let λ(u)=(λ1,…,λn) and λ1 is the maximum eigenvalue. For any T′<T, we consider the following function
W(x,t)=logλ1+φ(|∇u(x,t)|2)+ψ(u(x,t)−u_(x,t)), (x,t)∈M×[0,T′], | (4.1) |
where φ and ψ are determined later. We want to apply the maximum principle to the function W. Since the eigenvalues of the matrix {Xi¯j} with respect to ω need not be distinct at the point where W achieves its maximum, we will perturb {Xi¯j} following the technique of [20]. Let W achieve its maximum at (x0,t0)∈M×[0,T′]. Near (x0,t0), we can choose local coordinates such that {Xi¯j} is diagonal with X1¯1≥⋯≥Xn¯n, and λ(u)=(X1¯1,⋯,Xn¯n). Let D be a diagonal matrix such that D11=0 and 0<D22<⋯<Dnn are small, satisfying Dnn<2D22. Define the matrix ˜X=X−D. At (x0,t0), ˜X has eigenvalues
˜λ1=λ1, ˜λi=λi−Dii, n≥i≥2. |
Since all the eigenvalues of ˜X are distinct, we can define near (x0,t0) the following smooth function
˜W=log˜λ1+φ(|∇u|2)+ψ(u−u_), | (4.2) |
where
φ(s)=−12log(1−s2K), 0≤s≤K−1, |
ψ(s)=−Elog(1+s2L), −L+1≤s≤L−1, |
K=supM×[0,T′]|∇u|2+1, |
L=supM×[0,T′]|u|+supM×[0,T′]|u_|+1, |
E=2L(C1+1), |
and C1>0 is to be chosen later. Direct calculation yields
0<14K≤φ′≤12K, φ″=2(φ′)2>0, | (4.3) |
and
C1+1≤−ψ′≤2(C1+1), ψ″≥4ϵ1−ϵ(ψ′)2,∀ ϵ≤14E+1. | (4.4) |
Without loss of generality, we can assume that λ1>1. From here on, all calculations are done at (x0,t0). From the maximum principle, calculating the first and second derivatives of the function ˜W gives
0=˜Wi=˜λ1,iλ1+φ′(|∇u|2)i+ψ′(u−u_)i,1≤i≤n, | (4.5) |
0≥˜Wi¯i=˜λ1,i¯iλ1−˜λ1,i˜λ1,¯iλ21+φ′(|∇u|2)i¯i+φ″|(|∇u|2)i|2+ψ′(u−u_)i¯i+ψ″|(u−u_)i|2. | (4.6) |
0≤˜Wt=˜λ1,tλ1+φ′(|∇u|2)t+ψ′(u−u_)t. | (4.7) |
Define
L:=Fi¯j∇∂∂¯zj∇∂∂zi−∂t. |
Obviously,
0≥L˜W=Llog˜λ1+Lφ(|∇u|2)+Lψ(u−u_). | (4.8) |
Next, we will estimate the terms in (4.8). Direct calculation shows that
Llog˜λ1=Fi¯i˜λ1,i¯iλ1−Fi¯i|˜λ1,i|2λ21−˜λ1,tλ1. | (4.9) |
According to Inequality (78) in [20], we have
˜λ1,i¯i≥Xi¯i1¯1−2Re(Xi¯11¯T1i1)−C0λ1, | (4.10) |
where C0 depending χ, ω, |ϕ|C2(M×[−C,C]), |u_|C2(M×[0,T)), |∂tu_|C0(M×[0,T)) and |u0|C2(M)). From here on, C0 can always absorb the constant it represents before, and can change from one line to the next, but it does not depend on the parameter we choose later. By calculating the covariant derivatives of (4.7) in the direction ∂∂z1 and ∂∂¯z1, we obtain
ut1=Fi¯iXi¯i1−(logϕ)1−(logϕ)uu1, | (4.11) |
and
ut1¯1=Fi¯j,p¯qXi¯j1Xp¯q¯1+Fi¯iXi¯i1¯1−(logϕ)1¯1−(logϕ)1uu¯1−(logϕ)u¯1u1−(logϕ)uu|u1|2−(logϕ)uu1¯1. | (4.12) |
Notice that
X1¯1i=χ1¯1i+u1¯1i=(χ11i−χi11+Tpi1χp¯1)+Xi¯11−T1i1λ1, | (4.13) |
therefore
|X1¯1i|2≤|Xi¯11|2−2λ1Re(Xi¯11¯T1i1)+C0(λ21+|X1¯1i|). | (4.14) |
Combining (4.14) with
˜λ1,i=X1¯1i−(D11)i, |
gives
−Fi¯i|˜λ1,i|2λ21=−Fi¯i|X1¯1i|2λ21+2λ21Fi¯iRe(X1¯1i(D11)¯i)−Fi¯i|(D11)i|2λ2i≥−Fi¯i|X1¯1i|2λ21−C0λ21Fi¯i|X1¯1i|−C0F≥−Fi¯i|Xi¯11|2λ21+2λ1Fi¯iRe(Xi¯11¯T1i1)−C0λ21Fi¯i|X1¯1i|−C0F. | (4.15) |
Let's set λ1≥K, otherwise the proof is completed. Putting 4.10–4.12 and (4.15) into (4.9) yields
Llog˜λ1≥−Fi¯j,p¯qXi¯j1Xp¯q¯1λ1−Fi¯i|Xi¯11|2λ21−C0λ1Fi¯i|X1¯1i|λ1−C0F+(logϕ)1¯1+(logϕ)1uu¯1+(logϕ)u¯1u1+(logϕ)uu|u1|2−(logϕ)uχ1¯1λ1≥−Fi¯j,p¯qXi¯j1Xp¯q¯1λ1−Fi¯i|Xi¯11|2λ21−C0λ1Fi¯i|X1¯1i|λ1−C0F−C0 | (4.16) |
A simple computation gives
Lφ(|∇u|2)=φ′Fi¯i(|∇u|2)i¯i+φ″Fi¯i|(|∇u|2)i|2−φ′(|∇u|2)t. | (4.17) |
Next, we estimate the formula (4.17). Differentiating Eq (2.2), we have
(ut)p=Fi¯iXi¯ip−(logϕ)p−(logϕ)uup, | (4.18) |
and
(ut)¯p=Fi¯iXi¯i¯p−(logϕ)¯p−(logϕ)uu¯p. | (4.19) |
It follows from (4.18) and (4.19) that
∂t|∇u|2=∑putpu¯p+∑pupu¯t¯p=Fi¯i∑p(Xi¯ipu¯p+Xi¯i¯pup)−∑p(logϕ)pu¯p−∑p(logϕ)¯pup−2∑p(logϕ)u|∇u|2. | (4.20) |
By commuting derivatives, we have the identity
Fi¯iXi¯ip=Fi¯iui¯ip+Fi¯iχi¯ip=Fi¯iupi¯i−Fi¯iTqpiuq¯i−Fi¯iuqR qi¯ip+Fi¯iχi¯ip. | (4.21) |
Direct calculation gives
Fi¯i(|∇u|2)i¯i=∑pFi¯i(upi¯iu¯p+u¯pi¯iup)+∑pFi¯i(upiu¯p¯i+up¯iu¯pi). | (4.22) |
It follows from (4.21) that
Fi¯iupi¯iu¯p−Fi¯iXi¯ipu¯p=Fi¯iTqpiuq¯iu¯p+Fi¯iu¯puqR qi¯ip−Fi¯iχi¯ipu¯p≥−C0K12Fi¯iXi¯i−C0K12F−C0KF. | (4.23) |
Noticing that
Fi¯iXi¯i=σk−1(λ|i)σkλi−σl−1(λ|i)σlλi=k−l, | (4.24) |
from this and (4.23), we obtain
Fi¯iupi¯iu¯p−Fi¯iXi¯ipu¯p≥−C0K12−C0K12F−C0KF. | (4.25) |
In the same way, we can get
Fi¯iu¯pi¯iup−Fi¯iXi¯i¯pup≥−C0K12−C0K12F−C0KF. | (4.26) |
Using (4.20)–(4.26) in (4.17), we have
Lφ(|∇u|2)≥φ″Fi¯i|(|∇u|2)i|2+φ′(−C0K12−C0K12F−C0KF)+φ′∑p((logϕ)pu¯p+(logϕ)¯pup+2(logϕ)u|∇u|2)+∑pFi¯i(|upi|2+|u¯pi|2)≥φ″Fi¯i|(|∇u|2)i|2+∑pFi¯i(|upi|2+|u¯pi|2)−C0−C0F. | (4.27) |
A simple calculation gives
Lψ(u−u_)=ψ″Fi¯i|(u−u_)i|2+ψ′[Fi¯i(u−u_)i¯i−∂t(u−u_)]. | (4.28) |
Substituting (4.27), (4.28) and (4.16) into (4.8),
0≥−Fi¯j,p¯qXi¯j1Xp¯q¯1λ1−Fi¯i|Xi¯11|2λ21−C0λ1Fi¯i|X1¯1i|λ1+φ″Fi¯i|(|∇u|2)i|2+φ′∑pFi¯i(|upi|2+|u¯pi|2)−C0−C0F+ψ″Fi¯i|(u−u_)i|2+ψ′[Fi¯i(u−u_)i¯i−∂t(u−u_)]. | (4.29) |
Let δ>0 be a sufficiently small constant to be chosen later and satisfy
δ≤min{11+4E,12}. | (4.30) |
We separate the rest of the calculations into two cases.
Case 1: λn<−δλ1.
Using(4.5), we find that
−Fi¯i|X1¯1i|2λ21=−Fi¯i|φ′(|∇u|2)i+ψ′(u−u_)i−(D11)iλ1|2≥−2(φ′)2Fi¯i|(|∇u|2)i|2−2Fi¯i|ψ′(u−u_)i−(D11)iλ1|2≥−2(φ′)2Fi¯i|(|∇u|2)i|2−C0|ψ′|2KF−C0F. | (4.31) |
From (4.13), we have
|Xi¯11|2λ21≤|X1¯1i|2λ21+C0(1+|X1¯1i|λ1). | (4.32) |
Combining (4.31) with (4.32), we conclude that
−Fi¯i|Xi¯11|2λ21≥−2(φ′)2Fi¯i|(|∇u|2)i|2−C0|ψ′|2KF−C0F−C0Fi¯i|X1¯1i|λ1. | (4.33) |
Note that the operator F is concave, which implies that
−Fi¯j,p¯qXi¯j1Xp¯q¯1λ1≥0. | (4.34) |
Applying (4.33) and (4.34) to (4.29) and using φ″=2(φ′)2 yield that
0≥−C0Fi¯i|X1¯1i|λ1−C0λ1Fi¯i|X1¯1i|λ1+φ′∑pFi¯i(|upi|2+|u¯pi|2)−C0|ψ′|2KF−C0F−C0+ψ″Fi¯i|(u−u_)i|2+ψ′[Fi¯i(u−u_)i¯i−∂t(u−u_)]. | (4.35) |
Note that the fact
|X1¯1i|λ1=|−φ′(upiu¯p+upu¯pi)−ψ′(u−u_)i+(D11)iλ1|, |
It follows that
−C0Fi¯i|X1¯1i|λ1≥−C0φ′K−12Fi¯i(|upi|+|u¯pi|)+C0ψ′K12F−C0F. | (4.36) |
Using the following inequality
K−12(|upi|+|u¯pi|)≤14C0(|upi|2+|u¯pi|2)+C0K. |
deduces
−C0Fi¯i|X1¯1i|λ1≥−14φ′Fi¯i(|upi|2+|u¯pi|2)+C0ψ′K12F−C0F. | (4.37) |
Note that λ1>1, we have
−C0λ1Fi¯i|X1¯1i|λ1≥−14φ′Fi¯i(|upi|2+|u¯pi|2)+C0ψ′K12F−C0F. | (4.38) |
Since ψ″>0, which implies that
ψ″Fi¯i|(u−u_)i|2≥0. | (4.39) |
According to Lemma 2.3, there are at most two possibilities:
(1) If (2.8) holds true, then
ψ′[Fi¯i(u−u_)i¯i−∂t(u−u_)]≥θ(1+F)|ψ′|. | (4.40) |
Substituting (4.37)—(4.40) into (4.35) and using φ′≥14K yield that
0≥18K∑pFi¯i(|upi|2+|u¯pi|2)−C0|ψ′|2KF−C0(F+1)+θ(1+F)|ψ′|≥18KFi¯iλ2i−C0(C1+1)2KF+θ(C1+1)(F+1)−C0(F+1)≥δ2λ218nKF−C0(C1+1)2KF+θ(C1+1)(F+1)−C0(F+1). | (4.41) |
We may set θC1≥C0. It follows from (4.41) that λ1≤˜CK.
(2) If (2.9) holds true,
F1¯1>θN(1+F). | (4.42) |
According to ψ′<0 and the concavity of the operator F, we have
ψ′[Fi¯i(u−u_)i¯i−∂t(u−u_)]=ψ′[Fi¯i(Xi¯i−X_i¯i)−∂t(u−u_)]≥ψ′[F(χu)−F(χu_)−∂tu+∂tu_]=ψ′[ϕ(x,u)+∂tu_−F(χu_)]≥C0ψ′. | (4.43) |
Using (4.37)—(4.39) and (4.43) in (4.35), together with (4.42), we find that
0≥18KFi¯iλ2i−C0|ψ′|2KF+C0ψ′−C0(F+1)≥θλ218NK(1+F)+δ2λ218nKF−C0(C1+1)2KF−C0(C1+1)−C0(F+1). | (4.44) |
Let λ1 be sufficiently large, so that
θλ218NK(1+F)−C0(C1+1)−C0(F+1)≥0, |
It follows from (4.44)that \lambda_1\leq \widetilde{C}K .
Case 2: \lambda_{n}\geq-\delta \lambda_{1} .
Let
I = \big\{i\in\{1,\cdots, n\}|F^{i\overline{i}} > \delta^{-1}F^{1\overline{1}}\big\}. |
Let us first treat those indices which are not in I . Similar to (4.31), we obtain
\begin{align} -\sum\limits_{i\notin I}\frac{F^{i\overline{i}}|X_{1\overline{1}i}|^2}{\lambda_{1}^2} \geq -2(\varphi^{'})^2\sum\limits_{i\notin I}F^{i\overline{i}}\big|(|\nabla u|^2)_{i}\big|^2 -\frac{C_0K}{\delta}|\psi^{'}|^2F^{1\overline{1}}-C_0\mathcal{F}. \end{align} | (4.45) |
Using (4.32) yields that
\begin{align} -\sum\limits_{i\notin I}\frac{F^{i\overline{i}}|X_{i\overline{1}1}|^2}{\lambda_{1}^2} \geq& -2(\varphi^{'})^2\sum\limits_{i\notin I}F^{i\overline{i}}\big|(|\nabla u|^2)_{i}\big|^2 -\frac{C_0K}{\delta}|\psi^{'}|^2F^{1\overline{1}}\\ &-C_0\sum\limits_{i\notin I}\frac{F^{i\overline{i}}|X_{1\overline{1}i}|}{\lambda_{1}} -C_0\mathcal{F}. \end{align} | (4.46) |
Since
-F^{i\overline{1},1\overline{i}} = \frac{F^{i\overline{i}}-F^{1\overline{1}}} {X_{1\overline{1}}-X_{i\overline{i}}} \ \rm{and}\quad \lambda_{i}\geq\lambda_{n}\geq-\delta \lambda_{1}, |
which implies that
\begin{equation} -\sum\limits_{i\in I}F^{i\overline{1},1\overline{i}}\geq \frac{1-\delta}{1+\delta}\frac{1}{\lambda_{1}}\sum\limits_{i\in I}F^{i\overline{i}},\notag \end{equation} |
It follows that
\begin{align} -\frac{F^{i\overline{1},1\overline{i}}|X_{i\overline{1}1}|^2} {\lambda_{1}}\geq& \frac{1-\delta}{1+\delta}\sum\limits_{i\in I}F^{i\overline{i}}\frac{|X_{i\overline{1}1}|^2}{\lambda_1^2}. \end{align} | (4.47) |
Recalling \varphi'' = 2(\varphi')^2 and 0 < \delta\leq\frac{1}{2} , we obtain from (4.5) that
\begin{align} &\sum\limits_{i\in I}\varphi^{''}F^{i\overline{i}}\big|(|\nabla u|^2)_i\big|^2\\ = & 2\sum\limits_{i\in I}F^{i\overline{i}}\bigg|\frac{X_{i\overline{1}1}}{\lambda_1}+\psi'(u-\underline{u})_i +\frac{\chi_{11i}-\chi_{i11}+T^p_{i1}\chi_{p\overline{1}}-(D^{11})_i}{\lambda_1}\bigg|^2\\ \geq& 2 \sum\limits_{i\in I}F^{i\overline{i}}\bigg(\delta\bigg|\frac{X_{i\overline{1}1}}{\lambda_{1}}\bigg|^2 -\frac{2\delta}{1-\delta}(\psi^{'})^2|(u-\underline{u})_i|^2-C_0\bigg)\\ \geq& 2\delta \sum\limits_{i\in I}F^{i\overline{i}}\bigg|\frac{X_{i\overline{1}1}}{\lambda_{1}}\bigg|^2 -\frac{4\delta}{1-\delta}(\psi^{'})^2F^{i\overline{i}}|(u-\underline{u})_i|^2-C_0\mathcal{F}. \end{align} | (4.48) |
Notice that \psi^{''}\geq\frac{4\epsilon}{1-\epsilon}(\psi^{'})^2 if \epsilon = \frac{1}{4E+1} . Since \frac{1}{4E+1}\geq \delta , we get that
\begin{equation} \psi''F^{i\overline{i}}|(u-\underline{u})_i|^2 -\frac{4\delta}{1-\delta}(\psi^{'})^2F^{i\overline{i}}|(u-\underline{u})_i|^2 \geq 0. \end{equation} | (4.49) |
Take (4.46)–(4.49) into (4.29),
\begin{align} 0\geq& -C_0\sum\limits_{i\notin I}F^{i\overline{i}}\frac{|X_{1\overline{1}i}|}{\lambda_1} -\frac{C_0}{\lambda_1}F^{i\overline{i}}\frac{|X_{1\overline{1}i}|}{\lambda_1} +\varphi'\sum\limits_pF^{i\overline{i}}(|u_{pi}|^2+|u_{\overline{p}i}|^2) \\ &-C_0\mathcal{F}-C_0-\frac{C_0K}{\delta}|\psi^{'}|^2F^{1\overline{1}} +\psi'[F^{i\overline{i}}(u-\underline{u})_{i\overline{i}}-\partial_t(u-\underline{u})]. \end{align} | (4.50) |
Similar to (4.37) and (4.38), by using the third term of (4.50) to absorb the first two terms of it, We get that
\begin{align} 0\geq&\frac{1}{8K}\sum\limits_pF^{i\overline{i}}(|u_{pi}|^2+|u_{\overline{p}i}|^2)-\frac{C_0K}{\delta}|\psi^{'}|^2F^{1\overline{1}} \\ &-C_0\mathcal{F}-C_0 +\psi'[F^{i\overline{i}}(u-\underline{u})_{i\overline{i}}-\partial_t(u-\underline{u})]\\ \geq&\frac{1}{8K}\sum\limits_pF^{i\overline{i}}\lambda_i^2-\frac{C_0K}{\delta}|\psi^{'}|^2F^{1\overline{1}} -C_0(\mathcal{F}+1) \\ & +\psi'[F^{i\overline{i}}(u-\underline{u})_{i\overline{i}}-\partial_t(u-\underline{u})] . \end{align} | (4.51) |
According to Lemma2.3, there are at most two possibilities:
(1) If (2.8) holds true, then
\begin{align} \psi'[F^{i\overline{i}}(u-\underline{u})_{i\overline{i}}-\partial_t(u-\underline{u})] \geq \theta(1+\mathcal{F})|\psi'|. \end{align} | (4.52) |
Put (4.52) into (4.51)
\begin{align} 0\geq&\frac{1}{8K}\sum\limits_pF^{i\overline{i}}\lambda_i^2-\frac{C_0K}{\delta}|\psi^{'}|^2F^{1\overline{1}} -C_0(\mathcal{F}+1)+\theta(1+\mathcal{F})|\psi'|\\ \geq&\frac{1}{8K}\sum\limits_pF^{1\overline{1}}\lambda_1^2-\frac{C_0K}{\delta}(1+C_1)^2F^{1\overline{1}}-C_0(\mathcal{F}+1)+\theta(1+\mathcal{F})(1+C_1). \end{align} | (4.53) |
Here, C_1 is determined finally, such that
\theta C_1\geq C_0. |
It follows from (4.53) that
\lambda_1\leq \widetilde{C}K. |
(2) If (2.9) holds true,
\begin{align} F^{1\overline{1}} > \frac{\theta}{N}(1+\mathcal{F}). \end{align} | (4.54) |
Substituting (4.43) into (4.51) and using (4.54) give that
\begin{align} 0\geq& \frac{1}{8K}\lambda_{1}^2 -\frac{C_0K}{\delta}|(1+C_1)|^2-\frac{N}{\theta}C_0(1+C_1)-\frac{N}{\theta}C_0 \end{align} | (4.55) |
It follows that
\lambda_1\leq \widetilde{C}K. |
To obtain the gradient estimates, we adapt the blow-up method of Dinew and Kolodziej [7] and reduce the problem to a Liouville type theorem which is proved in [7].
Proposition 5.1. Under the assumption of Theorem 1.3, let u(x, t) be a solution to (1.1). Then there exists a uniform constant \widetilde{C} such that
\begin{equation} \sup\limits_{M\times [0,T)}|\nabla u|\leq \widetilde{C}. \end{equation} | (5.1) |
Proof. Suppose that the gradient estimate (5.1) does not hold. Then there exists a sequence (x_m, t_m)\in M\times [0, T) with t_m\rightarrow T such that
\sup\limits_{M\times [0,t_m]}|\nabla u(x,t)| = |\nabla u(x_m,t_m)|\ \rm{and} \lim\limits_{m\rightarrow \infty}|\nabla u(x_m,t_m)| = \infty. |
After passing to a subsequence, we may assume that \lim_{m\rightarrow \infty}x_m = x_0\in M . We choose a coordinate chart \{U, (z_1, \cdots, z_n)\} at x_0 , which we identify with an open set in \mathbb{C}^n , and such that \omega(0) = \beta = \sqrt{-1}\sum_{i}dz^i\wedge d\overline{z}^i . We may assume that the open set contains \overline{B_1(0)} and m is sufficiently large so that z_m = z(x_m)\in B_1(0) . Define
|\nabla u(x_m,t_m)| = C_m,\ \ \widetilde{u}_{m}(z) = u(\frac{z}{C_m},t_m). |
From this and Proposition 4.1, we have
\sup\limits_M|\nabla \widetilde{u}_m| = 1 ,\ \sup\limits_M |\sqrt{-1}\ \overline{\partial}\partial \widetilde{u}_m|\leq \widetilde{C}. |
This yields that \widetilde{u}_{m} is contained in the Hölder space C^{1, \gamma(\mathbb{C}^n)} with a uniform. Along with a standard application of Azela-Ascoli theorem, we may suppose \widetilde{u}_{m} has a limit \widetilde{u}\in C^{1, \gamma}(\mathbb{C}^n) with
\begin{equation} |\widetilde{u}|+|\nabla \widetilde{u}| < C \ \rm{and}\ |\nabla \widetilde{u}(0)|\neq 0, \end{equation} | (5.2) |
in particular \widetilde{u} is not constant. On the other hand, similar to the method of Dinew and Kolodziej [7], we have
\begin{align*} &\bigg[\chi_{u}(\frac{z}{C_m}) \bigg]^k \wedge\bigg[\omega(\frac{z}{C_m})\bigg]^{n-k}\notag\\ = & e^{\partial_t u}\phi_m(\frac{z}{C_m},u) \bigg[\chi_{u}(\frac{z}{C_m}) \bigg]^l \wedge\bigg[\omega(\frac{z}{C_m})\bigg]^{n-l}.\notag\\ \end{align*} |
Fixing z , we obtain
\begin{align} &C_m^{2(k-l)}\bigg[O(\frac{1}{C^2_m})\beta+\sqrt{-1}\ \overline{\partial}\partial \widetilde{u}_m(z)\bigg]^k \wedge\bigg[(1+O(\frac{|z|^2}{C^2_m}))\beta\bigg]^{n-k}\\ = & e^{\partial_t u_m}\phi_m(\frac{z}{C_m},u_m) \bigg[O(\frac{1}{C^2_m})\beta+\sqrt{-1}\ \overline{\partial}\partial \widetilde{u}_m(z)\bigg]^l \wedge\bigg[(1+O(\frac{|z|^2}{C^2_m}))\beta\bigg]^{n-l}. \end{align} | (5.3) |
Lemma 2.2 gives \partial_t u is bounded uniformly. Since
\phi_m(\frac{z}{C_m},u_m)\leq \sup\limits_{M\times [-C,C]}\phi, |
which implies that \phi_m(\frac{z}{C_m}, u_m) is bounded uniformly. Taking the limits on both sides of 5.3 by m\rightarrow \infty yields that
\begin{equation} (\sqrt{-1}\partial\overline{\partial} \widetilde{u})^k\wedge \beta^{n-k} = 0. \end{equation} | (5.4) |
which is in the pluripotential sense. Moreover, a similar reasoning tells us that for any 1\leq p\leq k ,
\begin{equation} (\sqrt{-1}\partial\overline{\partial} \widetilde{u})^p\wedge \beta^{n-p}\geq0. \end{equation} | (5.5) |
Then, (5.4) and (5.5) imply \widetilde{u} is a k -subharmonic. By a result of Blocki [1], \widetilde{u} is a maximal k -subharmonic function in \mathbb{C}^n . Applying the Liouville theorem in [7], we find that \widetilde{u} is a constant, which contradicts with (5.2).
In this section, we shall give a proof of the long-time existence to the flow and its convergence, that is Theorem 1.3.
From Lemma 2.2, Proposition 3.1, Propositions 4.1 and 5.1, we conclude that Eq 1.1 is uniformly parabolic. Therefore by Evans-Krylov's regularity theory [8,13,19,27] for uniformly parabolic equation, we obtain higher order derivative estimates. By the a priori estimates which don't depend on time, one can prove that the short time existence on [0, T) extends to [0, \infty) , that is the smooth solution exists at all time t > 0 . After proving C^{\infty} estimates on [0, \infty) , we are able to show the convergence of the solution flow.
Let v = e^{\gamma t}u_t , where 0 < \gamma < c_{\phi} . Commuting derivative of v with respect to t and using (2.4), we obtain
\begin{align*} v_t = &e^{\gamma t}u_{tt}+\gamma v\notag\\ = &\gamma v+e^{\gamma t}(F^{i\overline{j}}u_{ti\overline{j}} -\frac{\phi_z}{\phi}u_t)\notag\\ = &F^{i\overline{j}}v_{i\overline{j}}+(\gamma-\frac{\phi_z}{\phi})v. \end{align*} |
Using the condition (1.5) yields \gamma-\frac{\phi_z}{\phi} < 0 , According to the parabolic maximum principle, it follows that
\sup\limits_{M\times [0,\infty)}|v(x,t)|\leq\sup\limits_{M}|u_t(x,0)| \leq\sup\limits_{M}|F(\lambda(u_0))-\phi(x,u_0)|\leq C, |
which means that |u_t| decreases exponentially, in particular
\partial_t(u+\frac{C}{\gamma}e^{-\gamma t})\leq 0. |
According to Proposition 3.1, it follows that u+\frac{C}{\gamma}e^{-\gamma t} is bounded uniformly and decreasing in t . Thus it converges to a smooth function u_{\infty} . From the higher order prior estimates, we can see that the function u(x, t) converges smoothly to u_{\infty} . Letting t\rightarrow \infty in Eq (2.1),
\begin{equation*} \frac{\sigma_k(\lambda(u_\infty))}{\sigma_l(\lambda(u_\infty))} = \phi(x,u_{\infty}). \end{equation*} |
In this paper, we have considered the parabolic Hessian quotient equation (1.1), in which the right hand side function \phi depends on u . Firstly, we prove C^0 estimates of Eq (1.1) by the parabolic \mathcal{C} -subsolution condition and the Alexandroff-Bakelman-Pucci maximum principle. Secondly, we establish the C^2 estimate for Eq (1.1) by using the parabolic \mathcal{C} -subsolution condition. Thirdly, we obtain the gradient estimate by adapting the blowup method. Finally we give the proof of the long-time existence of the solution to the parabolic equation and its convergence. As an application, we show the solvability of a class of complex Hessian quotient equations, which generalizes the relevant results.
This work was supported by the Natural Science Foundation of Anhui Province Education Department (Nos. KJ2021A0659, gxgnfx2018017); Quality Enginering Project of Anhui Province Education Department (Nos. 2018jyxm0491, 2019mooc205, 2020szsfkc0686); Science Research Project of Fuyang Normal University (No. 2021KYQD0011)
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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