Analytically stable Higgs bundles on some non-Kähler manifolds

Abstract

In this paper, we study Higgs bundles on non-compact Hermitian manifolds. Under some assumptions for the underlying Hermitian manifolds which are not necessarily Kähler, we solve the Hermitian–Einstein equation on analytically stable Higgs bundles.

Appendix

Let ${\mathbb {C}}$ be the plane, $\omega _{{\mathbb {C}}}$ be the Euclidean metric and $\varphi _{{\mathbb {C}}}(w)=(1+|w|^{2})^{-1-\delta }$, where w is the complex coordinate of ${\mathbb {C}}$ and $\delta >0$. By using Simpson’s result ([31], Proposition 2.4), Mochizuki showed that $({\mathbb {C}}, \omega _{{\mathbb {C}}}, \varphi _{{\mathbb {C}}})$ meets the Assumption 1, i.e., we have the following lemma.

Lemma 5.1

([27], Lemma 3.5) Let $({\mathbb {C}}, \omega _{{\mathbb {C}}}, \varphi _{{\mathbb {C}}})$ be defined as above. For any nonnegative bounded function f satisfying

$$\begin{aligned} \sqrt{-1}\Lambda _{\omega _{{\mathbb {C}}}}\partial \overline{\partial }f\ge -B\varphi _{{\mathbb {C}}} , \end{aligned}$$

(5.1)

we have

$$\begin{aligned} \sup _{w\in {\mathbb {C}}} f(w)\le C(B)a\left( \int _{{\mathbb {C}}}f \cdot \varphi _{{\mathbb {C}}}\omega _{{\mathbb {C}}}\right) , \end{aligned}$$

(5.2)

where $a: [0, \infty ) \rightarrow [0, \infty )$ is an increasing function with $a(0)=0$ and $a(x) = x$ $(x\ge 1)$, C(B) is a positive constant depending only on B. Moreover, if $\sqrt{-1}\Lambda _{\omega _{{\mathbb {C}}} }\partial \overline{\partial }f\ge 0$, then f is a constant.

Set

$$\begin{aligned} (M, \omega _{M})=({\mathbb {C}}, \omega _{{\mathbb {C}}})\times (Y, \omega _{Y}), \end{aligned}$$

(5.3)

$\varphi _{M}(w, \cdot ) =\varphi _{{\mathbb {C}}}(w)$, where $(Y, \omega _{Y})$ is any compact Gauduchon manifold without boundary. Since $\omega _{{\mathbb {C}}}$ is Kähler and $\omega _{Y}$ is Gauduchon, the product metric $\omega _{M}$ is also Gauduchon. Following Mochizuki’s argument [27] and running Moser’s iteration procedure, we can prove the following proposition.

Proposition 5.2

$(M, \omega _{M}, \varphi _{M})$ satisfies the Assumption 1.

Proof

For any nonnegative bounded function $f:M\rightarrow {\mathbb {R}}$ satisfying

$$\begin{aligned} \sqrt{-1}\Lambda _{\omega _{M}}\partial {\overline{\partial }}f\ge -B\varphi _{M} , \end{aligned}$$

(5.4)

we set

$$\begin{aligned} {\tilde{f}}(w)=\int _{Y}f(w, \cdot )\frac{\omega _{Y}^{n}}{n!}, \end{aligned}$$

(5.5)

where n is the complex dimension of Y. By direct calculation, we derive

$$\begin{aligned} \begin{aligned}&\sqrt{-1}\Lambda _{\omega _{{\mathbb {C}}}}\partial ^{{\mathbb {C}}}{\overline{\partial }}^{{\mathbb {C}}}{\tilde{f}}(w)=\int _{Y}\sqrt{-1}\Lambda _{\omega _{{\mathbb {C}}}}\partial ^{{\mathbb {C}}}{\overline{\partial }}^{{\mathbb {C}}}f(w, \cdot )\frac{\omega _{Y}^{n}}{n!}\\&\qquad =\int _{Y}\sqrt{-1}\left( \Lambda _{\omega _{M}}\partial {\overline{\partial }}f(w, \cdot )-\Lambda _{\omega _{Y}}\partial ^{Y}{\overline{\partial }}^{Y}f(w, \cdot )\right) \frac{\omega _{Y}^{n}}{n!}\\&\qquad =\int _{Y}\sqrt{-1}\Lambda _{\omega _{M}}\partial {\overline{\partial }}f(w, \cdot )\frac{\omega _{Y}^{n}}{n!}\\&\qquad \ge -B\cdot \varphi _{{\mathbb {C}}}(w)\cdot \text {Vol}(Y, \omega _{Y}), \end{aligned} \end{aligned}$$

(5.6)

where we have used the condition that $(Y, \omega _{Y})$ is a compact Gauduchon manifold. Then (5.2) implies

$$\begin{aligned} \sup _{w\in {\mathbb {C}}} {\tilde{f}}(w)\le C(B\cdot \text {Vol}(Y, \omega _{Y}))a\left( \int _{M}f \cdot \varphi _{M}\frac{\omega _{M}^{n+1}}{(n+1)!}\right) . \end{aligned}$$

(5.7)

For any point $w_{0}\in {\mathbb {C}}$, set $B_{r}(w_{0}):=\{w\in {\mathbb {C}}\ | \ |w-w_{0}|<r\}$. Then $(X=B_{2}(w_{0})\times Y, \omega _{M})$ is a compact Riemannian manifold with smooth boundary (it doesn’t depend on $w_{0}$) and we have the following Sobolev inequality

$$\begin{aligned} \left( \int _{X}|\nu | ^{\frac{2(n+1)}{n}}\frac{\omega _{M}^{n+1}}{(n+1)!}\right) ^{\frac{n}{2(n+1)}}\le C_{S}\left[ \left( \int _{X}|d\nu | ^{2}\frac{\omega _{M}^{n+1}}{(n+1)!}\right) ^{\frac{1}{2}}+\left( \int _{X}|\nu | ^{2}\frac{\omega _{M}^{n+1}}{(n+1)!}\right) ^{\frac{1}{2}}\right] \end{aligned}$$

(5.8)

for any compactly supported function $\nu \in L_{1}^{2}(X)$. On the other hand, there holds

$$\begin{aligned} \sqrt{-1}\Lambda _{\omega _{M}}\partial {\overline{\partial }}f\ge -B\varphi _{M} \ge -B \end{aligned}$$

(5.9)

on X. In the following, by using the Sobolev inequality (5.8) and the inequality (5.9), we will run Moser’s iteration procedure to obtain a mean value inequality.

Take $1\le r \le r_{2}<r_{1} \le R \le 2$ and let $\psi _{1}\in C_{0}^{\infty }(X)$ be the cutoff function such that

$$\begin{aligned} \psi _{1}(x)=\left\{ \begin{aligned} \ 1,&\quad x\in B_{r_{2}}(w_{0})\times Y,\\ \ 0,&\quad x\in X\setminus B_{r_{1}}(w_{0})\times Y, \end{aligned} \right. \end{aligned}$$

(5.10)

$0\le \psi _{1} (x) \le 1$ and $|d\psi _{1} |_{\omega _{X}}\le 4(r_{1}-r_{2})$. Set ${\hat{f}}=f+1$. Of course (5.9) implies

$$\begin{aligned} \sqrt{-1}\Lambda _{\omega _{M}}\partial \overline{\partial }{\hat{f}}\ge -B\cdot {\hat{f}}. \end{aligned}$$

(5.11)

Multiplying ${\hat{f}}^{q-1}\psi _{1}^{2}$ on both sides of the inequality (5.11) ($q\ge 2$), and integrating it over X, we know

$$\begin{aligned} \begin{aligned}&-B\int _{X}{\hat{f}}^{q} \cdot \psi _{1}^{2} \frac{\omega _{M}^{n+1}}{(n+1)!}\le \int _{X}\sqrt{-1}{\hat{f}}^{q-1} \cdot \psi _{1}^{2}\partial {\overline{\partial }}{\hat{f}}\wedge \frac{\omega _{M}^{n}}{n!}\\ =\,&\int _{X}\sqrt{-1}\partial ({\hat{f}}^{q-1} \cdot \psi _{1}^{2} {\overline{\partial }}{\hat{f}}\wedge \frac{\omega _{M}^{n}}{n!})-\int _{X}\sqrt{-1}\partial ({\hat{f}}^{q-1} \cdot \psi _{1}^{2})\wedge {\overline{\partial }}{\hat{f}}\wedge \frac{\omega _{M}^{n}}{n!}\\&+\int _{X}\sqrt{-1}{\hat{f}}^{q-1} \cdot \psi _{1}^{2} {\overline{\partial }}{\hat{f}}\wedge \partial \left( \frac{\omega _{M}^{n}}{n!}\right) \\ =\,&-\int _{X}\sqrt{-1}(q-1){\hat{f}}^{q-2} \cdot \psi _{1}^{2} \partial {\hat{f}}\wedge {\overline{\partial }}{\hat{f}}\wedge \frac{\omega _{M}^{n}}{n!} -\int _{X}2\sqrt{-1}{\hat{f}}^{q-1} \cdot \psi _{1} \partial \psi _{1}\wedge {\overline{\partial }}{\hat{f}}\wedge \frac{\omega _{M}^{n}}{n!}\\&+\int _{X}\sqrt{-1}{\hat{f}}^{q-1} \cdot \psi _{1}^{2} {\overline{\partial }}{\hat{f}}\wedge \partial \omega _{M}\wedge \frac{\omega _{M}^{n-1}}{(n-1)!}.\\ \end{aligned} \end{aligned}$$

(5.12)

Because the manifold Y is compact, there is a uniform constant $C_{1}$ such that $\sup _{X}|\partial \omega _{M}|_{\omega _{M}}\le C_{1}$, and then

$$\begin{aligned} \begin{aligned}&\left| \frac{{\overline{\partial }}{\hat{f}}\wedge \partial \omega _{M}\wedge \left( \frac{\omega _{M}^{n-1}}{(n-1)!}\right) }{\frac{\omega _{M}^{n+1}}{(n+1)!}}\right| \\ \le&C_{2}|{\overline{\partial }}{\hat{f}}|_{\omega _{M}}|\partial \omega _{M}|_{\omega _{M}}\le C_{3}|{\overline{\partial }}{\hat{f}}|_{\omega _{M}}.\\ \end{aligned} \end{aligned}$$

(5.13)

On the other hand, applying Cauchy’s inequality, we have

$$\begin{aligned} |{\overline{\partial }}{\hat{f}}|_{\omega _{M}}{\hat{f}}^{q-1}\psi _{1}^{2}\le \epsilon |{\overline{\partial }}{\hat{f}}|_{\omega _{M}}^{2}{\hat{f}}^{q-2}\psi _{1}^{2}+\frac{1}{4\epsilon }{\hat{f}}^{q}\psi _{1}^{2} \end{aligned}$$

(5.14)

and

$$\begin{aligned} 2|{\overline{\partial }}{\hat{f}}|_{\omega _{M}}|\partial \psi _{1}|_{\omega _{M}}{\hat{f}}^{q-1}\psi _{1}\le \epsilon |{\overline{\partial }}{\hat{f}}|_{\omega _{M}}^{2}{\hat{f}}^{q-2}\psi _{1}^{2}+\frac{1}{\epsilon }{\hat{f}}^{q}|\partial \psi _{1}|_{\omega _{M}}, \end{aligned}$$

(5.15)

where $\epsilon$ is a positive constant which will be chosen later. By (5.12), (5.13), (5.14) and (5.15), we conclude that

$$\begin{aligned} \begin{aligned}&\int _{X}|\partial {\hat{f}}^{\frac{q}{2}}|^{2} \cdot \psi _{1}^{2} \frac{\omega _{M}^{n+1}}{(n+1)!}\le \frac{q^{2}}{4(q-1)}\left( B+\frac{C_{3}}{4\epsilon }\right) \int _{X} {\hat{f}}^{q} \cdot \psi _{1}^{2} \frac{\omega _{M}^{n+1}}{(n+1)!}\\&+\frac{q^{2}}{4\epsilon (q-1)}\int _{X} {\hat{f}}^{q} |\partial \psi _{1}|^{2} \frac{\omega _{M}^{n+1}}{(n+1)!}+\frac{\epsilon (C_{3}+1)}{q-1}\int _{M} \psi _{1}^{2} |\partial {\hat{f}}^{\frac{q}{2}}|^{2} \frac{\omega _{M}^{n+1}}{(n+1)!}.\\ \end{aligned} \end{aligned}$$

(5.16)

Choose $\epsilon =\frac{q-1}{2(C_{3}+1)}$, then

$$\begin{aligned} \begin{aligned}&\int _{X}|\partial {\hat{f}}^{\frac{q}{2}}|^{2} \cdot \psi _{1}^{2} \frac{\omega _{M}^{n+1}}{(n+1)!}\le \frac{q^{2}}{2(q-1)}\left( B+\frac{C_{3}}{2\epsilon }\right) \int _{X} {\hat{f}}^{q} \cdot \psi _{1}^{2} \frac{\omega _{M}^{n+1}}{(n+1)!}\\&+\frac{2q^{2}(C_{3}+1)}{(q-1)^{2}}\int _{X} {\hat{f}}^{q} |\partial \psi _{1}|^{2} \frac{\omega _{M}^{n+1}}{(n+1)!}\\ \end{aligned} \end{aligned}$$

(5.17)

and

$$\begin{aligned} \begin{aligned}&\int _{X}|\partial \left( {\hat{f}}^{\frac{q}{2}}\cdot \psi _{1}\right) |^{2} \frac{\omega _{M}^{n+1}}{(n+1)!}\le 2\int _{X}|\partial {\hat{f}}^{\frac{q}{2}}|^{2} \cdot \psi _{1}^{2} \frac{\omega _{M}^{n+1}}{(n+1)!}+2\int _{X} {\hat{f}}^{q} |\partial \psi _{1}|^{2} \frac{\omega _{M}^{n+1}}{(n+1)!}\\ \le&\frac{q^{2}}{q-1}\left( B+\frac{C_{3}}{2\epsilon }\right) \int _{X} {\hat{f}}^{q} \cdot \psi _{1}^{2} \frac{\omega _{M}^{n+1}}{(n+1)!} +\frac{4q^{2}(C_{3}+1)}{(q-1)^{2}}\int _{X} {\hat{f}}^{q} |\partial \psi _{1}|^{2} \frac{\omega _{M}^{n+1}}{(n+1)!}.\\ \end{aligned} \end{aligned}$$

(5.18)

Using (5.18) and the Sobolev inequality (5.8) ($\nu ={\hat{f}}^{\frac{q}{2}}\cdot \psi _{1}$), one can get

$$\begin{aligned} \begin{aligned}&\left( \int _{B_{r_{2}}(w_{0})\times Y}{\hat{f}}^{q\cdot \frac{n+1}{n}} \frac{\omega _{M}^{n+1}}{(n+1)!}\right) ^{\frac{n}{n+1}}\le \left( \int _{X}\left( {\hat{f}}^{q}\psi _{1}^{2}\right) ^{ \frac{n+1}{n}} \frac{\omega _{M}^{n+1}}{(n+1)!}\right) ^{\frac{n}{n+1}}\\ \le&2C_{S}^{2}[\int _{M}2|\partial ({\hat{f}}^{\frac{q}{2}}\cdot \psi _{1})|^{2} \frac{\omega _{M}^{n+1}}{(n+1)!}+ \int _{X}{\hat{f}}^{q}\psi _{1}^{2} \frac{\omega _{M}^{n+1}}{(n+1)!} ]\\ \le&2C_{S}^{2}\left( \frac{2q^{2}}{q-1}\left( B+\frac{C_{3}}{2\epsilon }\right) +1\right) \int _{M} {\hat{f}}^{q} \cdot \psi _{1}^{2} \frac{\omega _{M}^{n+1}}{(n+1)!}\\&+2C_{S}^{2}\frac{4q^{2}(C_{3}+1)}{(q-1)^{2}}\int _{M} {\hat{f}}^{q} |\partial \psi _{1}|^{2} \frac{\omega _{M}^{n+1}}{(n+1)!}\\ \le&C_{4}q^{2}((r_{1}-r_{2})^{-2}+1)\int _{B_{r_{1}}(w_{0})\times Y}{\hat{f}}^{q} \frac{\omega _{M}^{n+1}}{(n+1)!} \end{aligned} \end{aligned}$$

(5.19)

and then

$$\begin{aligned}&\left( \int _{B_{r_{2}}(w_{0})\times Y}{\hat{f}}^{q\cdot \frac{n+1}{n}} \frac{\omega _{M}^{n+1}}{(n+1)!}\right) ^{\frac{n}{q\cdot (n+1)}}\nonumber \\&\quad \le (C_{4}q^{2}((r_{1}-r_{2})^{-2}+1))^{\frac{1}{q}}\left( \int _{B_{r_{1}}(w_{0})\times Y}{\hat{f}}^{q} \frac{\omega _{M}^{n+1}}{(n+1)!}\right) ^{\frac{1}{q}}, \end{aligned}$$

(5.20)

where $C_{4}$ is a positive constant depending only on B, $C_{S}$ and $\sup _{Y}|\partial \omega _{Y}|$. Let $R_{i}=r+2^{-i}\cdot (R-r)$, $q_{i}=2(\frac{n+1}{n})^{i}$. Substituting $r_{2}=R_{i+1}$, $r_{1}=R_{i}$ and $q=q_{i}$ into (5.20), we obtain

$$\begin{aligned} \begin{aligned}&\left( \int _{B_{R_{i+1}}(w_{0})\times Y}{\hat{f}}^{q_{i+1}} \frac{\omega _{M}^{n+1}}{(n+1)!}\right) ^{\frac{1}{q_{i+1}}}\\ \le&C_{5}^{\left( \frac{n+1}{n}\right) ^{-i}}\left( \frac{2n+2}{n}\right) ^{i \left( \frac{n+1}{n}\right) ^{-i}}((R-r)^{-2}+1)^{ \frac{1}{2}\left( \frac{n+1}{n}\right) ^{-i}}\left( \int _{B_{R_{i}}(w_{0})\times Y}{\hat{f}}^{q_{i}} \frac{\omega _{M}^{n+1}}{(n+1)!}\right) ^{\frac{1}{q_{i}}}.\\ \end{aligned} \end{aligned}$$

(5.21)

Iterating the inequality (5.21) and using $\sum _{i=0}^{\infty }k^{-i}=\frac{k}{k-1}$, $\sum _{i=0}^{\infty }(i+1)k^{-i}=\frac{k^{2}}{(k-1)^{2}}$, we conclude that

$$\begin{aligned} \begin{aligned} \sup _{B_{r}(w_{0})\times Y}{\hat{f}}&\le \lim _{i\rightarrow +\infty }\left( \int _{B_{R_{i+1}}(w_{0})\times Y}{\hat{f}}^{q_{i+1}} \frac{\omega _{M}^{n+1}}{(n+1)!}\right) ^{\frac{1}{q_{i+1}}}\\&\le C_{6}((R-r)^{-2}+1)^{\frac{n+1}{2}}\left( \int _{B_{R}(w_{0})\times Y}{\hat{f}}^{2} \frac{\omega _{M}^{n+1}}{(n+1)!}\right) ^{\frac{1}{2}},\\ \end{aligned} \end{aligned}$$

(5.22)

where $C_{6}$ is a positive constant depending only on B, n, $C_{S}$ and $\sup _{Y}|\partial \omega _{Y}|$. For any $0< {\tilde{r}} \le 2$ and $0< \delta <1$, let $h_{0}=\delta {\tilde{r}}$, $h_{i}=h_{i-1}+2^{-i}(1-\delta ){\tilde{r}}$ for each $i=1, 2, 3, \cdots$. Putting $r=h_{i}$ and $R=h_{i+1}$ into (5.22), we have

$$\begin{aligned} \begin{aligned}&\sup _{B_{h_{i}}(w_{0})\times Y}{\hat{f}}\\ \le&C_{6}\left( (1-\delta )^{-2}{\tilde{r}}^{-2}+1\right) ^{\frac{n+1}{2}}2^{i(n+1)}\left( \int _{B_{h_{i+1}}(w_{0})\times Y}{\hat{f}} \frac{\omega _{M}^{n+1}}{(n+1)!}\right) ^{\frac{1}{2}}\left( \sup _{B_{h_{i+1}}(w_{0})\times Y}{\hat{f}}\right) ^{\frac{1}{2}}.\\ \end{aligned} \end{aligned}$$

(5.23)

Denote $A(i):=\sup _{B_{h_{i}}(w_{0})\times Y}{\hat{f}}$. Then (5.23) yields

$$\begin{aligned} \begin{aligned} A(0)&\le \prod _{i=0}^{j-1}\left\{ C_{6}((1-\delta )^{-2}{\tilde{r}}^{-2}+1)^{\frac{n+1}{2}}2^{i(n+1)}\left( \int _{B_{{\tilde{r}}}(w_{0})\times Y}{\hat{f}} \frac{\omega _{M}^{n+1}}{(n+1)!}\right) ^{\frac{1}{2}}\right\} ^{2^{-i}}A(j)^{2^{-j}}\\&\le C_{7}((1-\delta )^{-2}{\tilde{r}}^{-2}+1)^{n+1}\int _{B_{{\tilde{r}}}(w_{0})\times Y}{\hat{f}} \frac{\omega _{M}^{n+1}}{(n+1)!}, \end{aligned} \end{aligned}$$

(5.24)

where $C_{7}$ is a positive constant depending only on B, n, $C_{S}$ and $\sup _{Y}|\partial \omega _{Y}|$. Take ${\tilde{r}}=2$ and $\delta =\frac{1}{2}$. Clearly (5.24) gives us that

$$\begin{aligned} \sup _{B_{1}(w_{0})\times Y}{\hat{f}}\le C_{8}\int _{B_{2}(w_{0})\times Y}{\hat{f}} \frac{\omega _{M}^{n+1}}{(n+1)!} \end{aligned}$$

(5.25)

and

$$\begin{aligned} \begin{aligned} \sup _{B_{1}(w_{0})\times Y}f&\le C_{8}\left( \int _{B_{2}(w_{0})\times Y}f \frac{\omega _{M}^{n+1}}{(n+1)!}+\text {Vol}(B_{2}(w_{0}))\cdot \text {Vol}(Y, \omega _{Y})\right) \\&\le C_{8}\left( \int _{B_{2}(w_{0})}{\tilde{f}} \omega _{{\mathbb {C}}}+\text {Vol}(B_{2}(w_{0}))\cdot \text {Vol}(Y, \omega _{Y})\right) \\&\le C_{9}\left( a\left( \int _{M}f\cdot \varphi _{M} \frac{\omega _{M}^{n+1}}{(n+1)!}\right) +1\right) ,\\ \end{aligned} \end{aligned}$$

(5.26)

where $C_{8}$ and $C_{9}$ are positive constants depending only on B, n, $C_{S}$ and $\sup _{Y}|\partial \omega _{Y}|$. Since $w_{0}$ is arbitrary, we have

$$\begin{aligned} \sup _{M}f \le C_{9}\left( a\left( \int _{M}f\cdot \varphi _{M} \frac{\omega _{M}^{n+1}}{(n+1)!}\right) +1\right) . \end{aligned}$$

(5.27)

Suppose that $\sqrt{-1}\Lambda _{\omega _{M}}\partial {\overline{\partial }} f \ge 0$ on M. From the definition of ${\tilde{f}}$ and the condition $\omega _{Y}$ is Gauduchon, it is easy to see that ${\tilde{f}}$ is a bounded function on $({\mathbb {C}}, \omega _{{\mathbb {C}}})$ and satisfies $\sqrt{-1}\Lambda _{\omega _{{\mathbb {C}}}}\partial ^{{\mathbb {C}}} {\overline{\partial }}^{{\mathbb {C}}} {\tilde{f}} \ge 0$. According to Lemma 5.1, we know that ${\tilde{f}}\equiv {\tilde{C}}$. Set

$$\begin{aligned} f_{1}(w, \cdot )=f(w, \cdot )-\frac{{\tilde{C}}}{\text {Vol} (Y, \omega _{Y})}. \end{aligned}$$

(5.28)

Then

$$\begin{aligned} \int _{\{w\}\times Y}f_{1}\frac{\omega _{Y}^{n}}{n!}=0 \end{aligned}$$

(5.29)

and

$$\begin{aligned} \begin{aligned}&\sqrt{-1}\Lambda _{\omega _{{\mathbb {C}}}}\partial ^{{\mathbb {C}}} {\overline{\partial }}^{{\mathbb {C}}}\int _{\{w\}\times Y}|f_{1}|^{2}\frac{\omega _{Y}^{n}}{n!}=\sqrt{-1}\Lambda _{\omega _{{\mathbb {C}}}}\partial ^{{\mathbb {C}}} {\overline{\partial }}^{{\mathbb {C}}}\int _{\{w\}\times Y}|f|^{2}\frac{\omega _{Y}^{n}}{n!}\\&\quad \ge \int _{\{w\}\times Y}\sqrt{-1}\Lambda _{\omega _{M}}\partial ^{M} {\overline{\partial }}^{M}|f|^{2}\frac{\omega _{Y}^{n}}{n!}\\&\quad \ge \int _{\{w\}\times Y}2|\partial f_{1}|^{2}\frac{\omega _{Y}^{n}}{n!}\\&\quad \ge C_{p}\int _{\{w\}\times Y}| f_{1}|^{2}\frac{\omega _{Y}^{n}}{n!},\\ \end{aligned} \end{aligned}$$

(5.30)

where we have used the Poincaré inequality on the compact Riemannian manifold $(Y, \omega _{Y})$. Applying Lemma 5.1 again, we have $f_{1}\equiv 0$ and then f is constant.

$\square$