Curvature estimate of the Yang-Mills-Higgs flow on Kähler manifolds

1.   Introduction

Let $({\cal{E}},\theta)$ be a Higgs sheaf over a compact Kähler manifold $(M,\omega)$. Higgs bundles and Higgs sheaves, which were studied by Hitchin^[1] and Simpson^[2,3], play an important role in many different areas including gauge theory, Kähler and hyperkähler geometry, group representations, and nonabelian Hodge theory. A Higgs sheaf on $(M,\omega)$ is a pair $({\cal{E}},\theta)$, where ${\cal{E}}$ is a coherent sheaf on M, and the Higgs field $\theta\in \varOmega^{1,0}({\rm End}({\cal{E}}))$ is a holomorphic section such that $\theta\wedge \theta = 0$. A torsion-free Higgs sheaf $({\cal{E}},\theta)$ is $\omega$-stable (resp. $\omega$-semistable) if for every $\theta$-invariant proper coherent sub-sheaf ${\cal{F}}\hookrightarrow {\cal{E}}$ such that

where $\mu_{\omega}({\cal{F}})$ is called the $\omega$-slope of ${\cal{F}}$, and $\omega$-degree of ${\cal{F}}$ is defined as follows:

where $c_1({\cal{F}})$ is the first Chern class of ${\cal{F}}$.

Let Σ_ε be a set of singularities, where $({\cal{E}},\theta)$ is not locally free. If $({\cal{E}},\theta)$ is locally free on the whole M, i.e., Σ_ε$= \emptyset$, there is a Higgs bundle $(E,\bar{\partial}_E,\theta)$ on M such that the Higgs sheaf $({\cal{E}},\theta)$ is generated by the local holomorphic sections of $(E,\bar{\partial}_E,\theta)$. A locally free Higgs sheaf $({\cal{E}},\theta)$ can be considered as a Higgs bundle, i.e., $({\cal{E}},\theta) = (E,\bar{\partial}_E,\theta)$.

Given an unstable torsion-free coherent Higgs sheaf $({\cal{E}},\theta)$, one can associate a filtration by $\theta$-invariant coherent subsheaves as follows:

such that the quotient ${\cal{Q}}_i = {\cal{E}}_i/{\cal{E}}_{i-1}$ is torsion-free, $\omega$-semistable, and $\mu_{\omega}({\cal{Q}}_i)>\mu_{\omega}({\cal{Q}}_{i+1})$, which is called the Harder-Narasimhan filtration of $({\cal{E}},\theta)$. The associated graded object ${\rm Gr}^{\rm {hn}}({\cal{E}},\theta) = \oplus_{i = 1}^k{\cal{Q}}_i$ is uniquely determined by the isomorphism class of $({\cal{E}},\theta)$ and the Kähler class $[\omega]$. Moreover, for every quotient ${\cal{Q}}_i$, a further filtration by sub-sheaves exists,

such that the quotients ${\cal{Q}}_{i,j} = {\cal{E}}_{i,j}/{\cal{E}}_{i,j-1}$ are torsion-free and $\omega$-stable, and $\mu_{\omega}({\cal{Q}}_{i,j}) = \mu_{\omega}({\cal{Q}}_{i})$ for each $j$. The double filtration $({\cal{E}}_{i,j})$ is called the Harder-Narasimhan-Seshadri (HNS) filtration of the Higgs sheaf $({\cal{E}},\theta)$. The associated graded object

is uniquely determined by the isomorphism class of $({\cal{E}},\theta)$ and Kähler class $[\omega]$. The number $\displaystyle \sum\limits_{i = 1}^kk_i-1$ is the length of the HNS filtration.

Given a Hermitian metric $H$ on the Higgs bundle $(E,\bar{\partial}_E,\theta)$, we consider the Hitchin-Simpson connection:

where $D_H$ is the Chern connection with respect to the Hermitian metric $H$ and $\theta^{*H}$ is the adjoint of $\theta$ with respect to $H$. The curvature of the Hitchin-Simpson connection is expressed as follows:

where $F_H$ is the curvature of the Chern connection, denoted by $D_H$. A Hermitian metric on the Higgs bundle $(E,\bar{\partial}_E,\theta)$ is said to be $\omega$-Hermitian-Einstein if it satisfies the following Einstein condition on $M$, i.e.,

where $\lambda_{E,\omega} = \dfrac{2\pi}{{\rm Vol}(M,\omega)}\mu_{\omega}(E)$ and $\varLambda_{\omega}$ denotes the contraction with the Kähler metric $\omega$. Hitchin^[1] and Simpson^[2] proved that a Higgs bundle admits a Hermitian-Einstein metric if and only if it is Higgs poly-stable. Many interesting and important generalizations and extensions can also be found in the literature (see Refs. [1, 4 –12] ).

Let $H_0$ be a Hermitian metric on the complex vector bundle $E$, ${\cal{A}}_{H_0}$ be the space of connections of $E$ compatible with the metric $H_0$, and ${\cal{A}}_{H_0}^{1,1}$ be the space of the unitary integrable connection of $E$. A pair $(A,\theta)\in {\cal{A}}_{H_0}^{1,1}\times \varOmega^{1,0}({\rm End}(E))$ is a Higgs pair if $\bar{\partial}_A\theta = 0$ and $\theta\wedge \theta = 0$. Let ${\cal{B}}_{(E,H_0)}$ denote the space of all Higgs pairs on the Hermitian vector bundle $(E,H_0)$. We consider the following Yang-Mills-Higgs flow on the Hermitian vector bundle $(E,H_0)$ with initial data $(A_0,\theta_0)\in {\cal{B}}_{(E,H_0)}$:

The Yang-Mills flow was first introduced by Atiyah-Bott in Ref. [13]. Simpson^[2] induces the following Hermitian-Yang-Mills-Higgs (HYMH) flow for Hermitian metrics $H(t)$ on the Higgs bundle $(E,A_0,\theta_0)$ with initial metric $H_0$:

Simpson^[2] proved the long-time existence of the Hermitian-Yang-Mills-Higgs flow and demonstrated convergence under the condition that the Higgs bundle is stable. In Ref. [14], Li and Zhang showed that by choosing complex gauge transformations $\sigma(t)$ that satisfy $\displaystyle \sigma(t)^*\displaystyle \sigma(t) = H_0^{-1}H(t)$, $(A(t),\;\theta(t)) =$$\left( {\displaystyle \sigma(t)(A_0),\;\displaystyle \sigma(t)(\theta_0)} \right)$ is the unique long-time solution of the Yang-Mills-Higgs flow Eq. (8).

According to Uhlenbeck's compactness^[15,16], for any sequence $A(t_i)$ along the flow, there is a subsequent that weakly converges to a Yang-Mills connection $A_{\infty}$ in the gauge transformation sense outside a closed subset Σ_an of Hausdorff complex co-dimension of at least 2. We call Σ_an the bubbling set or the analytic singular set. In contrast, we denote Σ_alg as the singular set of the associated graded object ${\rm Gr}_{\omega}^{\rm hns}(E,\partial_{A_0},\theta_0)$, i.e., ${\rm Gr}_{\omega}^{\rm hns}(E,\partial_{A_0},\theta_0)$ is locally free away from Σ_alg, which is a complex analytic sub-variety of complex co-dimension of at least 2 that we call algebraic singular set. According to the results in Refs. [17, 18], it is not difficult to obtain that Σ_alg $\;\subset\;$ Σ_an. It is an interesting problem to demonstrate that Σ_an $\;\subset\;$ Σ_alg. This problem was solved by Daskalopoulos and Wentworth in Ref. [19] for Kähler surfaces, and by Sibley and Wentworth in Ref. [20] for Kähler manifolds. In this paper, we derive the curvature estimate in the case where $(E,\bar{\partial}_{A_0},\theta_0)$ is non-semistable and the Harder-Narasimhan-Seshadri filtration has no singularities with length one, which generalizes Li, Zhang, and Zhang's result^[21] in the Higgs bundle case. In fact, we obtain the following theorem:

Theorem 1.1. Let $(E,H_0)$ be a Hermitian vector bundle on a compact Kähler manifold $(M,\omega)$, and $(A(t),\theta(t))$ be the solution of the Yang-Mills-Higgs flow (8) with an initial Higgs pair $(A_0,\theta_0) \in {\cal{A}}_{H_0}^{1,1}\times \varOmega^{1,0}({\rm End}(E))$. If the Higgs bundle $({\cal{E}},\theta_0) =$$(E,A_0,\theta_0)$ is non-semistable and the HNS filtration has no singularities with length one, then there exists a uniform constant $C$ such that

If the algebraic singular set Σ_alg $\displaystyle \neq\emptyset$, we hypothesize that the theorem holds away from the algebraic singular set. The proof is complicated, and we mainly adapt some Li, Zhang, and Zhang's techniques to our cases of interest. However, the proof is even more complicated in the Higgs bundle case. We next provide an overview of the proposed proof. Let $H(t)$ be the long-time solution of the HYMH flow (9). It is well known that

Therefore, we only need to estimate the curvature tensor $F_{H(t),\theta_0}$ of the Hitchin-Simpson connection $D_{H(t),\theta_0}$ with respect to the evolved Hermitian metric $H(t)$. For simplicity, we denote $\theta_0$ as $\theta$ as a fixed Higgs field. According to the assumption of Theorem 1.1, there exists an exact sequence

such that $(S,\bar{\partial}_S,\theta_S)$ and $(Q,\bar{\partial}_Q,\theta_Q)$ become torsion-free, $\omega$-stable Higgs bundles, and

Let $H_S(t)$ and $H_{Q}(t)$ be the Hermitian metrics on the Higgs sub-bundle $(S,\theta_S)$ and quotient bundle $(Q,\theta_Q)$ , respectively, induced by the evolved metric $H(t)$ on $(E,\theta)$, $\gamma(t)$ as the second fundamental form, and let $\beta(t)$ be the $(1,0)$ form induced by the Higgs field $\theta$ and $H(t)$.We derive the evolution equations for $H_{S}(t)$, $H_Q(t)$, $\gamma(t)$, and $\beta(t)$ (see Eqs. (20), (21), (23), and (24)). In Ref. [2], Simpson generalized the Hermitian-Yang-Mills flow to the Higgs bundle case. Under the assumption of stability, using the results of Uhlenbeck and Yau^[22], Simpson obtained a uniform $C^0$-estimate on $H(t)$. This implies uniform higher-order estimates including the uniform curvature estimate. When $(E,\bar{\partial}_E,\theta)$ is unstable, the $C^0$-norm of the evolved metric $H(t)$ may be unbounded. In our case, we first obtain a uniform $C^0$-bound on the rescaled metrics, i.e., $\tilde{H}_S(t) = {\rm e}^{2(\lambda_S-\lambda_E)t}H_{S}(t)$ and $\tilde{H}_{Q}(t) = {\rm e}^{2(\lambda_Q-\lambda_E)t}H_Q(t)$. This is crucial in the proof of the proposed theorem, where we use the stabilities of $(S,\theta_S)$ and $(Q,\theta_Q)$ , and the property $\mu_{\omega}(S)>\mu_{\omega}(Q)$. Then, we prove that the norms of $|T_S(t)|_{H_S(t)}$, $|T_Q(t)|_{H_Q(t)}$, $|\gamma(t)|_{H(t)}$, and $|\beta(t)|_{H(t)}$ are uniformly bounded using the uniform $C^0$-estimate. By choosing suitable test functions and using the maximum principle, we obtain a uniform estimate of $|F_{H(t)}+[\theta,\theta^*]+\bar{\partial}_{E}\theta^*+\partial_H\theta|_{H(t)}$.

The remainder of this paper is organized as follows. In Section 2, we derive the evolution equations for the induced metrics $H_S(t)$ and $H_Q(t)$, the second fundamental forms $\gamma(t)$, and the induced $(1,0)$ forms $\beta(t)$. We also obtain the decomposition of Donaldson's functional, which is used in Section 3, where we derive a uniform $C^0$-bound for the rescaled metrics. In Section 4, we obtain the uniform $C^1$-estimate and complete the proof of Theorem 1.1.

2.   Evolution of the second fundamental form and the Higgs field

Let $(M,\omega)$ be a Kähler manifold that may be non-compact, and let $(E,\bar{\partial}_E,\theta)$ be a Higgs bundle over $M$. That is, $(E,\bar{\partial}_E)$ is a holomorphic vector bundle together with a Higgs field $\theta \in \varOmega_M^{1,0}({\rm End}(E))$ satisfying

Let $S$ be a $\theta$-invariant holomorphic sub-bundle of $(E,\bar{\partial}_E,\theta)$. We assume that an exact sequence of Higgs bundles exists

where $\theta_S$ and $\theta_Q$ are the Higgs fields on the $\theta$-invariant holomorphic sub-bundle $S$ and the quotient bundle $Q = E/S$ induced by $\theta$, $\bar{\partial}_S$, respectively, and $\bar{\partial}_Q$ are holomorphic structures on the holomorphic sub-bundle $S$ and the quotient bundle $Q$ induced by $\bar{\partial}_E$.

Given a Hermitian metric $H$ on $E$, we obtain the following bundle isomorphism:

where $X\in S,\; Y\in E,\; i:S\hookrightarrow E$ is the inclusion map and $\pi_H: E\rightarrow S$ is the orthogonal projection into $S$ with respect to metric $H$. Then, the pull-back metric, holomorphic structure, and Higgs field on $S\oplus Q$ respectively are

where $\gamma(t)\in \varOmega_{M}^{0,1}(S\otimes Q^*)$ and $\beta \in \varOmega_M^{1,0}(S\otimes Q^*)$. The corresponding Gauss-Codazzi equation for the Hitchin-Simpson connection is

Let $H(t)$ be the solution of the HYMH flow (9) on the Higgs bundle $(E,\bar{\partial}_E,\theta)$ with initial metric $H_0$. Now, we split the HYMH flow into the sub-bundle $S$ and quotient bundle $Q$. Recall from Ref. [21] that

and

Using the flow (9) and Gauss-Codazzi (17) equations, we have

Now, we consider the evolution of the second fundamental forms $\gamma(t)$ and $\beta(t)$:

Lemma 2.1. Let $H(t)$ be the solution of the HYMH flow (9) with initial metric $H_0$ and $\gamma(t)$ be the second fundamental form, and let $\beta(t)$ be induced by $\theta$ and $H(t)$. Then we have

and

Proof. For simplicity, we let $\varPsi(t) = 2\sqrt{-1}\varLambda_{\omega}(\partial_{S\otimes Q^{*}}\gamma+\theta_S^{*H_S} \wedge$$\beta+\beta\wedge \theta_Q^{*H_Q})$ and denote

and then

For the first evolution equation of $\gamma$, taking the derivative of the above equation with respect to $t$, we have that

Thus, we obtain the first evolution Eq. (23).

Concerning the second evolution equation of $\beta$, taking the derivative of $f^*_H(\theta)$ with respect to $t$, we have that

As a result, we obtain that $\dfrac{\partial \beta(t)}{\partial t} = \theta_S\circ \Psi(t)-\Psi(t)\circ \theta_Q$.

Lemma 2.2. Let $f_{H(t)}$ be the bundle isomorphism defined in (15). Then, we have that

where $G(t) \in \Gamma(S\otimes Q^*)$ and $G(0) = 0$. Furthermore,

and

Proof. Similar to the proof in Ref. [21], we have Eq. (27). For the sake of simplicity, we denote $\varPsi(t) = 2\sqrt{-1}\varLambda_{\omega}(\partial_{S\otimes Q^{*}}\gamma+$$\theta_S^{*H_S}\wedge\beta+\beta \wedge \theta_Q^{*H_Q})$.

Taking the derivative of Eq. (27) with respect to t and using Eq. (22), we have that

As a result, we obtain Eq. (28).

Finally, taking the derivative of Eq. (28) with respect to $\bar{\partial}_{S\otimes Q^*}$, we have that

Integrating Eq. (30) from $0$ to $t$, we obtain Eq. (29).

In the proof of the $C^0$-estimate, we need to decompose Donaldson's functional in the Higgs bundle case, which was proved by Donaldson (see Ref. [23]) for the holomorphic vector bundle case.

Let us recall Donaldson's functional in the Higgs bundle case:

and

where $H(s)$ is the path connecting the metrics $H_0$ and $H$ on $E$. Donaldson proved that the integral above is independent of the path when the base manifold $M$ is Kähler.

If we have an exact sequence of Higgs bundles, then

where $S$ is $\theta$-invariant and $\theta_S$ and $\theta_Q$ are the Higgs fields on $S$ and $Q$, respectively, induced by $\theta$. Recall that a Hermitian metric $H$ on $E$ induces Hermitian metrics $H_S$, $H_Q$ on $S$, $Q$, respectively. Note also that $\gamma_{H}\in \varOmega_M^{0,1}(S\otimes Q^*)$ is the second fundamental form, and $\beta_H\in \varOmega_{M}^{1,0}(S\otimes Q^{*})$ is determined by the Higgs field $\theta$ and Hermitian metric $H$. Then we have the following lemma:

Lemma 2.3. For any exact sequence of Higgs bundles (33), Donaldson's functional ${\cal{M}}_{E}(H_0,H(t))$ has the following decomposition:

where $H_{0,S}$ and $H_{0,Q}$ are the Hermitian metrics on $S$ and $Q$ induced by $H_0$ on $E$. $H(t)$ is a path-connecting metric between $H_0$ and $H(t)$.

Proof. Taking derivative of ${\cal{M}}_{E}^0$ with respect to $t$, we have

Then, we obtain Donaldson's functional of the pullback metric $f_{H(t)}^{*}(H(t))$:

Recall that

Using Eq. (36) and Stokes formula, we have

Given that

we obtain

Given that

We can obtain

Thus, we have

Given that

we obtain

Combining Eqs. (38) and (42), we have

Integrating Eq. (43) from $0$ to $t$, we obtain Eq. (34).

At the end of this section, we consider parabolic inequalities for $|\gamma(t)|_{H(t)}^2 = -\sqrt{-1}\varLambda_{\omega}{\rm tr}(\gamma\wedge \gamma^{*H(t)})$ and $|\beta(t)|_{H(t)}^2 =$$\sqrt{-1}\varLambda_{\omega}{\rm tr}(\beta\wedge\beta^{*H(t)})$, which will be used in the next section. Through direct calculation, we obtain

and

3.   C⁰-estimate of the rescaled metrics

Note that the Higgs bundles $(S,\bar{\partial}_S,\theta_S)$ and $(Q,\bar{\partial}_Q,\theta_Q)$ are $\omega$-stable. According to the Donaldson-Uhlenbeck-Yau theorem, we can suppose that $K_S$ and $K_Q$ are $\omega$-Hermitian-Einstein metrics on $(S,\bar{\partial}_S,\theta_S)$ and $(Q,\bar{\partial}_{Q},\theta_Q)$, i.e.,

and

where $\lambda_S = \dfrac{2\pi}{{\rm Vol}(M,\omega)}\mu_{\omega}(S)$ and $\lambda_Q = \dfrac{2\pi}{{\rm Vol}(M,\omega)}\mu_{\omega}(Q)$.

Let us denote $h_S(t) = K_S^{-1}H_S(t)$, $h_Q(t) = K_Q^{-1}H_Q(t)$, and set $\tilde{h}_S(t) =$${\rm e}^{2(\lambda_S-\lambda_E)t}h_S(t)$ and $\tilde{h}_Q(t) = {\rm e}^{2(\lambda_Q-\lambda_E)t}h_Q(t)$. Using (20) and (21), we have that

and

where we have applied the non-negativity of $\sqrt{-1}\varLambda_{\omega}\gamma^*\wedge \gamma$ and $\sqrt{-1}\varLambda_{\omega}\beta\wedge \beta^*$. We also used the following equalities:

and

Using inequalities (48) and (49), and the maximum principle, we can obtain a uniform bound on ${\rm tr}\tilde{h}_{S}+{\rm tr}\tilde{h}_{Q}^{-1}$. In fact, we obtain the following lemma:

Lemma 3.1. There exists a uniform constant $C_0$ such that

for all $t\in [0,+\infty)$.

In the following, we will derive uniform upper bounds on $\det(\tilde{h}_{S}^{-1}(t))$ and $\det(\tilde{h}_{Q}(t))$, which imply uniform upper bounds on ${\rm tr}\tilde{h}_{S}^{-1}(t)$ and ${\rm tr}\tilde{h}_{Q}(t)$ using (52). Then, we will obtain uniform $C^0$-bounds on $\tilde{h}_{S}(t)$ and $\tilde{h}_{Q}(t)$. At the beginning of the proof, the following proposition are required:

Proposition 3.2. Along the heat flow (9), we have

where $v(t)\geqslant 0$ satisfies $\int_1^{+\infty}v(t){\rm d}t \leqslant C_{1}$, and $C_{1}$ is a uniform constant.

Proof. From Lemma 2.3, for any exact sequence of Higgs bundles,

we have proved that

According to the flow equations (20) and (21), we have that

and

Then,

Furthermore, from the definition of Donaldson's functional and Gauss-Codazzi Eq. (17), it follows that

Now, we set

Given that $(S,\bar{\partial}_S,\theta_S)$ and $(Q,\bar{\partial}_Q,\theta_Q)$ are $\omega$-stable Higgs bundles over $M$, Donaldson's functional ${\cal{M}}_{S}(H_{S,0},\cdot)$ and ${\cal{M}}_{Q}(H_{Q,0},\cdot)$ are bounded from below. Together with the above Eq. (59), we obtain

where $\tilde{C}_{1}$ is a uniform positive constant.

By contrast, along with the heat flow Eq. (9), we have

According to the estimate of the heat kernel $K(x,y,t)$ by Cheng and Li in Ref. [24] (or see Theorem 3.2 in Ref. [25]), a positive constant $C_{K}$ exists such that

Applying the maximum principle, we have

for any $t>0$, $s>0$. Using the Gauss-Codazzi equation (17) and (64), we have

Let us set

Combining formulas (61) and (65), and noting that $n\geq 1$, we conclude that $v(t)$ is the targeted function.

Using Proposition 3.2, we obtain a uniform $C^0$-bound on the rescaled metrics $\tilde{H}_S(t) = {\rm e}^{2(\lambda_S-\lambda_E)t}H_S(t)$ and $\tilde{H}_Q(t) =$${\rm e}^{2(\lambda_Q-\lambda_E)t} H_Q(t)$.

Theorem 3.3. Let $H(t)$ be the solution of the Hermitian-Yang-Mills-Higgs flow (9) on the Higgs bundle $(E,\bar{\partial}_E,\theta)$ with initial metrics $H_0$, $H_S(t)$, and $H_Q(t)$ be the induced Hermitian metrics on $(S,\theta_S)$ and $(Q,\theta_Q)$. Set $\hat{h}_S(t) = {\rm e}^{2(\lambda_{S}-\lambda_{E})t}H_{S}^{-1}(0)H_{S}(t)$ and $\hat{h}_{Q}(t) = {\rm e}^{2(\lambda_{Q}-\lambda_{E})t}H_{Q}^{-1}(0)H_{Q}(t)$. Then, there exists a uniform constant $\hat{C}_0$ such that

for all $t\geqslant 0$.

Proof. Noting that the metrics $H_{S}(0)$ and $H_{Q}(0)$ are fixed Hermitian metrics, we can check that

and

for the entire $M$, where $C_{1}$ is a uniform constant. Then, we obtain a uniform constant $C_2$ such that

and

According to formulas (70), (71) and Moser's iteration, we have the following mean inequalities, including a uniform constant $C_{3}$ such that

and

From the uniform $L^1$-estimate in Ref. [26, Lemma 5.1], we know that there exists a uniform constant $C_{4}$ such that

Set

and

From (52) and (74), we can obtain that

where $C_{5}$ denotes a uniform constant. According to (20), it follws that

From (62), the Gauss-Codazzi equation (17), and the maximum principle, we have

and

where $C_6$ denotes a uniform constant. Then, (53) implies that there exists a uniform constant $C_{7}$ such that

As a result, we have

and then

for all $(x,t)\in M\times [0,+\infty)$, where $C_{8}$ denotes a uniform constant. Combining (75), (79) and (81), we can easily conclude that there exists a constant $\hat{C}_0$ such that

4.   Uniform curvature estimate

Using the above uniform $C^0$-estimate of $\tilde{H}_S(t)$ and $\tilde{H}_Q(t)$, we first prove that the norms of $|T_S(t)|_{H_S(t)}$, $|T_Q(t)|_{H_Q(t)}$, $|\gamma(t)|_{H(t)}$, and $|\beta(t)|_{H(t)}$ are uniformly bounded.

Let us set

and

where $h_S(t) = H_{0,S}^{-1}H_{S}(t)$ and $h_Q(t) = H_{0,Q}^{-1}H_{Q}(t)$. Direct calculations yield

and

However, we can also obtain the following inequality (see (2.5) in Ref. [14] for further details):

From the local $C^0$-estimate in (67) and the maximum principle, it can be concluded that $|\theta|_{H(t)}^2$ is also uniformly bounded on $M\times [0,+\infty)$, i.e.,

Together with the local uniform $C^1$-estimate in Ref. [21, Proposition 4.2], we obtain the following proposition:

Proposition 4.1. Let $H(t)$ be the solution of the heat flow (9) with initial metric $H_0$ on $(E,\bar{\partial}_E,\theta)$, let $T_S(t)$ and $T_Q(t)$ be defined by $(83)$ and $(84)$, and let $\gamma$ be the second fundamental form. Then, there exists a constant $\tilde{C}_{2}$ such that

According to Eqs. (44), (85), and (86), we obtain

where constant $\tilde{C}_i$ depends only on the uniform local $C^0$ bound of $\hat{h}_S$ and $\hat{h}_Q$, $H_{S,0}$, $H_{Q,0}$, and the lower bound of Ricci curvature of $(M,\omega)$. Let us consider

Then, it follows that

where $\tilde{C}_6$ is a constant, depending only on the complex dimension $n$ and rank $r$, and Rm is the Riemann curvature of M. The proof of inequality (92) can be found in Ref. [26].

Proof of Theorem 1.1. For the sake of simplicity, we denote

By estimating (89), we can choose a constant $\tilde{C}_7$ such that

for all $(x,t)\in M\times[0,+\infty)$.

Now we consider the following test function:

Let $\zeta_1(p,t_0) = \max\limits_{M\times [0,t_1]}\zeta_1$. Then, at the maximum point $(p,t_0)$, we have

and

Substituting (97) into (96), choosing the constants $\tilde{C}_7$ and $W$ sufficiently large, and using formulas (90) and (92), at the maximum point $(p,t_0)$ we have

where $\tilde{C}_8$ is a positive constant that depends only on the local uniform bound of $\nu$ and curvature of $(M,\omega)$. Thus, we obtain

Then, there exists a constant $\tilde{C}_9$ such that

This completes the proof of Theorem 1.1.

Acknowledgements

The author would like to express his gratitude to Prof. Xi Zhang for his advice on this study, which is supported by the National Key Research and Development Program of China (2020YFA0713100), the National Natural Science Foundation of China (12141104, 11801535, 11721101, 11625106), and the Fundamental Research Funds for the Central Universities.

Conflict of interest

The author declares that they have no conflict of interest.