Hermitian Calabi functional in complexified orbits

TL;DR

This paper studies the Hermitian Calabi functional on the space of compatible almost complex structures, providing explicit formulas for its Hessian at extremal metrics, proving semi-positivity at critical points, and analyzing flow properties.

Contribution

It derives an explicit Hessian formula for the Hermitian Calabi functional and establishes semi-positivity at extremal metrics within complexified orbits, extending K"ahler case results.

Findings

Hessian of Hermitian Calabi functional is semi-positive definite at critical points.

Explicit formula for the Hessian at extremal almost K"ahler metrics.

Weak parabolicity of the Hermitian Calabi flow.

Abstract

Let $(M,\omega)$ be a compact symplectic manifold. We denote by $\ac$ the space of all almost complex structure compatible with $\omega$. $\ac$ has a natural foliation structure with the complexified orbit as leaf. We obtain an explicit formula of the Hessian of Hermitian Calabi functional at an extremal almost K\"ahler metric in $\ac$. We prove that the Hessian of Hermitian Calabi functional is semi-positive definite at critical point when restricted to a complexified orbit, as corollaries we obtain some results analogy to K\"ahler case. We also show weak parabolicity of the Hermitian Calabi flow.

Equations

$$\omega$$

$$J\text{-invariant: }$$

$$g_J(\cdot ,\cdot ):=\omega (\cdot ,J\cdot ).$$

$${\mathcal{A}\mathcal{C}}_{\omega }:=\{J\in \mathrm{End}(TM):J^2=-1\text{ and J satisfies eqrefcompatibility}\},$$

$$\mathbb{J}(v):=Jv,⟨u,v{⟩}_J:={\int }_M{g}_J(u,v)\mathrm{vol},\mathrm{vol}:=\frac{{\omega }^m}{m!},\forall u,v\in T_J{\mathcal{A}\mathcal{C}}_{\omega }.$$

$$\mathcal{C}(J):={\int }_M[s^{\nabla }(J){]}^2\mathrm{vol}.$$

$${\mathcal{L}}_{\mathbb{K}}J=0,$$

$$P(f):=\frac{1}{2}{\mathcal{L}}_{{\mathrm{grad}}_{\omega }f}J.$$

$$D_J=\{P(f),JP(f):\text{ for all }f\in C^{\infty }(M,\mathbb{R})\},\forall J\in {\mathcal{A}\mathcal{C}}_{\omega }.$$

$$\mathrm{Hess}\mathcal{C}(u,v)=-⟨\frac{{\partial }^{2}J}{\partial t_1\partial t_2}{|}_{(0,0)},J{\mathcal{L}}_{\mathbb{K}}J⟩+⟨u,H(v)⟩-⟨u,v{\mathcal{L}}_{\mathbb{K}}J⟩.$$

$$H(u):=2JP(JP{)}^{\ast }u-J{\mathcal{L}}_{\mathbb{K}}u.$$

$$\mathrm{Hess}\mathcal{C}(u,v)=⟨H(u),v⟩,$$

$$\mathrm{Hess}\mathcal{C}(v,v)\ge 0,\forall v\in T_J\mathcal{D}.$$

$$\mathrm{Hess}\mathcal{C}(JP(f_1),JP(f_2))$$

$$=2⟨{\mathbb{L}}^{+}(f_1),{\mathbb{L}}^{-}(f_2)⟩,$$

$$\mathrm{Hess}\mathcal{C}(P(f_1),P(f_2))$$

$$\mathrm{Hess}\mathcal{C}(u,v)=2⟨(JP{)}^{\ast }u,(JP{)}^{\ast }v⟩,$$

$$\frac{d^2}{d{t}^2}\mathcal{C}(J_t)=⟨H(J^{\prime }),J^{\prime }⟩.$$

$$\mathrm{Im}P\cap \mathrm{Im}JP=\{0\},T_J{\mathcal{D}}_J=\mathrm{Im}P\oplus \mathrm{Im}JP.$$

$$\underset{\tilde{J}\in {\mathcal{A}\mathcal{C}}_{\omega }}{\max }\underset{J\in {\mathcal{D}}_{\tilde{J}}}{\min }\mathcal{C}(J).$$

$$\frac{d}{dt}J=\frac{1}{2}J{\mathcal{L}}_{\mathbb{K}}J.$$

$${\hat{\sigma }}_4(x,\xi )v=\frac{1}{2}(v,\Xi )\Xi ,$$

$$\mathbb{L}(f)=$$

$${\mathrm{D}}^{\pm }\alpha (X,Y)=\frac{1}{2}[({\mathrm{D}}_X\alpha )(Y)\pm ({\mathrm{D}}_{JX}\alpha )(JX)],$$

$$\mathrm{Ric}={\mathrm{Ric}}^{+},\delta ({\mathrm{D}}^{+}df-{\mathrm{D}}^{-}df)(X)=\mathrm{Ric}(\mathrm{grad}f,X).$$

$$\mathbb{L}(f)=\frac{1}{2}{\Delta }^{2}f-(\delta {\mathrm{Ric}}^{+},df)+(\rho ,d{d}^{c}f)=\frac{1}{2}{\Delta }^{2}f+\frac{1}{2}(ds,df)+(\rho ,d{d}^{c}f),$$

$$\{e_1,e_2,\dots ,e_m,J{e}_1=e_{m+1},J{e}_2=e_{m+2},\dots ,J{e}_m=e_n\}$$

$$\{e_1^{\ast },e_2^{\ast },\dots ,e_m^{\ast },J{e}_1^{\ast }=e_{m+1}^{\ast },J{e}_2^{\ast }=e_{m+2}^{\ast },\dots ,J{e}_m^{\ast }=e_n^{\ast }\}$$

$$\omega =\frac{1}{2}\sum _{i=1}^n{e}_i^{\ast }\wedge J{e}_i^{\ast }.$$

$$J\psi (X_1,\dots ,X_p):=\psi (J^{-1}{X}_1,\dots ,J^{-1}{X}_p),$$

$$d^c:=Jd{J}^{-1},{\delta }^c:=J\delta J^{-1}.$$

$$⟨S,T⟩={\int }_M(S,T)\mathrm{vol}={\int }_{M}g(S,T)\mathrm{vol},\forall S,T\in (TM{)}^{\otimes p}\otimes (T^{\ast }M{)}^{\otimes q}.$$

Full text

1. 1 Introduction

2. 2 Preliminaries

3. 2.1 Hamitonian group

4. 2.2 Hermitian scalar curvature

5. 2.3 Complexified orbit in ${\mathcal{AC}}_{\omega}$

6. 3 Operators

7. 3.1 Operators $P$ and $P^{\ast}$

8. 3.2 Lichnerowicz operator $\mathbb{L}$

9. 3.3 Operator $\mathcal{L}_{\mathbb{K}}$

10. 3.4 Calabi operators $\mathbb{L}^{\pm}$

11. 3.5 Decomposition of $C^{\infty}(M,{\mathbb{R}})$ and $T_{J}{\mathcal{AC}}_{\omega}$

12. 4 Hessian of the Hermitian Calabi functional

13. 4.1 Proof of (1) and (3) in Theorem 1.1

14. 4.2 Proof of (2) in Theorem 1.1

15. 4.3 Proof of Corollary 1.2

16. 4.4 Proof of Corollary 1.3

17. 4.5 Hermitian Calabi functional along geodesic in ${\mathcal{AC}}_{\omega}$

18. 5 Hermitian Calabi flow

19. 5.1 Linearisation operator of $J\mathcal{L}_{\mathbb{K}}J$

20. 5.2 Weak parabolicity of Hermitian Calabi flow

21. A Explicit expression of Lichnerowicz operator

1. Introduction

Extremal almost Kähler (EAK) metric extends Calabi’s extremal Kähler metric [5, 6] on a symplectic manifold. They are critical points of the Hermitian Calabi functional, which is the squared norm of the Hermitian scalar curvature. Hermitian Calabi flow is the gradient flow of the Hermitian Calabi functional.

In this paper, we compute Hessian of the Hermitian Calabi functional and prove weak parabolicity of the Hermitian Calabi flow.

Before we state results explicitly, we recall some notions. We let $(M,\omega)$ be a symplectic manifold, where $\omega$ is a given symplectic form on $M$. An almost Kähler structure consists a symplectic manifold $(M,\omega)$ and an $\omega$-compatible almost-complex structure $J$, which means they satisfy two conditions

[TABLE]

The compatibility conditions leads to a $J$-invariant Riemannian metric

[TABLE]

From now on, we always fix the symplectic form $\omega$ and we use the almost complex structure $J$ and the Riemannian metric $g$ interchangeable.

We collect all $\omega$-compatible almost-complex structures in the set

[TABLE]

which is an infinite-dimension Kähler manifold equipped with the complex structure $\mathbb{J}$ and the $L^{2}$ Riemannian metric

[TABLE]

Let $s^{\nabla}(J)$ be the Hermitian scalar curvature of an almost-complex structure $J\in\mathcal{AC}_{\omega}$. According to Fujiki [12] and Donaldson [9], the Hermitian scalar curvature is the moment map under the action of Hamiltonian symplectomorphisms ${\rm Ham}(M,\omega)$ on ${\mathcal{AC}}_{\omega}$. The Hermitian Calabi functional $\mathcal{C}:{\mathcal{AC}}_{\omega}\to\mathbb{R}$ is interpreted as the squared norm of the moment map,

[TABLE]

The extremal almost Kähler (EAK) metrics satisfy the equation

[TABLE]

where, $\mathbb{K}=:J{\rm{grad}}s^{\nabla}$ is the extremal vector field and $\mathcal{L}$ is the Lie derivative. Clearly, almost Kähler metric with constant Hermitian scalar curvature is EAK.

Donaldson [9, 10, 11] initialed a programme on the study of EAK metrics. There are many explicit examples of EAK (non-Kähler) metrics provided in [1] by Apostolov, Calderbank, Gauduchon and Tønnesen-Friedman. Recently, many results on extremal Kähler metrics have been extended to the EAK metrics. Lejmi [23] generalised the Futaki invariant and the extremal vector field to the almost Kähler setting. Keller-Lejmi [18] obtained the lower bound of the Hermitian Calabi functional. The deformation problem for the EAK metrics has been studied in [24, 19]. Vernier [36] constructed almost Kähler metric with constant Hermitian scalar curvature by the gluing method. Legendre [22] proved that under toric symmetry, the existence of EAK metric implies the existence of extremal Kähler metric. In general, the existence of EAK metrics are expected. We refer interested readers to the survey [2] of Apostolov and Drăghici on almost Kähler geometry.

The Hamiltonian symplectomorhisms group ${\rm Ham}(M,\omega)$ has a natural action on ${\mathcal{AC}}_{\omega}$. We denote by $\mathcal{O}_{J}$ the orbit of ${\rm Ham}(M,\omega)$ action through $J$. Any element ${\rm{grad}}_{\omega}f$ in the Lie algebra $\mathfrak{ham}$ of ${\rm Ham}(M,\omega)$ defines a vector field on the tangent space of $\mathcal{O}$:

[TABLE]

We also define $JP=J\circ P:C^{\infty}(M,{\mathbb{R}})\to T_{J}{\mathcal{AC}}_{\omega}$ and their adjoint operators $P^{\ast}$ and $(JP)^{*}$, see Definition 3.2 for an accurate statement.

Thanks to Donaldson’s observation [9], the Lie algebra $\mathfrak{ham}$ could be complexified and it induces a distribution $D$ in the tangent space $T{\mathcal{AC}}_{\omega}$ as follows

[TABLE]

Actually, $D$ is holomorphic and integrable, it generates an integral submanifold $\mathcal{D}\subset{\mathcal{AC}}_{\omega}$, which is called a complexified orbit. We denote by $\mathcal{D}_{J}$ the complexified orbit through $J$ (we may omit the lower index of $\mathcal{D}_{J}$ for convenience).

Using the notations above, we could explicitly state the Hessian ${\rm Hess}\mathcal{C}$ of the Hermitian Calabi functional.

Theorem 1.1**.**

For any $J\in{\mathcal{AC}}_{\omega}$ and for $u,v\in T_{J}{\mathcal{AC}}_{\omega}$, we choose a two-parameter family of $J(t_{1},t_{2})$ such that $J(0,0)=J,\quad\partial_{t_{1}}J(0,0)=u,\quad\partial_{t_{2}}J(0,0)=v$. Then we have

[TABLE]

Here, we introduce the operator

[TABLE]

Furthermore, we have the following applications.

1. (1)

If $J$ is EAK, then we have

[TABLE]

moreover, the operator $J\mathcal{L}_{\mathbb{K}}$ is self-adjoint on $T_{J}{\mathcal{AC}}_{\omega}$ and semi-positive on $T_{J}\mathcal{D}$, see Lemma 3.10. 2. (2)

The EAK metric $J$ is a local minimum of the Hermitian Calabi functional on the complexified orbit $\mathcal{D}_{J}$.

If $J$ is EAK, then ${\rm Hess}\mathcal{C}$ restricted to $\mathcal{D}_{J}$ is semi-positive

[TABLE]

Moreover, ${\rm Hess}\mathcal{C}$ is strictly positive on the subspace $\operatorname{Im}JP$, and vanishes on the subspace $\operatorname{Im}P$. Precisely, for any $f_{1},f_{2}\in C^{\infty}(M,{\mathbb{R}})$, we have

[TABLE]

In which, $\mathbb{L}$ is the Lichnerowicz operator $\mathbb{L}=P^{*}P$, c.f. Definition 3.3 and the Calabi operators $\mathbb{L}^{\pm}$ are self-adjoint and semi-positive, c.f. Definition 3.5. 3. (3)

The almost Kähler metric of constant Hermitian scalar curvature is a local minimum of the Hermitian Calabi functional on ${\mathcal{AC}}_{\omega}$.

Actually, if $J$ has constant Hermitian scalar curvature, then

[TABLE]

which is semi-positive on $T_{J}{\mathcal{AC}}_{\omega}$ and vanishes iff $v\in\ker(JP)^{*}$. 4. (4)

If $J(t)$ is a geodesic in terms of the Riemannian metric (1.2) in ${\mathcal{AC}}_{\omega}$, then the geodesic equation satisfies $J^{\prime\prime}=JJ^{\prime}J^{\prime}$ and the second order derivative of the Hermitian Calabi functional along $J(t)$ obeys

[TABLE]

By (2) in Theorem 1.1, we can get a structure property of the tangent space of the complexified orbit.

Corollary 1.2**.**

If $J$ is EAK, then

[TABLE]

Corollary 1.3**.**

On an almost Kähler manifold $(M,\omega,J)$,

• •

The Hermitian Calabi functional is invariant under the action of ${\rm Ham}(M,\omega)$.

• •

If we restrict the Hermitian Calabi functional $\mathcal{C}$ to a complexified orbit $\mathcal{D}$, $J$ is a critical point of $\mathcal{C}$ iff it is a local minimum of $\mathcal{C}$.

• •

The space of EAK metrics in $\mathcal{D}_{J}$ is a submanifold whose each connected component is an orbit of the Hamiltonian group ${\rm Ham}(M,\omega)$.

Remark 1.1*.*

Due to (1) in Theorem 1.1, it could be possible that the EAK metric becomes a saddle point of the Hermitian Calabi functional on ${\mathcal{AC}}_{\omega}$. While, it depends on the eigenvalues of the operators $2JP(JP)^{*}$ and $J\mathcal{L}_{\mathbb{K}}$. If so, it would be interesting to search the saddle point by applying the minimax approach

[TABLE]

Remark 1.2*.*

When the background manifold is Kähler, a complexified orbit can be identified with a Kähler class via Moser’s lemma. Thus (2) in Theorem 1.1 is a generalisation of Calabi’s classical results [6, Theorem 2]. That is the Hessian of the Calabi functional is semi-positive at an extremal Kähler metric. Our proof of (2) in Theorem 1.1 applies substantial results in Gauduchon’s book [17], where he used Mohsen formula to characterise the EAK condition, made a very detailed study of ${\mathcal{AC}}_{\omega}$ and the Hermitian Calabi functional. Our Theorem 1.1 follows his work to compute the Hessian of the Hermitian Calabi functional.

Remark 1.3*.*

In [37], Lijing Wang gave a different proof of Calabi’s result [6] on the Hessian of the Calabi functional. His method is based on reductive group action admitting a moment map on a Kähler manifold.

Remark 1.4*.*

If the Hermitian scalar curvature metric $s^{\nabla}_{J}$ of $J$ is constant, García-Prada and Salamon ([15, Remark 2.10], [14, Corollary 1.12]) showed that $\operatorname{Im}P\perp\operatorname{Im}JP$, so $T_{J}\mathcal{D}_{J}=\operatorname{Im}P\oplus\operatorname{Im}JP$. Corollary 1.2 is a generalisation of their result in the EAK case.

Remark 1.5*.*

Calabi [6] proved that the Hessian of Calabi functional is strictly positive along the directions transversal to the identity component of automorphism group ${\rm Aut}_{0}(M,J)$ orbit. Corollary 1.3 is an analogue to Calabi’s result, while the Hamiltonian group plays the role as automorphism group in the Kähler case.

In Section 5, we study the Hemritian Calabi flow. The Hermitian Calabi flow has appeared in the convergence problem of the Calabi flow [11, 25] and uniqueness of the adjacent constant scalar Kähler metrics [7].

The Hermitian Calabi flow is the negative gradient flow of the Hermitian Calabi functional

[TABLE]

Alternatively, we have $\frac{d}{dt}J=JP(s^{\nabla}(J))$, which suggests that the Hermitian Calabi flow $J(t)$ would stay in the distribution $\mathcal{D}$ as long as it exists.

Theorem 1.4**.**

The Hermitian Calabi flow is a 4th order weakly parabolic system. For any $\xi\in T^{*}M$, the principal symbol of its linearisation is given by

[TABLE]

where $\Xi\in T_{J}{\mathcal{AC}}_{\omega}$ is given by $\Xi=\xi^{\sharp}\otimes(J\xi)+(J\xi^{\sharp})\otimes\xi$.

In Appendix A, we compute an explicit expression of the Lichnerowicz operator $\mathbb{L}$, which appears in (2) in Theorem 1.1.

Theorem 1.5**.**

On an almost Kähler manifold $(M,\omega,J)$, the Lichnerowicz operator has the following explicit expression:

[TABLE]

for all $f\in C^{\infty}(M,\mathbb{R})$. Where $\delta$ is the formal adjoint of Levi-Civita connection ${\rm D}$ and ${\rm D}^{\pm}\alpha$ is the $J$-invariant(resp J-anti-invariant) part of ${\rm D}\alpha$, i.e.

[TABLE]

${\rm Ric}^{+}$* is the $J$-invariant part of Ricci curvature and $\rho(X,Y)={\rm Ric}^{+}(JX,Y)$.*

Remark 1.6*.*

Our result (1.7) in Theorem 1.5 is a continued calculation of Vernier’s formula [36, Equation (11)], where the Lichnerowicz operator is a 4th order elliptic operator plus an error term. We write the error term in an explicit way.

Remark 1.7*.*

On Kähler manifolds, we have

[TABLE]

So the formula in (1.7) becomes

[TABLE]

which is exactly Gauduchon’s real Lichnerowicz operator [17, Lemma 1.23.5], and twice of the real part of Calabi’s Lichnerowicz operator [6, Proof of Theorem 2], see also [33, 17].

Acknowledgements

Jie He would like to thank Youde Wang for his help and support. K. Zheng is partially supported by NSFC grant No. 12171365.

2. Preliminaries

In this section, we will introduce the basic materials in almost Kähler geometry. From now on, $(M,\omega)$ is always a compact symplectic manifold. Let $J$ be an $\omega$-compatible almost complex structure and $g$ be the Riemannian metric determined by (1.1).

We choose a local orthogonal frame of $TM$ as

[TABLE]

with the dual frame

[TABLE]

such that

[TABLE]

The almost complex structure is extended to any $p$-form $\psi\in\Omega^{p}(M)$ as

[TABLE]

for all $X_{1},\ldots,X_{p}\in\Gamma(TM)$, where $J^{-1}:=-J$. We define $Jf:=f$ if $f$ is a function.

The twisted differential operator $d^{c}$ and the twisted codifferential operator $\delta^{c}$ are defined by changing the differential operator $d$ and the codifferential operator $\delta$ under the extended almost complex structure $J$

[TABLE]

We denote by $\bm{\langle},\bm{\rangle}$ the $L^{2}$ inner product of any tensor of type $(p,q)$ over $M$, and by $(,)$ the inner product on fibre induced by $g$, i.e.

[TABLE]

Denote by ${\rm D}$ the Levi-Civita connection of $g$ and $\delta$ its formal adjoint. We have (see [4, 1.55]),

[TABLE]

For any $(p,q)$-tensor $T$ and $(p,q-1)$-tensor $S$ with $q\geq 1$, the adjointness implies

[TABLE]

For any $J\in{\mathcal{AC}}_{\omega}$, the Nijenhuis tensor $N$ of $J$ satisfies the formula

[TABLE]

By its very definition, we have

[TABLE]

The symplectic condition $d\omega=0$ yields a relation between ${\rm D}J$ and $N$.

Lemma 2.1** ([17, Lemma 9.3.1],[20, Proposition 4.2]).**

On any almost Kähler manifold $(M,\omega,J)$, it holds

[TABLE]

A direction consequence of (2.4) and (2.5) is the following corollary.

Corollary 2.2**.**

Let $(M,\omega,J)$ be an almost Kähler manifold. We have

[TABLE]

Proof.

It follows from compatible condition and (2.5) that

[TABLE]

It follows from (2.4) and (2.5) that

[TABLE]

Thus these two identities establish the relation (2.6). ∎

Corollary 2.3**.**

Let $(M,\omega,J)$ be an almost Kähler manifold. It holds $\delta J=0$, where $\delta$ is the formal adjoint of ${\rm D}^{*}$ as we defined in (2.2).

Proof.

It follows from (2.6) that

[TABLE]

Consequently, $\delta J=0$. ∎

Lemma 2.4**.**

On an almost Kähler manifold $(M,\omega,J)$, it holds

[TABLE]

Proof.

With the help of the definition of $d^{c}$, we see $d^{c}f$ is a $1$-form. Direct computation shows

[TABLE]

It follows from Corollary 2.3 that

[TABLE]

i.e.

[TABLE]

Substituting (2.8) into (2.7) yields

[TABLE]

Denote ${\rm Hess}f(e_{i},e_{j})={\rm Hess}_{ij},Je_{i}=J^{j}_{i}e_{j}$. Then we have ${\rm Hess}_{ij}={\rm Hess}_{ji},J_{i}^{j}=-J^{i}_{j}$. So we get

[TABLE]

i.e., $\delta d^{c}f=\sum_{i=1}^{n}{\rm Hess}f(e_{i},Je_{i})=0$. ∎

2.1. Hamitonian group

Denote ${\rm Ham}(M,\omega)$ the symplectic Hamiltonian group of $(M,\omega)$, and $\mathfrak{ham}$ the corresponding Lie algebra, then

[TABLE]

Writing $\iota_{X}\omega=-df$, the function $f$ is called the momentum of $X$ regarding to $\omega$, and

[TABLE]

where ${\rm{grad}}f$ is the Riemannian gradient of $f$. We call ${\rm{grad}}_{\omega}f$ the symplectic gradient of $f$.

A Hamiltonian vector field has many momentums which may differ by a constant. If we convent that the integral of the momenta function is 0, then it is unique. Under the corresponding

[TABLE]

$\mathfrak{ham}$ is identified to the set of smooth function with zero average which is denoted by $\mathfrak{ham}_{\omega}$,

[TABLE]

This identification is in fact a Lie algebra identification: if we define the Poisson bracket over $C^{\infty}_{0}(M,\mathbb{R})$

[TABLE]

then

[TABLE]

where $[,]$ is the Lie bracket of vector fields.

Definition 2.1**.**

On an almost Kähler manifold $(M,\omega,J)$, we say a real vector field $X$ is holomorphic if

[TABLE]

The real holomorphic vector fields constitutes a Lie subalgebra under the Lie bracket of vector fields.

2.2. Hermitian scalar curvature

On an almost Kähler manifold $(M,\omega,J)$, the Hermitian connection $\nabla$(c.f.[16]) is defined by

[TABLE]

When $(M,\omega,J)$ is Kähler, ${\rm D},\nabla$ coincide with each other.

We denote $R$ the Levi-Civita curvature tensor and $R^{\nabla}$ the canonical Hermitian curvature tensor, i.e.

[TABLE]

We denote by ${\rm Ric}$ the Riemann Ricci curvature and by ${\rm Ric}^{+}$ the $J$-invariant part of ${\rm Ric}$,

[TABLE]

The compatibility of ${\rm Ric}^{+}$ and $J$ determines a 2-form $\rho$

[TABLE]

We define another 2-form $\rho^{*}$ via contracting $R$ in terms of $\omega$,

[TABLE]

The (0,2)-tensor ${\rm Ric}^{+},\rho,\rho^{*}$ will be used in the computation of Lichnerowicz operator in Section A. The Hermitian Ricci form is the contraction of $R^{\nabla}$ by $\omega$,

[TABLE]

The Hermitian scalar curvature is defined by

[TABLE]

The averaged Hermitian scalar curvature

[TABLE]

is a topological constant which does not depend on the $J\in{\mathcal{AC}}_{\omega}$.

We denote by $\mathbb{K}$ the symplectic gradient of the Hermitian scalar curvature, i.e.

[TABLE]

When $J$ is EAK, $\mathbb{K}$ is exactly the extremal vector field (EVF). In general, EVF is defined to be ${\rm{grad}}_{\omega}(\Pi_{\omega}s^{\nabla}_{J})$, where $\Pi$ is the $L^{2}$-orthogonal projection in $\mathfrak{ham}_{\omega}$ and $\Pi_{\omega}s^{\nabla}_{J}$ is independent of $J$. EVF was first introduced by Mabuchi and Futaki [13] in Kähler geometry. Lejmi [23, Section 3.2] generalised this notion to almsot Kähler manifolds. $\mathbb{K}$ is very important in the variation of Calabi functional.

2.3. Complexified orbit in ${\mathcal{AC}}_{\omega}$

Recall that ${\mathcal{AC}}_{\omega}$ consists of all $\omega$-compatible almost complex structures, and its tangent space is

[TABLE]

By the compatible condition, $\omega(JX,vY)+\omega(vX,JY)=0$ is equivalent to $(X,vY)=(vX,Y)$, i.e., $v$ is self-adjoint. So we have an equivalent characterisation of $T_{J}{\mathcal{AC}}_{\omega}$,

[TABLE]

The Hamiltonian group ${\rm Ham}(M,\omega)$ has a natural action on ${\mathcal{AC}}_{\omega}$ by

[TABLE]

The tangent space of the resulting orbit $\mathcal{O}=\{\phi_{*}J\phi^{-1}_{*}:\phi\in{\rm Ham}(M,\omega)\}$ through $J$ is

[TABLE]

Lemma 2.5**.**

For any $f\in C^{\infty}(M,{\mathbb{R}})$,

[TABLE]

Proof.

By (2.16) we need to prove that ${\mathcal{L}}_{{\rm{grad}}_{\omega}f}J,\ J{\mathcal{L}}_{{\rm{grad}}_{\omega}f}J$ satisfy

1. (1)

$vJ+Jv=0$; 2. (2)

$(vX,Y)=(X,vY),\ \ \forall X,Y\in\Gamma(TM).$

Taking Lie derivative on both sides of $J^{2}=-1$ gives us

[TABLE]

i.e. ${\mathcal{L}}_{{\rm{grad}}_{\omega}f}J$ satisfies condition (1). For $J{\mathcal{L}}_{{\rm{grad}}_{\omega}f}J$, we compute

[TABLE]

i.e. $J{\mathcal{L}}_{{\rm{grad}}_{\omega}f}J$ satisfies condition (1).

For the second condition, since $\omega(J.,.)=-g(.,.)$ and $\mathcal{L}_{{\rm{grad}}_{\omega}f}\omega=0$, we have

[TABLE]

That $g$ is symmetric implies that $\mathcal{L}_{{\rm{grad}}_{\omega}f}g$ is also symmetric. Thus $J{\mathcal{L}}_{{\rm{grad}}_{\omega}f}J$ is self-adjoint and $J{\mathcal{L}}_{{\rm{grad}}_{\omega}f}J$ satisfies condition (2). For ${\mathcal{L}}_{{\rm{grad}}_{\omega}f}J$, we compute

[TABLE]

∎

According to the Hamiltonian action on ${\mathcal{AC}}_{\omega}$, any function $f\in C^{\infty}_{0}(M,{\mathbb{R}})=\mathfrak{ham}_{\omega}$ induces a tangent vector on $T_{J}{\mathcal{AC}}_{\omega}$ by

[TABLE]

The Lie algebra $\mathfrak{ham}_{\omega}$ is complexified by using $C_{0}^{\infty}(M,\mathbb{C})=\mathfrak{ham}_{\omega}+\sqrt{-1}\mathfrak{ham}_{\omega}$. The imaginary part $\sqrt{-1}f$ in the complexified Lie algebra induces a tangent vector in $T_{J}{\mathcal{AC}}_{\omega}$

[TABLE]

By Lemma 2.5, there exists a distribution $D$ on $T{\mathcal{AC}}_{\omega}$ given by

[TABLE]

which can be viewed as the distribution induced by the complexified Lie algebra.

It is obvious that $D_{J}$ is a holomorphic distribution, that is

[TABLE]

In 1983, Donaldson[9, Page 408] first observed that

Lemma 2.6**.**

$D_{J}$* forms an integrable distribution on ${\mathcal{AC}}_{\omega}$.*

We denote by $\mathcal{D}$ the integral submanifold generated by $D_{J}$. For any $J\in{\mathcal{AC}}_{\omega}$, we call ${\mathcal{D}}_{J}$ the complexified orbit through $J$. ${\mathcal{AC}}_{\omega}$ has a natural foliation structure with $\mathcal{D}$ as leaf.

3. Operators

In this section, we will introduce some operators related to ${\mathcal{AC}}_{\omega}$.

3.1. Operators $P$ and $P^{\ast}$

Definition 3.1**.**

The operator $P:C^{\infty}(M,{\mathbb{R}})\to T_{J}{\mathcal{AC}}_{\omega}$ is defined by

[TABLE]

Remark 3.1*.*

Comparing with the original definition of $P$ by Donaldson [9](see also [35, Page 49], [34, Page 6]), we add a normalisation factor $1/2$ in our definition of $P$. The normalisation factor ensures $P$ and the following $P$ related operations are all natural generalisation of their Kähler counterparts.

In fact $P(f)\in T_{J}\mathcal{O}$. We define $JP=J\circ P$, then

[TABLE]

where $\mathbb{J}$ is defined in (1.2). So we have the decomposition

[TABLE]

Definition 3.2**.**

We define $P^{\ast}:\Gamma(\operatorname{End}(TM))\rightarrow C^{\infty}(M,{\mathbb{R}})$ the formal adjoint operator of $P$ via $L^{2}$ integral. That is, $P^{\ast}$ satisfies

[TABLE]

under the $L^{2}$-inner product $\bm{\langle}\cdot,\cdot\bm{\rangle}$ over $M$ induced by $g$ on $C^{\infty}(M,{\mathbb{R}})$ and $\operatorname{End}(TM)$.

We also define $(JP)^{*}$ the formal adjoint of $JP$, then by definition, we have

[TABLE]

i.e.

[TABLE]

The following Lemma is important in the description of variation of Hermitian Calabi functional and moment map. It is contained in the proof of (9.6.5) in [17, Theorem 9.6.1]. We collect it here and reformulate the proof.

Lemma 3.1**.**

For any $v\in T_{J}{\mathcal{AC}}_{\omega}$, we have

[TABLE]

where $\delta$ is defined in (2.2).

Proof.

For any vector fields $X,Y$ and symplectic vector field $Z$, it follows from the compatible condition $g(X,Y)=\omega(X,JY)$ and $\mathcal{L}_{Z}\omega=0$ that

[TABLE]

Using the formula

[TABLE]

and the compatible condition $\omega(.,.)=g(J.,.)$, we have

[TABLE]

In order to compute $\bm{\langle}v,P(f)\bm{\rangle}$, we choose an orthonormal basis as we did in (2.1). We let $Z={\rm{grad}}_{\omega}f,\ Y=e_{i},\ X=-Jv(e_{i})$ in (3.3) and compute

[TABLE]

Here we view ${\rm D}{\rm{grad}}_{\omega}f\in\operatorname{End}(TM)$ as $X\to{\rm D}_{X}{\rm{grad}}_{\omega}f$. Since $Jv\in T_{J}{\mathcal{AC}}_{\omega}$, $Jv$ is self-adjoint by (2.16) and

[TABLE]

It follows from (3.4) that

[TABLE]

Then we compute

[TABLE]

Taking adjoint, we thus obtain $P^{*}v=\delta J(\delta Jv)^{\flat}$. ∎

We introduce the Mohsen Formula [31](also see [15, Theorem 2.6]),

Lemma 3.2** (Moshsen Formula).**

For any $v\in T_{J}{\mathcal{AC}}_{\omega}$ and any curve $J(t)\in{\mathcal{AC}}_{\omega}$ satisfying $J(0)=J,J^{\prime}(0)=v$, the first variation of the Hermtian ricci form and the Hermitian scalar curvature is

[TABLE]

If we view Hermitian scalar curvature as functional on ${\mathcal{AC}}_{\omega}$, combining Lemma 3.2 and Lemma 3.1 we have

[TABLE]

We can immediately obtain the description of EAK condition, i.e. the Euler-Lagrange equation of the Hermitian Calabi functional

[TABLE]

Corollary 3.3** ([2, 17]).**

$J\in{\mathcal{AC}}_{\omega}$* is EAK iff $\mathbb{K}={\rm{grad}}_{\omega}s^{\nabla}_{J}$ is a real holomorphic vector field.*

Proof.

It follows from (3.7) that

[TABLE]

Thus $J\in{\mathcal{AC}}_{\omega}$ is a critical point of $\mathcal{C}$ iff $\mathcal{L}_{\mathbb{K}}J=0$, i.e. ${\rm{grad}}_{\omega}s^{\nabla}$ is a holomorphic vector field. ∎

Now we can see (3.7) yields Donaldson’s famous results: the Hermitian scalar curvature is a moment map $\mu:{\mathcal{AC}}_{\omega}\to\mathfrak{ham}^{*}_{\omega}$ for the Hamiltonian action on ${\mathcal{AC}}_{\omega}$ via the $L^{2}$-product

[TABLE]

For any $f\in\mathfrak{ham}$, the induced vector on $T_{J}{\mathcal{AC}}_{\omega}$ is $\rho(f)=\mathcal{L}_{{\rm{grad}}_{\omega}f}J$ we only need to prove that

[TABLE]

where ${\bm{\kappa}}(u,v)$ is the Kähler form of the Kähler manifold ${\mathcal{AC}}_{\omega}$ defined by

[TABLE]

We further compute with (3.7) to get

[TABLE]

3.2. Lichnerowicz operator $\mathbb{L}$

Lichnerowicz operator is a 4th order elliptic operator defined on Kähler manifolds. It was first introduced in 1958 by Lichnerowicz[26, Chapter V]. Later in 1985, Calabi([6]) used the complex version of Lichnerowicz operator, which is called Calabi operators by Gauduchon in [17, Section 4.5] , to calculate the variation of Calabi functional. Gauduchon gave a very detailed and comprehensive introduction of Lichnerowicz operator in his book [17]. In this section, we generalise this notation to almost Kähler manifolds.

Definition 3.3**.**

On an almost Kähler manifold $(M,\omega,J)$, the generalised Lichnerowicz oeprator $\mathbb{L}:C^{\infty}(M)\to C^{\infty}(M)$ is defined by

[TABLE]

Remark 3.2*.*

By Lemma 3.1, we know that $\mathbb{L}=\frac{1}{2}\delta J(\delta J\mathcal{L}_{{\rm{grad}}_{\omega}f}J)^{\flat}$. This formula was first studied by Vernier [36], who computed the principal term of $\mathbb{L}$ and proved that $\mathbb{L}$ is a 4th order elliptic operator.

Since the Riemannian metric on $M$ is $J$ invariant, we have

[TABLE]

So $\mathbb{L}$ has an equavilent expression:

[TABLE]

Its expression implies that $\mathbb{L}$ is a self-adjoint semi-positive operator, and we will see the explicit expression of Lichnerowicz operator in Section A. By Definition 3.3, we have the following description for the kernel of $\mathbb{L}$.

Proposition 3.4**.**

Let $(M,\omega,J)$ be an almost Kähler manifold, then $\mathbb{L}(f)=0$ iff $\mathcal{L}_{{\rm{grad}}_{\omega}f}J=0$, i.e. the symplectic gradient ${\rm{grad}}_{\omega}f$ is holomorphic.

In fact, Definition 3.3 is a natural generalisation of Lichnerowicz operator in Kähler case.

Proposition 3.5**.**

When $(M,\omega,J,g)$ is a Kähler manifold, Definition 3.3 coincides with the definition

[TABLE]

in the Kähler case, where ${\rm D}^{-}\alpha(X,Y)=\frac{1}{2}[(D_{X}\alpha)Y-(D_{JX}\alpha)(JY)],\forall\alpha\in\Omega^{1}(M),$ is the $J$ anti-invariant part of ${\rm D}\alpha$.

Proof.

According to ([17], Lemma 1.23.2), it holds

[TABLE]

The Kähler condition gives

[TABLE]

Hence we have

[TABLE]

It follows from (3.10) that

[TABLE]

Taking adjoint, we obtain that

[TABLE]

i.e. $\mathbb{L}=({\rm D}^{-}d)^{*}{\rm D}^{-}d$. ∎

3.3. Operator $\mathcal{L}_{\mathbb{K}}$

Another important operator related to Lichnerowicz operator is the Lie derivative along $\mathbb{K}$.

Definition 3.4**.**

For any tensor field $T$ on $M$, we define the operator $\mathcal{L}_{\mathbb{K}}$ of Lie derivative along $\mathbb{K}$, i.e. $\mathcal{L}_{\mathbb{K}}:T\to\mathcal{L}_{\mathbb{K}}T$.

The most important case is $T\in T_{J}{\mathcal{AC}}_{\omega}$ and $T$ is a function. When $T=f$ is a function, we have

[TABLE]

To further describe $\mathcal{L}_{\mathbb{K}}$ acting on functions, we consider the twisted Lichnerowicz operator $(JP)^{*}P$.

In fact, we see that $(JP)^{*}P$ is an anti self-adjoint operator on functions. The proof is a direct computation. It follows from (3.1) that

[TABLE]

By definition $(JP)^{*}P$ seems to be a 4th order operator. However, it is half of the operator $\mathcal{L}_{\mathbb{K}}$. To prove this fact we need a lemma of García-Prada and Salamon.

Lemma 3.6** ([15, Remark 2.10]).**

For a closed connected symplectic $2m$-manifold $(M,\omega)$ , an almost complex structure $J\in{\mathcal{AC}}_{\omega}$, and two Hamiltonian momentum functions $f,g:M\to\mathbb{R}$ we have

[TABLE]

Proposition 3.7**.**

Let $(M,\omega,J)$ be an almost Kähler manifold, then for any $f$ we have

[TABLE]

In particular, $\mathcal{L}_{\mathbb{K}}$ is anti-self-adjoint.

Proof.

First we see

[TABLE]

Using $\delta d^{c}f_{1}=0$ from Lemma 2.4, we get

[TABLE]

So we have

[TABLE]

It then follows from (3.11) that $\mathcal{L}_{\mathbb{K}}=2(JP)^{*}P=-2P^{*}JP$. ∎

Remark 3.3*.*

When $(M,\omega,J)$ is Kähler, Gauduchon([17, Lemma 1.23.5]) proved that

[TABLE]

where the $\delta$ operator is the formal adjoint of ${\rm D}$ as we defined in (2.2), and after taking two successive $\delta$ operation, the (0,2)-tensor ${\rm D}^{-}d^{c}f$ becomes a function.

In fact, Proposition 3.7 is a generalisation of Gauduchon’s result on almost Kähler manifolds. By definition, it holds

[TABLE]

Due to the fact ${\rm D}df_{2}={\rm D}^{-}df_{2}+{\rm D}^{+}df_{2}$ and $\bm{\langle}{\rm D}^{-}d^{c}f_{1},{\rm D}^{+}df_{2}\bm{\rangle}=0$, we have

[TABLE]

It follows from (3.10) that

[TABLE]

So, the anti self-adjointness of $(JP)^{*}P$ leads to

[TABLE]

Now we study the action of $\mathcal{L}_{\mathbb{K}}$ on $T_{J}{\mathcal{AC}}_{\omega}$.

Lemma 3.8**.**

When acting on $T_{J}{\mathcal{AC}}_{\omega}$, $\mathcal{L}_{\mathbb{K}}$ is an anti-self-adjoint operator.

Proof.

From (2.16), we know that any $u\in T_{J}{\mathcal{AC}}_{\omega}$ is symmetric. The metric on $T_{J}{\mathcal{AC}}_{\omega}$ could also be written as (see also [17, (9.2.10)])

[TABLE]

where $uv=u\circ v$ denote the composition of $u,v\in{\rm End}(TM)$ and ${\rm tr}$ is the trace operation. Since trace operation commutes with Lie derivative(for example, see [32, Exercise 2.5.10]), we have

[TABLE]

But Lemma 2.4 implies $\delta d^{c}s^{\nabla}=0$, we have

[TABLE]

We obtain

[TABLE]

Thus $\mathcal{L}_{\mathbb{K}}$ is anti self-adjoint.

∎

If $J$ is EAK, the operator $\mathcal{L}_{\mathbb{K}}$ have the following commutative relation.

Lemma 3.9**.**

If $J$ is EAK, then $\mathcal{L}_{\mathbb{K}}$ commutes with $P,P^{*},JP,(JP)^{*},\mathbb{L}$.

Proof.

Since $J$ is EAK, we have $\mathcal{L}_{\mathbb{K}}J=0$, which implies

[TABLE]

The Poisson bracket satisfies

[TABLE]

Combining (3.14) and (3.15) together, we arrive at

[TABLE]

It follows from (3.16) and Lemma 3.8 that

[TABLE]

for any $v\in{\rm End}(TM),f\in C^{\infty}(M,{\mathbb{R}})$, i.e.

[TABLE]

Since

[TABLE]

making use of the EAK condition $\mathcal{L}_{\mathbb{K}}J=0$ and (3.16), we get

[TABLE]

Applying Lemma 3.8 and (3.18), we obtain that

[TABLE]

i.e.

[TABLE]

Since $\mathbb{L}=PP^{*}$, it follows from (3.16) and (3.17) that $\mathcal{L}_{\mathbb{K}}$ commutes with $\mathbb{L}$. This completes the proof. ∎

Lemma 3.10**.**

When $J$ is EAK, the operator $J\mathcal{L}_{\mathbb{K}}$ is self-adjoint on $T_{J}{\mathcal{AC}}_{\omega}$ and semi-positive on $\operatorname{Im}P$ and $\operatorname{Im}JP$.

Proof.

According to Lemma 3.8 and the EAK condition $\mathcal{L}_{\mathbb{K}}J=0$, we have

[TABLE]

i.e. $J\mathcal{L}_{\mathbb{K}}$ is self-adjoint. Choosing any $v=P(\phi)\in\operatorname{Im}P$, Proposition 3.7 and the commutative relation in Lemma 3.9 gives

[TABLE]

Taking $v=JP(\phi)$, we conclude that

[TABLE]

∎

3.4. Calabi operators $\mathbb{L}^{\pm}$

Considering the $(0,1)$ and $(1,0)$ part of $P$,

[TABLE]

we can define Calabi operators on almost Kähler manifolds.

Definition 3.5**.**

On an almost Kähler manifold $(M,\omega,J)$, the Calabi operators are defined by

[TABLE]

If we extend $P$ and $P^{*}$ to the space of complex function on $M$, and consider the Hermitian $L^{2}$ inner products on $C^{\infty}(M,\mathbb{C})$ and $T_{J}{\mathcal{AC}}_{\omega}\otimes\mathbb{C}$, then both $\mathbb{L}^{\pm}$ are all self-adjoint semi-positive operators.

Proposition 3.11**.**

By the definition of  $\mathbb{L}$ and $\mathcal{L}_{\mathbb{K}}$, we obtain equivalent expressions of Calabi operators

[TABLE]

Proof.

Direct computation shows

[TABLE]

Similarly, we can obtain the expression of $\mathbb{L}^{-}$. ∎

Remark 3.4*.*

In the Kähler case, the most commonly used version of Lichnerowicz operator([6, Equation(1.2)], [21, Corollary 1]) is

[TABLE]

where $\partial^{\sharp}\psi=(\overline{\partial}\psi)^{\sharp}$. In fact, $(\overline{\partial}\partial^{\sharp})^{*}(\overline{\partial}\partial^{\sharp})$ is half of $\mathbb{L}^{+}$. Since ${\rm D}^{0,1}(\bar{\partial}\psi)\in\Omega^{0,1}(M)\otimes\Omega^{0,1}(M)$, we have

[TABLE]

Kähler condition implies ${\rm D}g={\rm D}J=0$. Thus ${\rm D}^{1,0}$ commutes with flatten operator and

[TABLE]

By the relation

[TABLE]

and Proposition 3.5, Remark 3.3, we have

[TABLE]

i.e.

[TABLE]

Proposition 3.12**.**

For any $F=\varphi+i\phi\in C^{\infty}(M,\mathbb{C})$, we have $F\in\ker\mathbb{L}^{+}$ if and only if

[TABLE]

And $F\in\ker\mathbb{L}^{-}$ if and only if

[TABLE]

In particular, if $F=\varphi\in C^{\infty}(M,\mathbb{R})$, then $F\in\ker\mathbb{L}^{+}$ iff  $\mathbb{L}(\varphi)=\mathcal{L}_{\mathbb{K}}(\varphi)=0$ iff $P(\varphi)=0$. The same conclusion holds for $\varphi\in\ker\mathbb{L}^{-}$.

Proof.

For $F=\varphi+\sqrt{-1}\phi\in C^{\infty}(M,\mathbb{C})$, we calculate

[TABLE]

So $\mathbb{L}^{+}(F)=0$ if and only if

[TABLE]

If $F=\varphi$, i.e. $\phi=0$, then $\mathbb{L}^{+}(F)=0$ if and only if

[TABLE]

Here $\mathbb{L}(\varphi)=0$ implies $P(\varphi)=0$, which also implies $\mathcal{L}_{\mathbb{K}}(\varphi)=0$ since $\mathcal{L}_{\mathbb{K}}(\varphi)=2(JP)^{*}P(\varphi)$. ∎

Remark 3.5*.*

When the background manifold is Kähler, $F=\varphi+i\phi\in\ker\mathbb{L}^{+}$ implies

[TABLE]

is holomorphic (see [17, Section 2.5]). In the Kähler case, if we denote $\mathfrak{h}_{\rm{red}}$ the set of real holomorphic vector fields whose zero set is non-empty, then any real vector field $X\in\mathfrak{h}_{\rm{red}}$ if and if and only if there exists $F=\varphi+i\phi\in\ker\mathbb{L}^{+}$ such that ([27, section 95])

[TABLE]

When $J$ is EAK, the Calabi operators have the following commutative relation.

Lemma 3.13**.**

If $J$ is EAK, then $\mathbb{L}^{+},\mathbb{L}^{-}$ commute. The composition $\mathbb{L}^{+}\mathbb{L}^{-}$ is self-adjoint and semi-positive and we have

[TABLE]

Proof.

Since $J$ is EAK, Lemma 3.9 implies that $\mathcal{L}_{\mathbb{K}}\mathbb{L}=\mathbb{L}\mathcal{L}_{\mathbb{K}}$. It follows from Proposition 3.11 that $\mathbb{L}^{+}\mathbb{L}^{-}=\mathbb{L}^{-}\mathbb{L}^{+}$. Since $\mathbb{L}^{+},\mathbb{L}^{-}$ are all semi-positive and self-adjoint, by commutativity, $\mathbb{L}^{+}\mathbb{L}^{-}$ is also semi-positive and self-adjoint. Applying the $L^{2}$-splitting theorem, we obtain that

[TABLE]

and $\mathbb{L}^{\pm}:\operatorname{Im}\mathbb{L}^{\pm}\to\operatorname{Im}\mathbb{L}^{\pm}$ is isomorhism. So we have

[TABLE]

But $\ker\mathbb{L}^{-}+\ker\mathbb{L}^{+}\subset\ker(\mathbb{L}^{+}\mathbb{L}^{-})$ is obvious, due to the commutativity of $\mathbb{L}^{+},\mathbb{L}^{-}$. ∎

3.5. Decomposition of $C^{\infty}(M,{\mathbb{R}})$ and $T_{J}{\mathcal{AC}}_{\omega}$

Lemma 3.14**.**

The functions space $C^{\infty}(M,{\mathbb{R}})$ has the following orthogonal decompostion

[TABLE]

and $\mathbb{L}:\operatorname{Im}P^{*}\to\operatorname{Im}P^{*}$ is an isomorphism.

Proof.

Since the Lichnerowicz operator $\mathbb{L}$ is an elliptic (see [36, equation (11)]) self-adjoint operator, the splitting theorem of elliptic operator tells us

[TABLE]

and $\mathbb{L}=P^{*}P:\operatorname{Im}\mathbb{L}\to\operatorname{Im}\mathbb{L}$ is an isomorphisom. Since $\ker\mathbb{L}=\ker P,\ \operatorname{Im}\mathbb{L}=(\ker P^{*}P)^{\perp}=(\ker P)^{\perp}=\operatorname{Im}P^{*}$, the decomposition becomes

[TABLE]

The isomorphism $\mathbb{L}=P^{*}P:\operatorname{Im}\mathbb{L}\to\operatorname{Im}\mathbb{L}$ can be decomposed as

[TABLE]

Similarly, if we consider $\mathbb{L}=(JP)^{*}JP$, we can obtain decompostion with repect to $JP$ and $(JP)^{*}$. ∎

Lemma 3.15**.**

The tangent space $T_{J}{\mathcal{AC}}_{\omega}$ has the following decompostion:

[TABLE]

and

[TABLE]

The map $PP^{*}:\operatorname{Im}P\to\operatorname{Im}P$ is an isomorphism.

Proof.

Since $(\operatorname{Im}P)^{\perp}=\ker P^{*}$ , we have

[TABLE]

By Lemma 3.14, we know that $PP^{*}:\operatorname{Im}P\to\operatorname{Im}P$ is an isomorphism, and

[TABLE]

Similarly we can obtain $T_{J}{\mathcal{AC}}_{\omega}=\operatorname{Im}JP\oplus\ker(JP)^{*}$ and that $JP(JP)^{*}:\operatorname{Im}JP\to\operatorname{Im}JP$ is an isomorphism. Therefore, (3.24) follows from the fact $T_{J}\mathcal{D}=\operatorname{Im}P+\operatorname{Im}JP$.

∎

According to the decomposition of $T_{J}{\mathcal{AC}}_{\omega}$ in Lemma 3.15, we discuss the variation of Hermitian Calabi functional in different directions.

Lemma 3.16**.**

If we take variation in different directions in Corollary 3.3, we have

1. (1)

If $v\in\ker(JP)^{*}$, $D\mathcal{C}(v)=0,\quad Ds^{\nabla}(v)=0$. 2. (2)

If $v\in\operatorname{Im}JP$, $v=JP(f)$, $D\mathcal{C}(v)=-2\bm{\langle}f,\mathbb{L}(s^{\nabla})\bm{\rangle},\quad Ds^{\nabla}(v)=-\mathbb{L}(f)$. 3. (3)

If $v\in\operatorname{Im}P$, $v=P(f)$, $D\mathcal{C}(v)=0,\quad Ds^{\nabla}(v)=-\frac{1}{2}\mathcal{L}_{\mathbb{K}}(f)$.

Proof.

The above are all direct consequences of Lemma 3.15,

1. (1)

For any $v\in\ker(JP)^{*}$, it follows immediately from (3.7)

[TABLE] 2. (2)

If we choose $v=JP(f)\in\operatorname{Im}JP$ for some $f\in C^{\infty}(M,{\mathbb{R}})$, by (3.7) we have

[TABLE]

and by (3.8),

[TABLE] 3. (3)

If we choose $v=P(f)\in\operatorname{Im}(P)$ for some $f\in C^{\infty}(M,{\mathbb{R}})$, then

[TABLE]

Since

[TABLE]

we get

[TABLE]

∎

4. Hessian of the Hermitian Calabi functional

In this section, we study the second variation of Hermitian Calabi functional.

4.1. Proof of (1) and (3) in Theorem 1.1

Theorem 4.1**.**

For any $J\in{\mathcal{AC}}_{\omega}$ and $u,v\in T_{J}{\mathcal{AC}}_{\omega}$, we choose $J(t_{1},t_{2})$ satisfying $J(0,0)=J,J^{\prime}_{t_{1}}(0,0)=u,J^{\prime}_{t_{2}}(0,0)=v$. Then

[TABLE]

In particular,

1. (1)

Assuming that $J$ is EAK, then

[TABLE] 2. (2)

Assuming that $J$ has constant Hermitian scalar curvature, then

[TABLE]

${\rm Hess}\mathcal{C}$* is semi-positive on $T_{J}{\mathcal{AC}}_{\omega}$ and ${\rm Hess}\mathcal{C}(v,v)=0$ iff $v\in\ker(JP)^{*}$.*

Proof.

Taking derivative on (3.8) leads to

[TABLE]

It follows from Lemma 3.3 and the identity: $\iota_{\mathbb{K}}\omega=-ds^{\nabla}$ that

[TABLE]

So we have

[TABLE]

which gives

[TABLE]

Then it follows

[TABLE]

If $J$ is EAK, then $\mathcal{L}_{K}J=0$, we obtain (4.1); if $J$ has constant Hermitian scalar curvature, then $\mathbb{K}=0$, we obtain (4.2).

The average scalar curvature

[TABLE]

dose not depend on $J\in{\mathcal{AC}}_{\omega}$(see [17, (9.5.15)]). Thus any $J\in{\mathcal{AC}}_{\omega}$ with constant Hermitian scalar curvature take $\underline{s^{\nabla}}$ as its Hermitian scalar curvature. Cauchy inequality implies

[TABLE]

i.e.

[TABLE]

Thus any $J\in{\mathcal{AC}}_{\omega}$ with constant Hermitian scalar curvature is the local minimum point of $\mathcal{C}$. ∎

Theorem 4.1 shows that the Calabi functional is convex at $J$ if $J$ has constant Hermitian scalar curvature. In fact, we can also obtain convexity result at EAK point if we restrict Hermitian Calabi functional to complexified orbit.

Tangent vectors of complexified orbits are all characterised by smooth functions on $M$, we will restrict Hermitian Calabi functional in complexified orbits as following.

Corollary 4.2**.**

Suppose that $J$ is EAK. Let $u_{1}=P(f_{1}),v_{1}=P(f_{2})$ in Theorem 4.1. Then

[TABLE]

Let $u=P(f_{1})$ and $v=JP(f_{2})$ in Theorem 4.1, we have

[TABLE]

Proof.

Taking $u_{1}=P(f_{1}),v_{1}=P(f_{2})$ in (4.1), we have

[TABLE]

It follows from the description of $\mathcal{L}_{\mathbb{K}}$ in Proposition 3.7

[TABLE]

By the commutativity $\mathcal{L}_{\mathbb{K}}P=P\mathcal{L}_{\mathbb{K}}$ in Lemma 3.9, we get

[TABLE]

So we see

[TABLE]

Similarly, if we take $u_{2}=P(f_{1}),v_{2}=JP(f_{2})$, we then obtain

[TABLE]

∎

Corollary 4.3**.**

In fact, if $J$ is EAK, then $\operatorname{Im}P$ annihilates the Hessian of Hermitian Calabi functional in the total space $T_{J}{\mathcal{AC}}_{\omega}$, i.e.

[TABLE]

Proof.

That’s because

[TABLE]

∎

Corollary 4.4**.**

Suppose that $J$ is EAK. We let $u_{1}=JP(f_{1}),v_{1}=JP(f_{2})$ in Theorem 4.1. Then

[TABLE]

Proof.

Taking $u=JP(f_{1}),u=JP(f_{2})$ in (4.1), we have

[TABLE]

Since $J$ is EAK, Lemma 3.9 implies $JP\mathcal{L}_{\mathbb{K}}=\mathcal{L}_{\mathbb{K}}JP$, we obtain

[TABLE]

On the other hand, we use Proposition 3.11 to compute

[TABLE]

Again the commuting relation $\mathcal{L}_{\mathbb{K}}\mathbb{L}=\mathbb{L}\mathcal{L}_{\mathbb{K}}$ in Lemma 3.9 infers that

[TABLE]

Thus we obtain

[TABLE]

∎

4.2. Proof of (2) in Theorem 1.1

By Corollary 4.2 we have known that ${\rm Hess}\mathcal{C}$ vanishes on $\operatorname{Im}P$.

We only need to show that ${\rm Hess}\mathcal{C}$ is strictly positive on the subspace $\operatorname{Im}JP$. It is proved in Corollary 4.4 that

[TABLE]

$\mathbb{L}^{+},\mathbb{L}^{-}$ are all self-adjoint(see Definition 3.5), hence

[TABLE]

Since $J$ is EAK, the commutativity of $\mathbb{L}^{+},\mathbb{L}^{-}$ in Lemma 3.13 implies that $\mathbb{L}^{-}\mathbb{L}^{+}$ is semi-positive and $\ker\mathbb{L}^{+}\mathbb{L}^{-}=\ker\mathbb{L}^{+}+\ker\mathbb{L}^{-}$. By Proposition 3.12, any real function $f\in\ker\mathbb{L}^{\pm}$ iff $JP(f)=0$, i.e. $JP(f)$ is zero. So ${\rm Hess}\mathcal{C}$ is strictly positive on the subspace $\operatorname{Im}JP$.

∎

4.3. Proof of Corollary 1.2

For any $v\in\operatorname{Im}P\cap\operatorname{Im}JP$, by Corollary 4.2, $v\in\operatorname{Im}P=T_{J}\mathcal{O}$ implies that ${\rm Hess}\mathcal{C}(v,v)=0$. But, from Corollary 4.4, ${\rm Hess}\mathcal{C}$ is strictly positive on the subspace $\operatorname{Im}JP=\mathbb{J}T_{J}\mathcal{O}$, this forces $v=0$, i.e. $\operatorname{Im}P\cap\operatorname{Im}JP=\{0\}$.

∎

Definition 4.1**.**

We define the operator $H:T_{J}{\mathcal{AC}}_{\omega}\to T_{J}{\mathcal{AC}}_{\omega}$ by

[TABLE]

Assuming that $J$ is EAK, then $H$ is self-adjoint by Lemma 3.10, and

[TABLE]

By Corollary 4.2 and Corollary 4.4, we know that the operator $H$ is semi-positive over $T_{J}\mathcal{D}$,

[TABLE]

and $H(JP(f))=0$ if and only if $JP(f)=0$.

In order the prove Corollary 1.3, we introduce the following lemma.

Lemma 4.5**.**

If $J$ is EAK, for any $v\in T_{J}\mathcal{D}$, if

[TABLE]

we have $v=P(f)$ for some $f\in C^{\infty}(M,{\mathbb{R}})$.

Proof.

Since $v\in T_{J}\mathcal{D},v=P(f)+JP(g)$ for some $f,g\in C^{\infty}(M,{\mathbb{R}})$, using Corollary 4.2 and Corollary 4.4, we have

[TABLE]

The proof given above implies $JP(g)=0$. Thus $v=P(f)$. ∎

4.4. Proof of Corollary 1.3

1. (1)

For any path $J_{t}$ in the orbit of ${\rm Ham}(M,\omega)$, we know that $J_{t}^{\prime}=P_{t}(f_{t})$ for some $f_{t}\in C^{\infty}(M,{\mathbb{R}})$, by (3.28) we have

[TABLE]

i.e. $\mathcal{C}(J_{t})$ is constant. Since ${\rm Ham}(M,\omega)$ is path connected(see [29, Proposition 10.2]), the Hermitian Calabi functional is invariant under the action of ${\rm Ham}(M,\omega)$. 2. (2)

If $J$ is EAK, we have

[TABLE]

Since $\mathcal{C}$ is ${\rm Ham}(M,\omega)$ invariant and ${\rm Hess}\mathcal{C}$ is strictly positive along the directions in $\operatorname{Im}JP$, thus every critical $J$ achieves a local, non-degenerate minimum value of $\mathcal{C}$ relative to the action of the gauge group ${\rm Ham}(M,\omega)$.

On the other hand, if $J$ is a local minimum, then $D\mathcal{C}(v)=0,\forall v\in T_{J}\mathcal{D}$. Thus for any $f\in C^{\infty}(M,{\mathbb{R}}),D\mathcal{C}(JP(f))=\bm{\langle}\mathbb{L}(s^{\nabla}),f\bm{\rangle}=0.$ Thus $\mathbb{L}(s^{\nabla})=0$, which implies $\mathcal{L}_{\mathbb{K}}J=0$. 3. (3)

Suppose that $J_{0}$ is EAK, denote by $\mathcal{E}_{J_{0}}$ the connected component of $J_{0}$ in the subset of extremal almost Kähler metrics in $\mathcal{D}_{J_{0}}$. Since ${\rm Ham}(M,\omega)$ is connected, $\mathcal{O}_{J_{0}}$ is already connected, we need to prove that $\mathcal{E}_{J_{0}}=\mathcal{O}_{J_{0}}$.

We first show that $\mathcal{E}_{J_{0}}\subset\mathcal{O}_{J_{0}}$, for any $J_{1}\in\mathcal{E}_{J_{0}}$, choose a curve $J_{t}:[0,1]\to\mathcal{E}_{J_{0}}$ such that $J(0)=J_{0},J(1)=J_{1}$. Since $J_{t}$ is EAK for all $t\in[0,1]$, we have $\frac{d}{dt}\mathcal{C}(J_{t})=0$, so

[TABLE]

is a constant function, any order derivative of $\mathcal{C}(t)$ is 0, so we have

[TABLE]

Lemma 4.5 implies that $J^{\prime}_{t}=P_{t}(f_{t})$ for some $f_{t}\in C^{\infty}(M,{\mathbb{R}})$, this implies that $J_{t}$ lies in the orbit of ${\rm Ham}(M,\omega)$ action.

On the other hand, any $J\in\mathcal{E}_{J_{0}}$ is a local minimum of $\mathcal{C}$, but $\mathcal{C}$ is constant on $\mathcal{O}_{J_{0}}$, thus $\mathcal{E}_{J_{0}}$ is an open subset of $\mathcal{O}_{J_{0}}$, but $\mathcal{E}_{J_{0}}$ is closed since it is characterised by the Euler-Lagrange equation

[TABLE]

So $\mathcal{E}_{J_{0}}=\mathcal{O}_{J_{0}}$. ∎

4.5. Hermitian Calabi functional along geodesic in ${\mathcal{AC}}_{\omega}$

In this section, we prove (4) in Theorem 1.1.

In order to deduce the Levi-Civita connection and geodesic equation in ${\mathcal{AC}}_{\omega}$, we introduce the space

[TABLE]

$\operatorname{End}(TM,\omega)$ is a trivial bundle over ${\mathcal{AC}}_{\omega}$, and $T{\mathcal{AC}}_{\omega}$ is a sub-bundle of $\operatorname{End}(TM,\omega)$ which is characterized by

[TABLE]

And we have

• •

Any secition $a\in\operatorname{End}(TM,\omega)$ determines a vector field $\hat{a}$, on ${\mathcal{AC}}_{\omega}$ by

[TABLE]

• •

Conversely for any $A\in T_{J}{\mathcal{AC}}_{\omega}$, $a=-\frac{1}{2}JA\in\operatorname{End}(TM,\omega)$ satisfies $A=\hat{a}$.

For any $a\in\operatorname{End}(TM,\omega)$, we denote by $a^{\pm}$ the $J$ commutative and $J$ anti-commutative part of $a$, i.e.,

[TABLE]

then we have $a^{-}\in T_{J}{\mathcal{AC}}_{\omega}$ and $\hat{a}=2Ja^{-}$.

Lemma 4.6** ([17, (9.2.7)]).**

For any $a,b\in{\rm End}(TM,\omega)$, we have

[TABLE]

where $[\hat{a},\hat{b}]$ denote the Lie bracket of vector fields on ${\mathcal{AC}}_{\omega}$, and $[a,b]=ab-ba$.

Lemma 4.7**.**

For any $a,b,c\in{\rm End}(TM,\omega)$, we have

[TABLE]

and

[TABLE]

Proof.

Direct computation shows

[TABLE]

applying (4.11), we obtain (4.12) .

By (4.9), the definition curve of $\hat{a}$ is

[TABLE]

According to definition, we have

[TABLE]

Direct computation shows

[TABLE]

It follows from (4.14) and (4.15) that

[TABLE]

∎

Now we turn to study the geodesic equation in ${\mathcal{AC}}_{\omega}$. Denote by $\mathfrak{D}$ the Levi-Civita connection on ${\mathcal{AC}}_{\omega}$, the following lemma is mentioned in [17, Section 9.2] and for sake of completeness we give a proof here.

Lemma 4.8**.**

For $\hat{b}\in\Gamma(T{\mathcal{AC}}_{\omega}),\forall A\in T_{J}{\mathcal{AC}}_{\omega}$, we have

[TABLE]

Proof.

By Koszul formula, we have

[TABLE]

It follows from (4.12) and (4.13) in Lemma 4.7 that

[TABLE]

Thus $\mathfrak{D}_{\hat{a}}\hat{b}$ is just the $J$-anti-commutative part of $[\hat{a},b]^{-}$, i.e.,

[TABLE]

Since $\hat{a},a\in{\rm End}(TM,\omega)$ can generate the whole space $T_{J}{\mathcal{AC}}_{\omega}$, (4.16) establishes. ∎

By the Levi-Civita connection on ${\mathcal{AC}}_{\omega}$, we can characterize the geodesic in ${\mathcal{AC}}_{\omega}$.

Lemma 4.9**.**

A curve $J_{t}:(a,b)\to{\mathcal{AC}}_{\omega}$ is a geodesic if and only if $(J_{t}^{\prime\prime})^{-}=0$, i.e.,

[TABLE]

Using the fact $J^{\prime}J+JJ^{\prime}=0$, we can get another equivalent conidtion

[TABLE]

Proof.

By definition $J_{t}$ is a geodesic in ${\mathcal{AC}}_{\omega}$ if and only if $\mathfrak{D}_{J_{t}^{\prime}}J_{t}^{\prime}=0$. For any $b\in\operatorname{End}(TM,\omega)$, we have

[TABLE]

∎

Proposition 4.10**.**

If $J(t)$ is geodesic in ${\mathcal{AC}}_{\omega}$, we have

[TABLE]

where $H$ is defined in Definition 4.1.

Proof.

Let $J(t)$ be an geodesic in ${\mathcal{AC}}_{\omega}$, it follows from Theorem 4.1 that

[TABLE]

Due to the geodesic condition that $J^{\prime\prime}$ is $J$-commutative and the fact that $J\mathcal{L}_{\mathbb{K}}J$ is $J$-anti-commutative, we have

[TABLE]

For any $u\in T_{J}{\mathcal{AC}}_{\omega}$, we have

[TABLE]

Simplifying (4.17) by (4.18) and (4.19) gives

[TABLE]

∎

Remark 4.1*.*

We don’t know whether $H$ is positive in the whole space ${\mathcal{AC}}_{\omega}$, so the convexity of Hermitain Calabi functional along geodesic in Proposition 4.10 is not clear. If $J$ is EAK, we know that the operator $H$ is semi-positive over $T_{J}\mathcal{D}$. It is natural to ask if a geodesic $J(t)$ lies in some complexified orbit $\mathcal{D}$, whether the Hermitian Calabi functional is convex along the geodesic.

5. Hermitian Calabi flow

According to the variation formula of Hermitian Calabi functional (3.8), we write down the gradient flow of Hermitian Calabi functional.

Definition 5.1**.**

The Hermitian Calabi flow(HCF), i.e., gradient flow of Hermitian Calabi functional, is defined by

[TABLE]

Since $JP(s^{\nabla}_{J})\in\operatorname{Im}JP\subset T_{J}\mathcal{D}$, we see the Hermitian Calabi flow starting from $J$ always lies in the complexified orbit $\mathcal{D}_{J}$.

Proposition 5.1**.**

Along the Hermitian-Calabi flow, we have

[TABLE]

So the Hermitain Calabi functional is strictly decreasing along the flow unless $J$ is EAK.

Remark 5.1*.*

In Kähler case, when the flow $J(t)$ is integrable, the Hermitian Calabi flow coincides with the classical Calabi flow

[TABLE]

up to a diffeomorphism (c.f. [7, Lemma 5.1]). In [25], Li-Wang-Zheng proved the convergence theorems of the Calabi flow on extremal Kähler surfaces, which partially confirm Donaldson’s conjectural picture [11] for the Calabi flow in complex dimension 2, see [25] for more references therein.

In this section, we choose a local coordinate system $(x_{1},\ldots,x_{n})$. For simplicity, we denote $\frac{\partial}{\partial x_{i}}$ by $\partial_{i}$ and we denote the covariant derivative ${\rm D}_{i}={\rm D}_{\frac{\partial}{\partial x_{i}}}$.

Lemma 5.2**.**

Let $X=X^{i}\partial_{i}$ be a vector field. Then

[TABLE]

If $X={\rm{grad}}_{\omega}f$, then

[TABLE]

where $\omega^{ij}=g^{kj}J_{k}^{i}$ and

[TABLE]

Proof.

It is a direct computation

[TABLE]

The local formula of $(J\mathcal{L}_{X}J)^{q}_{p}$ follows from substituting

[TABLE]

into in the last two terms in

[TABLE]

∎

Lemma 5.3**.**

The Hermitian Calabi flow defined in Definition 5.1 is a 4th order flow.

Proof.

First of all, we see $\mathbb{K}=\omega^{ij}\frac{\partial s^{\nabla}}{\partial x_{j}}\partial_{i}$ and $\omega^{ij}=g^{kj}J_{k}^{i}$ does not depend on $J$.

Secondly, inserting $X=\mathbb{K}$ and $f=s^{\nabla}$ in Lemma 5.2, we have the expression of $J\mathcal{L}_{\mathbb{K}}J$ in local coordinates

[TABLE]

where $L_{1}$ denotes the lower order derivative terms of $s^{\nabla}$.

In conclusion, since $s^{\nabla}$ depends on 2nd order derivative of $J$, we have the highest order derivative of $J$ involved in $(J\mathcal{L}_{\mathbb{K}}J)_{p}^{q}$ is 4, thus Hermitian Calabi flow is a 4th order flow. ∎

5.1. Linearisation operator of $J\mathcal{L}_{\mathbb{K}}J$

We define

Definition 5.2**.**

$\widetilde{H}:=D(J\mathcal{L}_{\mathbb{K}}J),\quad T_{J}{\mathcal{AC}}_{\omega}\to T_{J\mathcal{L}_{\mathbb{K}}J}T_{J}{\mathcal{AC}}_{\omega}.$

We first compute $D\mathbb{K}$.

Lemma 5.4**.**

Let $X={\rm{grad}}_{\omega}f$ a Hamiltonian vector field with $f$ depending on $J$. Then

[TABLE]

In particular, $D\mathbb{K}(v)=-{\rm{grad}}_{\omega}(JP)^{*}v.$

Proof.

We use the identity: $\iota_{X}\omega=-df$ to see

[TABLE]

So, the first identity is obtained. The variation of $\mathbb{K}$ follows from Lemma 3.3.

∎

Then we have $D(\mathcal{L}_{X}J)$.

Lemma 5.5**.**

Let $X$ as given above. Then

[TABLE]

Proof.

It follows from $D(\mathcal{L}_{X}J)(v)=\mathcal{L}_{{\rm{grad}}_{\omega}Df(v)}J+\mathcal{L}_{X}v$ and the definition of $P$. ∎

Now, we compute the linearisation operator $\widetilde{H}$.

Lemma 5.6**.**

$\widetilde{H}(v)=-2JP(JP)^{*}v+J\mathcal{L}_{\mathbb{K}}v+v\mathcal{L}_{\mathbb{K}}J.$**

Proof.

We continue to calculate by taking $X=\mathbb{K}$ in the previous lemmas

[TABLE]

Inserting the formula of $D\mathbb{K}$, we obtain

[TABLE]

Thus we prove the lemma. ∎

Lemma 5.7**.**

The principal terms of $\widetilde{H}$ lie in $-2JP(JP)^{*}v$ and

[TABLE]

In particular, if $J$ has constant Hermitian Calabi functional, then

[TABLE]

Proof.

Since $J\mathcal{L}_{\mathbb{K}}v$ is a first order derivative of $v$ and $v\mathcal{L}_{\mathbb{K}}J$ does not involve derivative terms of $v$, the principal term of $\widetilde{H}$ is contained in $-2JP(JP)^{*}v$.

Now we consider the kernel space

[TABLE]

Since the HCF always lies in the complexified orbit, we only need to consider $\ker(JP)^{*}$ in $T_{J}\mathcal{D}=\operatorname{Im}P+\operatorname{Im}JP$. For any $v=P(f)+JP(\phi)\in\ker(JP)^{*}$, we have

[TABLE]

If $J$ has constant Hermitian scalar curvature, then $\operatorname{Im}P\perp\operatorname{Im}JP$, which implies $(\operatorname{Im}JP)^{\perp}=\operatorname{Im}P$. So we have

[TABLE]

Hence any $v\in\ker JP(JP)^{*}$ takes the form $v=P(f)$ for some $f\in C^{\infty}(M,{\mathbb{R}})$. Thus we have shown that

[TABLE]

∎

5.2. Weak parabolicity of Hermitian Calabi flow

Lemma 5.8**.**

Any 1-form $\xi\in T^{*}M$ induces an element $\Xi\in T_{J}{\mathcal{AC}}_{\omega}$ by

[TABLE]

In local coordinates, we denote $\Xi=\Xi^{d}_{c}\partial_{d}\otimes dx_{c}$. Then

[TABLE]

and

[TABLE]

Proof.

We need to prove that $J\Xi(X)+\Xi J(X)=0$ and $(\Xi(X),Y)=(X,\Xi(Y))$, for any $X,Y\in TM$. Since

[TABLE]

and

[TABLE]

Thus $\Xi\in T_{J}{\mathcal{AC}}_{\omega}$.

In local coordinates $(x_{1},\ldots,x_{n})$, we have

[TABLE]

and

[TABLE]

Here we use the fact $g^{ic}g_{jd}v^{j}_{i}=v^{c}_{d}$, since $v$ is self-adjoint. ∎

We use $L_{i},i=1,2,\ldots$ to denote the terms containing the derivatives with orders strictly less than 2.

For $v=v^{p}_{l}\frac{\partial}{\partial x_{p}}\otimes dx_{l}\in T_{J}{\mathcal{AC}}_{\omega}$ we first compute $(JP)^{*}v=\delta J(\delta v)^{\flat}$ in local coordinates.

Lemma 5.9**.**

[TABLE]

Proof.

We compute that

[TABLE]

The 2rd derivative terms only lie in $-g^{ij}{\rm D}_{i}[J(\delta v)^{\flat}(\partial_{j})]$. We compute

[TABLE]

So the lemma is proved.

∎

Then we compute $JP$ in local coordinates in terms of covariant derivatives as well.

Lemma 5.10**.**

[TABLE]

Proof.

Due to Lemma 5.2, we have

[TABLE]

and ${\rm{grad}}_{\omega}f=(g^{ab}J^{l}_{b}{\rm D}_{a}f)\partial_{l}$, we get

[TABLE]

While $\sum_{l}J^{d}_{l}J^{l}_{k}=-\delta^{d}_{k}$, we thus prove this lemma.

∎

Proposition 5.11**.**

For a covector $\xi\in T_{x}^{*}M$, the principal symbol of $JP(JP)^{*}$ is

[TABLE]

where $\Xi\in T_{J}{\mathcal{AC}}_{\omega}$ is defined in Lemma 5.8. So we have

[TABLE]

and if we choose $v=J\Xi$, then $(\hat{\sigma}_{4}(x,\xi)v,v)=0$. Thus the Hermitian Calabi flow is a 4th order weakly parabolic system.

Proof.

According to Lemma 5.3, $JP(JP)^{*}$ is a 4th order operator, we only need to compute the 4th order derivative terms of $JP(JP)^{*}$.

Combined Lemma 5.9 and Lemma 5.10, the principal term of $2JP(JP)^{*}v$ becomes

[TABLE]

i.e., for any non-zero covector $\xi=\xi_{i}dx_{i}$, it holds that

[TABLE]

Since

[TABLE]

and

[TABLE]

we further have

[TABLE]

By Lemma 5.8, we thus obtain

[TABLE]

i.e.

[TABLE]

∎

Appendix A Explicit expression of Lichnerowicz operator

In this section, we give the explicit expression of Lichnerowicz operator:

[TABLE]

where ${\rm Ric}^{+}$ is defined in (2.10), $\rho$ is defined in (2.11), and

[TABLE]

Our result is in fact a continuous computation of Vernier [36], where the expression of $\mathbb{L}(f)$ was given by a $\Delta^{2}$ term plus an error term. We will write down the error term explicitly.

By Lemma 3.1, Lichnerowicz operator has an equivalent expression:

[TABLE]

We now begin to calculate $\delta(J\mathcal{L}_{{\rm{grad}}_{\omega}f}J)$. Since

[TABLE]

defining $\mathcal{N}_{f}(X):=N({\rm{grad}}_{\omega}f,X)$, we have

[TABLE]

Our whole proof of (A.1) is divided into two parts

[TABLE]

In the first part, we deal with $I$, and in the second part, we deal with $II$.

We first introduce the following lemmas.

Lemma A.1** ([30, Lemma 3.19]).**

Let $(M,g)$ be a Riemannian manifold, $\psi_{t}$ be the flow of the vector field $\xi\in\Gamma(TM)$, we have

[TABLE]

Lemma A.2** ([23, Lemma 2.2]).**

For any real 1-form $\alpha$,

[TABLE]

where the star Ricci form $\rho^{*}$ is defined in (2.12).

Lemma A.3**.**

For any $\xi\in\Gamma(TM)$,

[TABLE]

Proof.

For any 1-form $\alpha$ on $M$, we define $\gamma,\theta\in\Omega(M)$ by

[TABLE]

then we have

[TABLE]

where locally $({\rm D}\alpha,{\rm D}\xi)=g^{ij}{\rm D}_{i}\alpha({\rm D}_{j}\xi)$, since $\int_{M}{\rm div}\gamma=\int_{M}{\rm div}\eta=0$, we have

[TABLE]

∎

The first part has been partially calculated in [36] in a local orthonormal frame $\{e_{i},\ldots,e_{n}\}=\{e_{1},e_{2},\ldots e_{m},Je_{1},\ldots,Je_{m}\}$. We will follow the computation and give a global expression.

Lemma A.4**.**

For any $X\in\Gamma(TM)$,

[TABLE]

Proof.

Denote $\psi_{t}$ the flow of the vector field ${\rm{grad}}f$. By Corollary 2.3, we have $\psi^{*}_{t}(\delta J)=\delta^{g_{t}}J_{t}=0$ where $g_{t}=\psi_{t}^{*}g,J_{t}=\psi_{t}^{*}J$. Acting $\frac{d}{dt}|_{t=0}$ on $\delta^{g_{t}}J_{t}=0$ , we have

[TABLE]

It follows from (A.4) that

[TABLE]

By Lemma A.1, we have the expression of $\delta_{ij}\frac{d}{dt}|_{t=0}[{\rm D}^{t}_{e_{i}}(Je_{j})-J({\rm D}^{t}_{e_{i}}e_{j})]$, which is

[TABLE]

Thanks to the form of the local frame $\{e_{i}\}$, we have

[TABLE]

The Bianchi identity gives

[TABLE]

Since (A.7) and (A.8) add up to zero, (A.5) becomes

[TABLE]

Applying Lemma A.3 to the second term of the right hand side of (A.9), we get

[TABLE]

Using the Weitzenbock formula on 1-form, we further write

[TABLE]

Inserting Lemma A.2, we have the resulting identity. ∎

Now let’s compute the second part $II$, we first introduce the following lemma.

Lemma A.5**.**

Let $(M,g,J,\omega)$ be an almost Kähler manifold, let $\{e_{1},\ldots,e_{n}\}$ be an orthonormal frame, then for any vector field $X\in\Gamma(TM)$, we have

[TABLE]

Proof.

Direct computation shows

[TABLE]

Lemma 2.3 implies $\delta J=-\sum_{i=1}^{n}({\rm D}_{e_{i}}J)e_{i}=0.$ So we have

[TABLE]

The last line in (A.10) vanishes since $g(e_{i},e_{j})=\delta_{ij}$,

[TABLE]

∎

Lemma A.6**.**

For any $X\in\Gamma(TM)$,

[TABLE]

Proof.

Choosing an auxiliary local orthonormal frame $\{e_{1},\ldots,e_{n}\}$, we have

[TABLE]

Thus

[TABLE]

Here, we omit the summary notation $\sum_{i=1}^{n}$. Inserting (2.5), we get

[TABLE]

Define $H_{f}\in\Gamma({\rm End}(TM))$ by $H_{f}(X)={\rm D}_{X}{\rm{grad}}f$, i.e. $g(H_{f}(X),Y)={\rm Hess}f(X,Y).$ Thus $H_{f}$ is self-adjoint with respect to $g$, i.e.

[TABLE]

But ${\rm D}_{X}J$ is anti-self-adjoint since $J$ is anti-self-adjoint, we see

[TABLE]

Then we have

[TABLE]

We apply Lemma A.5 to conclude

[TABLE]

Using (A.8) again, we have

[TABLE]

∎

We now complete the proof of Theorem 1.5, it follows from Lemma A.4 and Lemma A.6 that

[TABLE]

Since

[TABLE]

we obtain

[TABLE]