概複素多様体上の放物型フローについて
On some parabolic flows on almost complex manifolds

Masaya Kawamura
National Institute of Technology, Kochi College
Abstract
We introduce some parabolic flows on almost complex manifolds. First, we
define an almost Hermitian flow which starts at an almost Hermitian metric. The
motivation is to extend the identifiabilty theorem for a pluriclosed flow, which wasproven by Streets and Tian. In this regard, we define another parabolic flow whichis called an almost Hermitian curvature flow and prove that an almost Hermitian
flow and an almost Hermitian curvature flow are equivalent. Second, we define analmost pluriclosed flow, which starts at an almost pluriclosed metric and preserves
the almost pluriclosedness. The diﬀerence between an almost pluriclosed flow andan almost Hermitian flow is that for the first one, we use the almost pluriclosedness
in the proof of the short-time existence of the flow, on the other hand, we can prove
the short-time existence without using the almost pluriclosedness for the second one.
Third, we introduce a scalar Calabi-type flow in almost Hermitian geometry. This isa parabolic flow of almost Hermitian metrics which evolves an initial metric along thesecond derivative of the Chern scalar curvature. We show that the flow has a unique
short-time solution and also show a stability result when the background metric isquasi-Kähler with constant scalar curvature. Finally, we introduce a parabolic flowof almost balanced metrics. We show that the flow has a unique solution on compact
almost Hermitian manifolds and that if the initial structure is Kähler, then the flow
reduces to the Calabi flow.

Notice that we assume the Einstein convention omitting the symbol of sum over repeated indexes in whole this article.

1

An almost Hermitian flow and an almost Hermitian curvature flow

In [12] and [14], Streets and Tian introduced a parabolic evolution equation of pluriclosed
metrics on a compact Hermitian manifold, which is called the pluriclosed flow. At first,
they asked whether or not it is possible to prove classification results in higher dimensions
for complex non-Kähler manifolds using geometric evolution equations as in the case that
the Ricci flow was used for proving uniformization of Riemann surfaces. They tried to havea parabolic flow such that it preserves Hermitianness and as much additional structure
as possible and also is as close to the Kähler-Ricci flow as possible. Since a pluriclosed
form ω is locally given by ω = ∂η + ∂¯η̄ for some η ∈ Λ0,1 (cf. [11, Lemma 3.9]), they
concluded that it is natural to define a flow of pluriclosed metrics using a second order
Keywords: almost Hermitian metric, parabolic evolution equation, Chern connection.
2010 Mathematics Subject Classification. 53C44 (primary); 53C55; 53C15 (secondary).
E-mail address: kawamura@kochi-ct.ac.jp

1

closed (1, 1)-form (the Chern curvature form) and a first-order (0, 1)-form. From thispoint of view, they defined the pluriclosed flow starting at a pluriclosed metric. They also
introduced the Hermitian curvature flow with a quadratic Q1 in the torsion of the Chern
connection in [12]: Let (M, J) be a compact complex manifold with pluriclosed metric g0 .
They have proven that the following evolution equation has a unique solution g(t) on M
(cf. [12, Theorem 1.2]).
∂
g(t) = −S(g(t)) + Q1 (g(t)),∂t(HCF)Q1g(0) = g0 .
Streets and Tian showed that the solution of the pluriclosed flow coincides with thesolution of (HCF)Q1 as follows (cf. [12, Proposition 3.3]):
Proposition 1.1. (Streets-Tian identifiability theorem) Let (M, J) be a compact complex
manifold with pluriclosed metric g̃. The solution to (HCF)Q1 with pluriclosed initialcondition g̃ is equivalent to a solution to the pluriclosed flow with the initial condition ω̃
on M , where ω̃ is the associated real (1, 1)-form with respect to g̃.
This result indicates that the flow (HCF)Q1 preserves the pluriclosed condition. It isbecause that the pluriclosed flow preserves the pluriclosed condition and the solution to
the pluriclosed flow solves (HCF)Q1 (Streets-Tian identifiability theorem), and moreoversince the solutions are unique, a solution to (HCF)Q1 with the pluriclosed initial condition
coincides with a solution to the pluriclosed flow with the same initial metric, and hence
the flow (HCF)Q1 must preserve the pluriclosed condition. Our approach is to try to
generalize their flow to almost Hermitian cases and to expect to obtain similar results
as in the complex cases. We would like to define two parabolic flows and show that theidentifiability theorem holds for these flows on compact almost Hermitian manifolds aswell.
Let (M, J) be a compact almost complex manifold and let g be an almost Hermitian
metric on M . Let {Zr } be an arbitrary local (1, 0)-frame around a fixed point p ∈ M andlet {ζ r } be the associated coframe. Then √the associated unique real (1, 1)-form ω withrespect to g takes the local expression ω = −1grk̄ ζ r ∧ ζ k̄ . We will also refer to ω as to analmost Hermitian metric. We would like to define a parabolic flow of almost Hermitian
metrics with an almost Hermitian initial metric ω0 on (M, J). We will call it the almost
Hermitian flow (AHF):

∂
∗
∗


ω(t) − P (ω(t)) =: −Φ(ω(t)),ω(t) + ∂¯∂¯g(t)ω(t) = ∂∂g(t)
∂t
(AHF)


ω(0) = ω0 ,∗∗are the L2 -adjoint operators with respect to metrics g(t), and P (ω)
and ∂¯g(t)where ∂g(t)is one of the Ricci-type curvatures of the Chern curvature. One has with an arbitrary(1, 0)-frame {Zr } with respect to g, Pij̄ = g kl̄ Ωij̄kl̄ , where Ω is the curvature of the Chern
connection ∇ on (M, J, g).

2

Our first goal is to prove that the operator ω 7→ Φ(ω) is a strictly elliptic operator
for an almost Hermitian metric ω, which means that the equation (AHF) with an almost
Hermitian initial metric is a strictly parabolic equation. Hence the short-time existence
and the uniqueness of the solution (AHF) follows from the standard parabolic theory since
the manifold is supposed to be compact. This flow (AHF) coincides with the pluriclosed
flow if J is integrable and also the initial metric is pluriclosed.
Theorem 1.1. Given a compact almost Hermitian manifold (M, J, ω0 ), there exists a
unique solution to (AHF) with initial condition ω0 on [0, ε) for some ε > 0.
The following result indicates that the parabolic flow (AHCF) introduced in the statement of Theorem 1.2 could play the same role as the flow (HCF)Q1 on complex manifolds.
We may expect to have some other similar results as in [12], [13] and [14] for (AHCF).
Our second goal is to show the following generalized Streets-Tian identifiability theorem.
We denote by S one of the Ricci-type curvatures of the Chern curvature, which is locallygiven by Sij̄ = g kl̄ Ωkl̄ij̄ .
Theorem 1.2. Let (M, J, g0 ) be a compact almost Hermitian manifold with fundamental
form ω0 . Then the metric gt of the solution ωt to (AHF) starting at ω0 evolves as∂g(t) = −S(g(t)) − Q7 (g(t)) − Q8 (g(t)) + BT ′ (g(t)) + Z̄(T ′ )(g(t)),∂t(AHCF)

g(0) = g0 ,where wi := Tirr̄ ,jprr
BTi′j̄ := Br̄p
Tirp̄ +Bp̄i
Tprj̄ +Br̄r
Tpij̄ +Bj̄iwr , Z̄(T ′ )ij̄ := −Zr̄ (Tris )gsj̄ −Zj̄ (wi )−g pq̄ Tpir Zj̄ (grq̄ ).

Here T is the torsion of the Chern connection and the components are computed withrespect to a unitary frame. Then a solution to (AHCF) with initial condition g0 is
equivalent to a solution to (AHF) starting at the same initial condition ω0 .
In the sequel we refer to (AHCF) as almost Hermitian curvature flow. Note that thisresult in Theorem 1.2 implies that there exists a unique short-time solution to (AHCF)
with initial almost Hermitian metric on a compact almost Hermitian manifold by applying
Theorem 1.1. The parabolic flow (AHCF) coincides with the flow (HCF)Q1 starting at apluriclosed metric ω0 if J is integrable .
Proposition 1.2. The parabolic flow (AHCF) coincides with the flow (HCF)Q1 startingat a pluriclosed metric if J is integrable.
Proof. Under our assumption that J is integrable, we have Q7 = Q8 = BT ′ = 0.
Also, since we have ∂r̄ Trij̄ = ∂j̄ Trir̄ for a pluriclosed metric on a Hermitian manifold (onecan check that (HCF) preserves the pluriclosedness (cf. [12, Theorem 3.4])) and then we
may choose a local (1, 0)-frame Zr = ∂z∂ r for some complex local coordinate {z1 , . . . , zn },we obtain
Z̄(T ′ )ij̄ = −∂r̄ (Tris )gsj̄ − ∂j̄ (wi ) − g pq̄ Tpir ∂j̄ grq̄ = −∂r̄ Trij̄ + Tris ∂r̄ gsj̄ − ∂j̄ Tirr̄ − g pq̄ Tpir Γk̄j̄ q̄ grk̄= −∂j̄ Trir̄ − ∂j̄ Tirr̄ + Tris Γk̄r̄j̄ gsk̄ − Tris Γk̄j̄ r̄ gsk̄ = Tirs̄ Tj̄ r̄s = Q1ij̄ ,3

where we are writing ∂r , ∂j̄ for ∂z∂ r , ∂z∂ respectively. Combining these yields the result.
j̄

The result in Proposition 1.2 tells us that our flow (AHCF) can be considered as a
generalized flow of the pluriclosed flow and (HCF)Q1 . Here, concerning the diﬀerence
between the flow (AHCF) and Vezzoni’s flow in [15], Vezzoni studied the parabolic flow
on a compact almost Hermitian manifold (M 2n , J, g0 ) such that∂g(t) = −S − Q7 − Q8 + Q =: −K∂twith g(0) = g0 , where Q1 , Q2 , Q3 , Q4 , Q7 , Q8 are quadratics in the torsion of the Chern
connection (cf. [15, pg. 712])
Q1ij̄ := Tikr̄ Tj̄ k̄r , Q2ij̄ := Tk̄r̄i Tkrj̄ ,Q3ij̄ := Tikk̄ Tj̄ r̄r ,and1
Q4ij̄ := (Trkk̄ Tr̄j̄i + Tr̄k̄k Trij̄ ), Q7ij̄ := Tirk Tr̄k̄j̄ , Q8ij̄ := Tirk Tj̄ k̄r̄
2
1 1
1 2
and Q := 2 Q − 4 Q − 12 Q3 +Q4 . These components are defined using an arbitrary unitary
frame. Vezzoni’s flow was defined for generalizing some studies on (HCF)Q in [18] andHermitian Hilbert functional:
∫
1−n
n
kdV,
F(g) = Vol(M )Mwhere k = trg K = s − 14 |T ′ |2 − 12 |w|2 , where s is the scalar curvature of the Chern
connection. Note that we have trg (Q7 + Q8 ) = 0. Vezzoni showed that a metric g is a
critical point of F if and only if k is constant and K = nk g.
Our approach is diﬀerent from Vezzoni’s one, which is to try to generalize StreetsTian identifiability theorem in [12]. Note that we have P = S + div∇ T ′ − ∇w̄ + Q7 + Q8
for any almost Hermitian metric g (cf. [15, Lemma 3.5]), where T ′ is the torsion of
the Chern connection ∇ associated to g, (div∇ T ′ )ij̄ = g kl̄ ∇l̄ Tkij̄ , (∇w̄)ij̄ = g kl̄ ∇i Tj̄ l̄k .
¯ − BT ′ − Z̄(T ′ ), ∂∂ ∗ ω = −∇w̄ for any
For this purpose, we show that div∇ T ′ = −∇wgalmost Hermitian metric ω, and by using these formulae, we generalized the identifiability
theorem. This generalized result indicates that (AHCF) could possibly play the same roleas the flow (HCF)Q1 and the pluriclosed flow. We may expect to have some other similar
results as in [12] for (AHCF).

2

An almost pluriclosed flow

Let (M, J) be a compact almost complex manifold and let g be an almost Hermitian
metric on M . Let {Zr } be an arbitrary local (1, 0)-frame around a fixed point p ∈ M andlet {ζ r } be the associated coframe.
associated real (1, 1)-form ω with respect to√ Then ther
g takes the local expression ω = −1grk̄ ζ ∧ ζ k̄ . We will also refer to ω as to an almost
Hermitian metric. We would like to define a parabolic flow of almost Hermitian metrics
with an almost pluriclosed initial metric ω0 on (M, J). We say a metric ω is almost4¯
pluriclosed if ω is an almost Hermitian metric and ∂ ∂-closed(cf. Definition 2.1). We willcall it the almost pluriclosed flow (APF): ∂∗∗ω(t) + ∂¯∂¯g(t)ω(t) − P (ω(t)) =: −Φ(ω(t)),
 ∂t ω(t) = ∂∂g(t)(APF)
ω(0) = ω0 ,∗∗where ∂g(t)and ∂¯g(t)are the L2 -adjoint operators with respect to metrics g(t), and P (ω)
is one of the Ricci-type curvatures of the Chern curvature. One has with an arbitrary(1, 0)-frame {Zr } with respect to g, Pij̄ = g kl̄ Ωij̄kl̄ = −g kl̄ Zj̄ Zi (gkl̄ ) + O(∂ω), where Ω isthe curvature of the Chern connection ∇ on (M, g, J) and O(∂ω) means an expression
which only depends on at most first derivatives of ω.
First, by using the almost pluriclosedness, we prove that the operator ω 7→ Φ(ω) is astrictly elliptic operator for an almost pluriclosed metric ω, which means that the equation(APF) with an almost pluriclosed initial metric is a strictly parabolic equation. Hence theshort-time existence and the uniqueness of the solution (APF) follows from the standard
parabolic theory since the manifold is supposed to be compact. This flow (APF) coincides
with the pluriclosed flow if J is integrable and also the initial metric is pluriclosed (cf.
[12], [14]). We define an almost pluriclosed metric on almost Hermitian manifolds.

Definition 2.1. Let (M, J) be an almost complex manifold. A metric g is called analmost pluriclosed metric on M if g is an almost Hermitian metric whose associatedreal (1, 1)-form (,which is √called the fundamental (1, 1)-form ω = g(J·, ·) of the almost¯ = 0.
Hermitian metric g,) ω = −1gij̄ ζ i ∧ ζ j̄ satisfies ∂ ∂ω
We will also refer to the associated real (1, 1)-form ω as an almost pluriclosed metric.
Theorem 2.1. Given a compact almost Hermitian manifold (M, J, ω0 ) with almost pluriclosed metric ω0 , there exists a unique solution ω(t) to (APF) with initial condition ω0
for t ∈ [0, ε) for some ε > 0. Moreover, if ω(0) is almost pluriclosed, the metric ω(t) isalmost pluriclosed for all t ∈ [0, ε).
Since the almost pluriclosedness can be preserved along the flow (AHF), we may use
the almost pluriclosedness for proving its short-time existence.
We denote by S one of the Ricci-type curvatures of the Chern curvature, which is
locally given by Sij̄ = g kl̄ Ωkl̄ij̄ . Second, we prove that a solution of the almost pluriclosed
flow with initial almost pluriclosed metric ω0 is equivalent to a solution to (AHCF) startingat the initial almost pluriclosed metric ω0 on a compact almost complex manifold.
Theorem 2.2. Given a compact almost Hermitian manifold (M, J, ω0 ) with almost pluriclosed metric ω0 , if (M, J, ω(t)) is a solution to (APF) starting at ω0 , then it coincides
with a solution to (AHCF) starting at ω0 .
Third, we prove that the equation (AHCF) is a strictly parabolic equation. Therefore, the short-time existence and the uniqueness of the solution to (AHCF) with initialcondition ω0 follow from the standard parabolic theory since the manifold is supposed tobe compact. Since the solution to (AHCF) are unique, the solution to (AHCF) exactly
coincides with the solution to (APF).
5

Theorem 2.3. Given a compact almost Hermitian manifold (M, J, ω0 ) with almost pluriclosed metric ω0 , there exists a unique solution ω(t) to (AHCF) with initial condition ω0
for t ∈ [0, ε) for some ε > 0. Moreover, if ω(0) is almost pluriclosed, the metric ω(t) isalmost pluriclosed for all t ∈ [0, ε).
Now we show that the almost pluriclosedness can be preserved along the solution to(APF). We need the following lemma.
Lemma 2.1. One has¯ = 0,∂ ∂∂∂ ∂¯2 = 0.

¯ = 0, AĀ + ∂ ∂¯ + ∂∂¯ + ĀA = 0, ∂ Ā + ∂¯2 + Ā∂ = 0, then
Proof. Using A∂¯ + ∂ 2 + ∂A
we compute¯ = −∂(AĀ + ∂ ∂¯ + ĀA) = A∂ Ā − ∂ 2 ∂¯ − ∂ ĀA = A∂ Ā + A∂¯2 + ∂A¯ ∂¯ − ∂ ĀA∂ ∂∂¯ ∂¯ − ∂ ĀA = −AĀ∂ + ∂A¯ ∂¯ − ∂ ĀA.
= A∂ Ā − A∂ Ā − AĀ∂ + ∂A¯¯ ∂A¯ − AĀ∂A − ∂ ĀAA = ∂A¯ ∂A¯ − AĀ∂A, equivalently
Since A2 = 0, we obtain ∂ ∂∂A= ∂A
we obtain¯ − ∂A¯ ∂¯ + AĀ∂)A = 0,(∂ ∂∂which implies that we have¯ = ∂A¯ ∂¯ − AĀ∂,∂ ∂∂∂ ĀA = 0.

¯ = ∂A¯ ∂¯ − AĀ∂ = ∂A¯ ∂¯ + ĀA∂ + ∂∂¯ 2 + ∂ ∂∂,¯ which tells us that
And then we compute ∂ ∂∂we have¯ ∂¯ = −ĀA∂ − ∂∂¯ 2.
∂AFrom the equality ∂ Ā + ∂¯2 + Ā∂ = 0, we obtain0 = ∂ ĀA + ∂¯2 A + Ā∂A = ∂¯2 A + Ā∂A,
where we used that ∂ ĀA = 0. Then we have ∂ Ā∂A + ∂ ∂¯2 A = (∂ Ā∂ + ∂ ∂¯2 )A = 0, which
implies that we have¯ ∂¯ = −∂∂¯ 2 , ĀA∂ = 0, ∂¯2 A = 0∂ Ā∂ = −∂ ∂¯2 , ∂A
and also we obtain from ∂¯2 A + Ā∂A = 0,A∂¯2 A + AĀ∂A = 0,equivalently(A∂¯2 + AĀ∂)A = 0,
which gives us thatA∂¯2 = −AĀ∂.
¯ = ∂A¯ ∂¯ − AĀ∂ = −∂∂¯ 2 + A∂¯2 , which gives
Therefore, ∂ ∂∂¯
¯ 2 A + A∂¯2 A = −∂∂¯ 2 A,
∂ ∂∂A
= −∂∂
6

¯ + ∂∂¯ 2 )A = 0, which implies thatequivalently (∂ ∂∂¯ = −∂∂¯ 2 , AĀ∂ = 0.
∂ ∂∂Finally, we obtain since ∂ 2 Ā = 0 and AĀ∂ = ĀA∂ = 0,
¯ Ā = −∂∂¯ 2 Ā − AĀ∂ Ā − ĀA∂ Ā = (−AĀ∂ − ĀA∂)Ā = 0 · Ā = 0,
∂ ∂∂which implies we have
¯ = 0,∂ ∂∂∂ ∂¯2 = 0.

Proposition 2.1. The almost pluriclosedness is preserved along the solution to (APF).
Proof. From the results in Lemma 3.4, a direct computation gives∂ ¯¯¯ ∗ ω(t) + ∂ ∂¯∂¯∂¯∗ ω(t) − ∂ ∂P
¯ (ω(t)) = 0,
∂ ∂ω(t) = −∂ ∂Φ(ω(t))= ∂ ∂∂∂
g(t)
g(t)
∂t
where we used that P (ω) is closed.

3

A scalar Calabi-type flow

In [2], Bedulli and Vezzoni introduced a geometric flow of Hermitian metrics which evolves
an initial metric along the second derivative of the Chern scalar curvature. They showedthat the flow has a unique short-time solution and provide a stability result when the
background metric is Kähler with constant scalar curvature. We extend thier results in
the Hermitian geometry to the almost Hermitian geometry.
Let (M, J) be a compact almost complex manifold and let g be an almost Hermitian
metric on M . Let {Zr } be an arbitrary local (1, 0)-frame around a fixed point p ∈ M andlet {ζ r } be the associated coframe.
associated real (1, 1)-form ω with respect to√ Then ther
g takes the local expression ω = −1grk̄ ζ ∧ ζ k̄ .
Given an almost Hermitian form ω on a compact almost complex manifold (M, J), we
consider the set√n−2¯Cω∞ (M ) := {φ ∈ C ∞ (M )|ω n−1 + −1∂ ∂(φω
) > 0},√
where the −1∂ ∂¯ is in the almost Hermitian geometry, given by for φ ∈ C ∞ (M ),√√¯ = 1 (dJdφ)(1,1) = −1(Zi Zj̄ − [Zi , Zj̄ ](0,1) )φζ i ∧ ζ j̄ .
−1∂ ∂φ2

Every φ ∈ Cω∞ (M ) induces an almost Hermitian form ωφ defined by√n−2¯
ωφn−1 = ω n−1 + −1∂ ∂(φω).
We denote by Cω (M ) the setCω (M ) := {ωφ |φ ∈ Cω∞ (M )}.
7

We investigate the solutions ω(t) ∈ Cω (M ) to the following parabolic flow;√∂¯ ω(t) ω n−2 ),ω(t)n−1 = −1∂ ∂(s
∂t
(†)


ω(0) = ω0 ,whose definition depends on the background almost Hermitian form ω, where ω0 is analmost Hermitian form and sω(t) is the Chern scalar curvature of the Chern connection
induced by ω(t). If the background almost Hermitian metric is quasi-Kähler, then almost
Hermitian metrics with constant Chern scalar curvature are stationary solutions to theflow (†). A quasi-Kähler structure is an almost Hermitian structure whose associated¯ = 0 (cf. [3]). A quasi-Kähler manifold with J
real (1, 1)-form ω satisfies (dω)(1,2) = ∂ωintegrable is a Kähler manifold.
The equation (†) can be reduced to the scalar equation;∂
φ(t) = sφ(t) ,
∂t
(†)′


φ(0) = φ0 ,where φ0 ∈ Cω∞ (M ) is such that ωφ0 = ω0 , and for φ ∈ Cω∞ (M ), sφ is the Chern scalar
curvature of ωφ . If φ(t) satisfies (†)′ , then ωφ(t) satisfies (†). Conversely, if ωφ(t) satisfies(†), then there exists φ̂(t) satisfying ωφ̂(t) = ωφ(t) and (†)′ .
We next define an almost balanced metric.
Definition 3.1. Let (M 2n , J) be an almost complex manifold. A metric g is called analmost balanced√metric on M if g is an almost Hermitian metric whose associated real
(1, 1)-form ω = −1gij̄ ζ i ∧ζ j̄ satisfies d(ω n−1 ) = 0. And when an almost Hermitian metricg is almost balanced, the triple (M 2n , J, g) will be called an almost balanced manifold.
We will show that the flow (†) preserves the almost balanced condition. We obtainthe following generalized result in the almost Hermitian geometry.
Theorem 3.1. The flow (†) has always a unique short-time solution {ω(t)}t∈[0,Tmax ) .
If the initial almost Hermitian metric ω0 satisfies almost balanced condition, then themetrics ω(t) are almost balanced for all t ∈ [0, Tmax ). Assume that the backgroundalmost Hermitian metric ω is quasi-Kähler with constant Chern scalar curvature. Then if
ω0 is close enough to ω in C ∞ -topology, the solution {ω(t)}t is defined for any t ∈ [0, ∞),and converges in C ∞ -topology to ω as t → ∞.
Let (M, g) be an oriented compact Riemannian manifold with the volume form dVg
with respect to g and let W+2r,2 (M ) be an open neighborhood of 0 in W 2r,2 (M ) which isinvariant by additive constants. For k > 2r, we define W+k,2 (M ) := W+2r,2 (M ) ∩ W k,2 (M )and C+∞ (M ) := C ∞ (M ) ∩ W+2r,2 (M ). Let Q : W+2r,2 (M ) → L2 (M ) be a smooth elliptic
operator of order 2r in a strong sense explained such as in [2], [4] and [9] (we will see
later), and we denote by L the diﬀerential of Q at 0. Assume further that Q satisfies thefollowing conditions:
8(I) Q(0) = 0 and Q(ψ) = Q(ψ + a) for every ψ ∈ W+2r,2 (M ), a ∈ R (the set of constantfunctions is identified with R);(II) The kernel of L is made only by constant functions and L(W02r,2 (M )) ⊆ L20 (M ),where the subscript 0 means that the elements have average 0 with respect to g;(III) L is symmetric and semi-negative definite with respect to the L2 -scalar productinduced by the fixed metric g on M , i.e., for every ψ1 , ψ2 ∈ W 2r,2 (M ),∫∫∫L(ψ1 )ψ2 dVg =L(ψ2 )ψ1 dVg andL(ψ1 )ψ1 dVg ≤ 0.
M

M

M

We note here the ellipticness in a strong sense for the operator Q as we mentioned before. Let (M, g) be a compact m-dimensional Riemannian manifold and let Q : C ∞ (M ) →C ∞ (M ) be a quasi-linear partial diﬀerential operator of order 2r. Then, Q(ψ) is locallygiven asQ(ψ)(x) = Ai1 ...i2r (x)(ψ(x), ∇ψ(x), . . . , ∇2r−1 ψ(x))∇2r
i1 ...i2r ψ(x)+b(x)(ψ(x), ∇ψ(x), . . . , ∇2r−1 ψ(x)),
where ∇ is the Levi-Civita connection of g and the functions Ai1 ...i2r and b are smooth intheir entries. We assume that Q is elliptic by requiring
Ai1 j1 ...ir jr = (−1)r−1 E1i1 j1 . . . Erir jr ,for some (2, 0)-tensors E1 , . . . , Er such that there exits a positive constant λ ∈ R suchthat each tensor Ek (k = 1, . . . , r) satisfies
Ekij (x)(v1 , . . . , v2r−1 )ξi ξj ≥ λ|ξ|2g for every ξ ∈ Tx∗ M ,when x ∈ M and vk ∈ ⊗k Tx∗ M . Given such a smooth elliptic quasi-linear operator Q oforder 2r and an initial datum φ0 ∈ C ∞ (M ), we consider the parabolic equation∂
φ(t) = Q(φ(t)),
∂t
(♯)


φ(0) = φ0 .
We notice that the following result can be found in [4] and [9].
Proposition 3.1. (cf. [4, Theorem 7.15], [9, Theorem 1.1]) For every φ0 ∈ C ∞ (M ),there exists a time Tmax > 0 such that the parabolic problem (♯) has a maximal solution
φ ∈ C ∞ (M × [0, Tmax )). Moreover, the solution φ is unique and depends continuously on
the initial datum φ0 in the C ∞ (M )-topology.
Under the assumptions (I), (II) and (III) for the smooth elliptic operator Q, Bedulli
and Vezzoni obtained the following stability result:9

Proposition 3.2. ([2, Theorem 1.2]) For every ε > 0, there exists δ > 0 such that if
φ0 ∈ C+∞ (M ) satisfies ||φ0 ||C ∞ < δ, then the parabolic problem∂
φ(t) = Q(φ(t)),∂tφ(0) = φ0
has a unique solution φ(t) ∈ C ∞ (M × [0, ∞)) such that φ(t) ∈ C+∞ (M ) for every t andsatisfies(I) ||φ(t)||C ∞ < ε for every t ∈ [0, ∞);(II) φ(t) converges in C ∞ -topology to a smooth function φ∞ such that Q(φ∞ ) = 0.
We need the following lemma for proving the preservation of the almost balancedness.
By applying some results in the proof of Lemma 2.1, we obtain the following equalities.
Lemma 3.1. One has¯ ∂¯ = 0.
∂ 2 ∂¯ = 0, ∂∂¯ + ĀA = 0, ∂ Ā + ∂¯2 + Ā∂ = 0, ∂A + A∂ = 0 and theProof. By using AĀ + ∂ ∂¯ + ∂∂¯ = 0, we obtain
results in the proof of [9, Lemma 3.4]; ∂ ĀA = 0, A∂¯2 = −AĀ∂ and ∂ ∂∂¯ + ĀA) = −∂AĀ − ∂ ∂∂¯ − ∂ ĀA
∂ 2 ∂¯ = ∂∂ ∂¯ = −∂(AĀ + ∂∂= A∂ Ā = −A∂¯2 − AĀ∂ = 0.

By checking conditions (I), (II) and (III) for the elliptic operator Q(ψ) = sψ − R,
ψ ∈ Cω∞ (M ), where R is the constant Chern scalar curvature of the quasi-Kähler metricω, and applying Proposition 3.1, 3.2, we can show Theorem 3.1.

4

A parabolic flow of almost balanced metrics

In [1], Bedulli and Vezzoni introduced a parabolic flow of balanced metrics and they
showed that the flow has a unique short-time solution. We extend their results in the
Hermitian geometry to the almost Hermitian geometry.
Let M n be a complex manifold and let ω be a fundamental (1, 1)-form associated toa Hermitian metric g on M . In the Kähler geometry, the Calabi flow is a well-known
gradient flow of the Calabi functional∫
ωn(sω )2
ω 7→n!
M
as it is restricted to the cohomology class of an initial Kähler metric ω0√¯ > 0|ϕ ∈ C ∞ (M, R)}Cω0 = {ω0 + −1∂ ∂ϕ10

where sω is the scalar curvature of the metric ω. The Calabi flow is as follows:√∂¯ ω(t) ,ω(t) = −1∂ ∂s∂tω(0) = ω0 .
The flow above can be alternatively expressed in terms of positive (n − 1, n − 1)-forms as

√
∂


φ(t) = −1∂ ∂¯ ∗t (Pt ∧ ∗t φ(t)),∂tφ(0) = φ0 ,where Pt is the Ricci form of φ(t) and φ0 = ∗ω0 ω0 .
Let (M 2n , J) be a real 2n-dimensional compact almost complex manifold and let g be
an almost Hermitian metric on M . Let {Zr } be an arbitrary local (1, 0)-frame arounda fixed point p ∈ M and let {ζ r } be the associated coframe.√Then the associated real
(1, 1)-form ω with respect to g takes the local expression ω = −1grk̄ ζ r ∧ ζ k̄ .
Let φ0 be a positive closed (n − 1, n − 1)-form on M . We investigate the following
parabolic flow of almost balanced structures φ on M ;

√
∂


φ(t) = −1∂ ∂¯ ∗t (Pt ∧ ∗t φ(t)) + (n − 1)∆BC φ(t),



 ∂t
(‡)

where

dφ(t) = 0,






φ(0) = φ0 ,∆BC := ∂ ∂¯∂¯∗ ∂ ∗ + ∂¯∗ ∂ ∗ ∂ ∂¯ + ∂¯∗ ∂∂ ∗ ∂¯ + ∂ ∗ ∂¯∂¯∗ ∂ + ∂¯∗ ∂¯ + ∂ ∗ ∂is the modified Bott-Chern Laplacian (cf. [10]) and ∗t and Pt are the Hodge star operator
and the Chern-Ricci form locally given by Pij̄ = g kl̄ Ωij̄kl̄ , Ω is the curvature form of theChern connection with respect to g.
We can show that the modified Bott-Chern Laplacian ∆BC is a fourth order elliptic
operator by considering the symbol of the operator in the base (ζI ∧ ζJ )|I|=p,|J|=q , where{ζI }I is the associated coframe with respect to the local (1, 0)-frame {ZI }I around a pointx ∈ M as in [10].
We introduce the definition of an almost balanced metric.
Definition 4.1. Let (M 2n , J) be an almost complex manifold. A metric g is called analmost balanced√metric on M if g is an almost Hermitian metric whose associated real
(1, 1)-form ω = −1gij̄ ζ i ∧ζ j̄ satisfies d(ω n−1 ) = 0. And when an almost Hermitian metricg is almost balanced, the triple (M 2n , J, g) will be called an almost balanced manifold.
An almost balanced structure can be alternatively regarded as a closed positive real(n − 1, n − 1)-form φ. We define the Bott-Chern cohomology in almost complex geometry
as follows:kerd
HBC (M ) =¯ .
Im(∂ ∂)
Our main result is as follows.
11

Theorem 4.1. Let (M, J, g) be a compact almost Hermitian manifold and let φ0 be aclosed positive real (n − 1, n − 1)-form on M . The flow (†) admits a unique solution in
the Bott-Chern class of [φ0 ] defined in a maximal interval [0, ε) on M . Moreover, if theinitial structure is Kähler, then (†) reduces to the Calabi flow.
Let ∆A denote the modified Aeppli Laplacian (cf. [10]), which is defined by¯ ∗ + ∂∂¯ ∗ ∂ ∂¯∗ + ∂∂ ∗ + ∂¯∂¯∗ .
∆A := ∂¯∗ ∂ ∗ ∂ ∂¯ + ∂ ∂¯∂¯∗ ∂ ∗ + ∂ ∂¯∗ ∂∂We can show that the modified Aeppli Laplacian ∆A is a fourth order elliptic operator in
the same way for proving that ∆BC is elliptic. We will need a Hodge-like decomposition
induced by ∆A . We can prove the Aeppli decomposition based on the result that ∆A is
elliptic.
Proposition 4.1. (cf. [10]) If (M, J, g) is a compact almost Hermitian manifold, thenwe have the following orthogonal decomposition for every (p, q)
p,q¯ ⊕ Im(∂ ∂)¯ ∗,C ∞ (M, Λp,q ) = H∆
(M ) ⊕ (Im∂ + Im∂)Ap,qwhere H∆(M ) = Ker∆A .
A

The following proposition is a crucial step in the proof of Theorem 4.1. The proof isthe similar to the one in [1, Proposition 2.4]. The diﬀerence appears only in the part ofusing Lemma 2.1 and 3.1.
Proposition 4.2. (cf. [1, Proposition 2.4]) Let GA be the Green operator associated to¯ ∞ (M, Λpq ), we havethe modified Aeppli Laplacian ∆A . Then for every ψ ∈ ∂ ∂C
¯ A (∂ ∂)¯ ∗ (ψ).
ψ = ∂ ∂G¯ p,q . By applying the Aeppli decomposition and
Proof. Choose arbitrary ψ ∈ ∂ ∂Λp,q∗∗¯ = 0 (cf. [10]), and ∂ ∂∂¯ = 0, ∂ ∂¯2 = 0 in Lemma 2.1, weu ∈ H∆A (M ) ⇔ ∂¯ u = ∂ u = ∂ ∂u
¯ with β ∈ Im(∂ ∂)¯ ∗ . Especially, since we have ∂ ∗ ∂¯∗ ∂ ∗ = 0, ∂¯∗ ∂¯∗ ∂ ∗ = 0 fromhave ψ = ∂ ∂β¯ ∗ ∂ ∂β,¯ which tells us that β = GA ((∂ ∂)¯ ∗ ∂ ∂β)¯ =Lemma 2.1 and 3.1, we have ∆A β = (∂ ∂)∗∗¯ ψ), and hence we obtain ψ = ∂ ∂G
¯ A (∂ ∂)¯ (ψ).
GA ((∂ ∂)Next we introduce a Hodge system.
Definition 4.2. A Hodge system on a manifold M consists of the following sequaenceD

C ∞ (M, E− ) −−−→ C ∞ (M, E)∆D y
C ∞ (M, E− ) ←−−− C ∞ (M, E)∗
D

where E− and E are fiber bundles over M with an assigned metric along their fibers, D isa diﬀerential operator, D∗ is the formal adjoint of D and ∆D is an elliptic operator suchthat ψ = DGD∗ ψ for every ψ ∈ ImD, where G is the Green operator of ∆D .
12

Consider a Hodge system on a compact manifold M as in Definition 2.1. Let O be an
open subset of E such that π(O) = M , where π : E → M is the projection. Consider anon-linear partial diﬀerential operator of order 2m
L : C ∞ (M, O) → C ∞ (M, E)
and a fixed initial datum φ0 ∈ C ∞ (M, O) such thatL(φ0 + Dγ) ∈ ImD
for every γ ∈ C ∞ (M, E− ). We consider the following evolution problem∂
φ(t) = L(φ(t)),
∂t
(⨿)


φ(0) = φ0 ,where φ(t) is in the following space
U = {φ0 + Dγ|γ ∈ C ∞ (M, E− )} ∩ C ∞ (M, O)
and φ(t) is required to depend smoothly on time.
Let D2m (E, E) denote the space of partial diﬀerential operators on E of order ≤ 2m,
which can be seen as the space of smooth sections of a vector bundle. A linear partial
operator Q of order 2m is said to be strongly elliptic if its principal symbol σQ (x, ξ)satisfies the following inequality:−⟨σQ (x, ξ)v, v⟩E ≥ λ|ξ|2m |v|2m
for some positive constant λ and for all (x, ξ) ∈ T M , ξ ̸= 0 and v ∈ Ex , whose definitiondose not depend on the metric ⟨·, ·⟩E along the fibers onf E. The principal symbol of Q
is defined by√( −1)2m
Q(f 2m u)(x)σQ (x, ξ)v =(2m)!
for f ∈ C ∞ (M ) with f (x) = 0, dx f = ξ and u ∈ C ∞ (E) with u(x) = v. We denote byL∗ |φ the derivative of the operator L at φ.
Theorem 4.2. ([1, Theorem 3.2]) Let (E− , E, D, ∆D ) be a Hodge system on a compact
Riemannian manifold M . Let L, φ0 and U be as above. Assume that there exists a
nonlinear partial diﬀerential operatorL̃ : C ∞ (M, O) → D2m (E, E),

φ 7→ L̃φsuch that(I) L̃φ is strong elliptic for every φ ∈ U ;(II) L∗ |φ (ψ) = L̃φ (ψ) for every φ ∈ U and ψ ∈ DC ∞ (M, E− ).
13Assume further thatL∗ |φ (Dθ) = Dlφ (θ)
for every θ ∈ C ∞ (M, E− ), where lφ is a strongly elliptic linear diﬀerential operator on E− .
Then there exists ε > 0 such that the system (⨿) has a unique solution φ ∈ C ∞ ([0, ε), U ).
We consider√
D= −1∂ ∂¯

E− = Λn−2,n−2
M −−−−−−→ E = Λn−1,n−1
M,RR
where Λp,p
R M is the bundle of real (p, p)-forms; the subset U is the set of smooth sections
lying in the same cohomology class as φ0 . Letof Λn−1,n−1+
M ) → C ∞ (M, ΛRn−1,n−1 M )L : C ∞ (M, Λn−1,n−1+√
be the operator L(φ) = −1∂ ∂¯ ∗ (P ∧ ∗φ) + (n − 1)∆BC φ.
M ) and every closed ψ ∈We can show that for every closed φ ∈ C ∞ (M, Λn−1,n−1+
n−1,n−1∞
M ), we have
C (M, ΛR√¯ φ (ψ),L∗ |φ (ψ) = (1 − n)∆BC ψ + −1∂ ∂Ψwhere Ψφ is a linear algebraic operator on ψ with coeﬃcients depending on the torsion of
φ in a universal way. Let us consider√¯
lφ = −(n − 1)(∆A )φ + −1Φφ ◦ ∂ ∂.
As we have confirmed that −∆A is strongly elliptic. In addition, we have the following:
Proposition 4.3. One has¯ A.
∆BC ∂ ∂¯ = ∂ ∂∆Proof of Proposition 4.3. By using Lemma 3.1, we obtain∆BC ∂ ∂¯ = (∂ ∂¯∂¯∗ ∂ ∗ + ∂¯∗ ∂ ∗ ∂ ∂¯ + ∂¯∗ ∂∂ ∗ ∂¯ + ∂ ∗ ∂¯∂¯∗ ∂ + ∂¯∗ ∂¯ + ∂ ∗ ∂)∂ ∂¯¯ ∂¯ + ∂¯∗ ∂∂ ∗ ∂∂¯ ∂¯ + ∂ ∗ ∂¯∂¯∗ ∂ 2 ∂¯ + ∂¯∗ ∂∂¯ ∂¯ + ∂ ∗ ∂ 2 ∂¯
= ∂ ∂¯∂¯∗ ∂ ∗ ∂ ∂¯ + ∂¯∗ ∂ ∗ ∂ ∂∂¯
= ∂ ∂¯∂¯∗ ∂ ∗ ∂ ∂.
On the other hand, using Lemma 2.1 we have¯ A = ∂ ∂(¯ ∂¯∗ ∂ ∗ ∂ ∂¯ + ∂ ∂¯∂¯∗ ∂ ∗ + ∂ ∂¯∗ ∂∂¯ ∗ + ∂∂¯ ∗ ∂ ∂¯∗ + ∂∂ ∗ + ∂¯∂¯∗ )∂ ∂∆¯ ∗ + ∂ ∂¯2 ∂ ∗ ∂ ∂¯∗ + ∂ ∂∂∂¯ ∗ + ∂ ∂¯2 ∂¯∗¯ ∂¯∂¯∗ ∂ ∗ + ∂ ∂∂¯ ∂¯∗ ∂∂= ∂ ∂¯∂¯∗ ∂ ∗ ∂ ∂¯ + ∂ ∂∂¯
= ∂ ∂¯∂¯∗ ∂ ∗ ∂ ∂.
¯ A.
Hence we get ∆BC ∂ ∂¯ = ∂ ∂∆□n−2,n−2∞
Then we have that L∗ |φ (Dψ) = Dlφ (ψ) for every closed ψ ∈ C (M, ΛR
M ). By
applying Theorem 4.2, we obtain the desired unique existence result.
If φ0 is the (n−1, n−1)-positive form of a Kähler structure, then the solution φ0 +β(t)to (‡) corresponds to a family of Kähler forms ω(t) solving the Calabi flow (cf. [1]).

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