Expansion formula for complex Monge–Ampère equation along cone singularities

Abstract

In this paper, we prove the asymptotic expansion of the solutions to some singular complex Monge–Ampère equation which arise naturally in the study of the conical Kähler–Einstein metric.

Appendix A: Estimate of some elliptic system

In this appendix, we prove some estimates for the following elliptic system defined on $B_1\subset {\mathbb {C}}^n$,

$$\begin{aligned} \triangle _g g_{i\bar{j}} - g^{k\bar{m}} g^{n\bar{l}} \frac{\partial g_{i\bar{m}}}{\partial z_n} \frac{\partial g_{k\bar{j}}}{\partial z_{\bar{l}}} = \lambda \rho _0^2 g_{i\bar{j}} + h_{\tilde{V},i\bar{j}} . \end{aligned}$$

(A.1)

When $h=0$, this is the equation satisfied by the metric tensor of Kähler–Einstein metric. We always assume

$$\begin{aligned} \left| \lambda \rho _0^2\right| + \left\| h_{\tilde{V},i\bar{j}}\right\| _{C^0(B_1)}\le \Lambda . \end{aligned}$$

(A.2)

The methods used here are from the book of Giaquinta [4] and are by now classical. In contrast to the theorems in [4], we prove effective estimates instead of just regularity statements.

Remark

Note that in this appendix, the real dimension of the domain is 2n, instead of n.

We first bound the $L^2$ norm of the gradient of $g_{i\bar{j}}$.

Lemma A.1

Suppose that $g_{i\bar{j}}$ are some smooth complex-valued functions defined on $B_1\subset {\mathbb {C}}^n$ solving (A.1) whose coefficients satisfy (A.2). If for some $\sigma >0$, we have

$$\begin{aligned} \sigma ^{-1} g_{i\bar{j}}\le \delta _{ij}\le \sigma g_{i\bar{j}} \quad \text{ on } \quad B_1, \end{aligned}$$

then

$$\begin{aligned} \left\| \nabla g_{i\bar{j}}\right\| _{L^2(B_{3/4})}\le C \left( \sigma ,\Lambda ,\left\| g_{i\bar{j}}\right\| _{C^\alpha (B_1)}\right) . \end{aligned}$$

Proof

For any point $x_0$ in $B_{3/4}$ and $R>0$ to be determined in the proof, let $\eta$ be some smooth cut-off function supported in $B_R(x_0)$ with $\eta \equiv 1$ in $B_{R/2}(x_0)$ and $\left| \nabla \eta \right| \le CR^{-1}$. Multiplying both sides of (A.1) by $(g_{i\bar{j}}-g_{i\bar{j}}(x_0)) \eta ^2$ and freezing the coefficients of the leading term in (A.1) gives

$$\begin{aligned} 0= & {} g^{k\bar{l}}(x_0) \partial _k\bar{\partial }_l g_{i\bar{j}} (g_{i\bar{j}}-g_{i\bar{j}}(x_0)) \eta ^2 \nonumber \\- & {} (g^{k\bar{l}}(x_0)-g^{k\bar{l}}) \partial _k \bar{\partial }_l g_{i\bar{j}} (g_{i\bar{j}}-g_{i\bar{j}}(x_0)) \eta ^2 \nonumber \\- & {} (g\cdot g\cdot D g\cdot D g +\lambda \rho _0^2 g +h) \cdot (g_{i\bar{j}}-g_{i\bar{j}}(x_0)) \eta ^2 . \end{aligned}$$

(A.3)

Note that we have omitted subscripts in the above computation when they are not essential to the proof.

By the Hölder continuity of $g_{i\bar{j}}$, we have

$$\begin{aligned} \left| g_{i\bar{j}}-g_{i\bar{j}}(x_0)\right| \le C R^{\alpha }. \end{aligned}$$

(A.4)

Integration by parts of (A.3), (A.4) and Young’s inequality imply that

$$\begin{aligned} \int \left| D g\right| ^2 \eta ^2\le & {} C \int R^\alpha \left| D g\right| \eta \left| \nabla \eta \right| + R^\alpha \left| D g\right| ^2 \eta ^2 \\&+\ C\int R^{2\alpha } \left| D g\right| \eta \left| \nabla \eta \right| + R^\alpha \left| D g\right| ^2 \eta ^2 + C R^{2n+\alpha } \\\le & {} \left( \frac{1}{2}+ CR^\alpha \right) \int \left| D g\right| ^2 \eta ^2 + CR^{2\alpha } \int \left| \nabla \eta \right| ^2 + C R^{2n+\alpha }. \end{aligned}$$

Now we can choose R so small (depending only on $\sigma , \alpha$ and the Hölder norm of $g_{i\bar{j}}$) that the first term in the right hand side is absorbed by the left hand side to give

$$\begin{aligned} \int _{B_{R/2}(x_0)} \left| D g\right| ^2 dx \le CR^{2n-2+2\alpha }\le C. \end{aligned}$$

(A.5)

The lemma then follows from the above inequality by covering $B_{3/4}$ by balls of radius R. $\square$

Next, we prove $C^\gamma$ estimate of $g_{i\bar{j}}$ for any $\alpha<\gamma <1$.

Lemma A.2

For $g_{i\bar{j}}$ as in Lemma A.1 and any $\alpha<\gamma <1$, we have

$$\begin{aligned} \left\| g_{i\bar{j}}\right\| _{C^{\gamma }(B_{3/4})} \le C\left( \sigma ,\Lambda ,\gamma ,\left\| g_{i\bar{j}}\right\| _{C^\alpha (B_1)}\right) . \end{aligned}$$

Proof

For any $x_0\in B_{3/4}$ and $R\le 1/4$, let $v_{i\bar{j}}$ be the solution of

$$\begin{aligned} \left\{ \begin{array}[]{l@{\quad }l} g^{k\bar{l}}(x_0) \partial _k \bar{\partial }_l v_{i\bar{j}} =0 &{} \quad \text{ on } \quad B_R(x_0) \\ v_{i\bar{j}}= g_{i\bar{j}} &{}\quad \text{ on } \quad \partial B_R(x_0). \end{array} \right. \end{aligned}$$

By Theorem 2.1 on page 78 of [4] (applied to $Dv_{i\bar{j}}$), there is a constant depending only on $\sigma$ such that for $0<\rho <R$,

$$\begin{aligned} \int _{B_{\rho }(x_0)} \left| Dv_{i\bar{j}}\right| ^2 \le C\left( \frac{\rho }{R}\right) ^{2n} \int _{B_R(x_0)} \left| Dv_{i\bar{j}}\right| ^2. \end{aligned}$$

(A.6)

Setting $w_{i\bar{j}}=g_{i\bar{j}}-v_{i\bar{j}}$, we get (using (A.6))

$$\begin{aligned}&\int _{B_{\rho }(x_0)} \left| Dg_{i\bar{j}}\right| ^2 \nonumber \\&\quad \le \int _{B_{\rho }(x_0)} \left| Dv_{i\bar{j}}\right| ^2+ \int _{B_{\rho }(x_0)} \left| Dw_{i\bar{j}}\right| ^2\nonumber \\&\quad \le C\left( \frac{\rho }{R}\right) ^{2n} \int _{B_{R}(x_0)} \left| Dv_{i\bar{j}}\right| ^2 + C \int _{B_{\rho }(x_0)} \left| Dw_{i\bar{j}}\right| ^2 \nonumber \\&\quad \le C\left( \frac{\rho }{R}\right) ^{2n} \int _{B_{R}(x_0)} \left| Dg_{i\bar{j}}\right| ^2 + C \int _{B_{R}(x_0)} \left| Dw_{i\bar{j}}\right| ^2. \end{aligned}$$

(A.7)

Using the equation satisfied by $v_{i\bar{j}}$, we may rewrite (A.1) as follows

$$\begin{aligned} g^{k\bar{l}}(x_0) \partial _k\bar{\partial }_l (g_{i\bar{j}}-v_{i\bar{j}})= -(g^{k\bar{l}}-g^{k\bar{l}}(x_0))\partial _k\bar{\partial }_l g_{i\bar{j}} + g\cdot g\cdot D g \cdot D g + \lambda \rho _0^2 g + h. \end{aligned}$$

Since $w_{i\bar{j}}$ vanishes on $\partial B_R(x_0)$, we can use it as the test function of the above equation to obtain

$$\begin{aligned} \int _{B_R(x_0)} \left| Dw_{i\bar{j}}\right| ^2\le & {} C \int _{B_R(x_0)} \partial _k \bar{\partial }_l g_{i\bar{j}} \left( w (g^{k\bar{l}}-g^{k\bar{l}}(x_0)) \right) + \left| w\right| \left| D g\right| ^2 + \left| w\right| \\\le & {} C \int _{B_R(x_0)} \left| D g\right| \left| \nabla w\right| \left| g^{k\bar{l}}-g^{k\bar{l}}(x_0)\right| + \left| w\right| \left| D g\right| ^2 + \left| w\right| . \end{aligned}$$

Using Young’s inequality, we get

$$\begin{aligned} \int _{B_R(x_0)} \left| Dw_{i\bar{j}}\right| ^2 \le C \int _{B_R(x_0)} \left| D g\right| ^2 \left( \left| g^{k\bar{l}}-g^{k\bar{l}}(x_0)\right| ^2 + \left| w\right| \right) + \left| w\right| . \end{aligned}$$

(A.8)

The maximum principle implies that $\text{ osc }_{B_R(x_0)} v_{i\bar{j}} \le osc_{B_R(x_0)} g_{i\bar{j}}$, which implies that

$$\begin{aligned} \left\| w\right\| _{C^0(B_R(x_0))}\le \text{ osc }_{B_R(x_0)} v_{i\bar{j}}+ \text{ osc }_{B_R(x_0)} g_{i\bar{j}}\le C R^\alpha . \end{aligned}$$

(A.9)

Putting (A.7), (A.8) and (A.9) together yields the following decay estimate

$$\begin{aligned} \int _{B_\rho (x_0)} \left| Dg\right| ^2 \le C\left( \left( \frac{\rho }{R}\right) ^{2n} + R^\alpha \right) \int _{B_R(x_0)} \left| Dg\right| ^2 + C R^{2n+\alpha }. \end{aligned}$$

Dividing both sides of the above equation by $\rho ^{2n-2}$ and setting $\rho /R=\tau$ give

$$\begin{aligned} \rho ^{2-2n}\int _{B_\rho (x_0)} \left| Dg\right| ^2 \le \left[ C (1+ \tau ^{-2n} R^{\alpha }) \right] \tau ^2 R^{2-2n}\int _{B_R(x_0)} \left| Dg\right| ^2 + C \tau ^{2-2n} R^2. \end{aligned}$$

By picking $\tau$ small so that $2C \tau ^{(2-2\gamma )}=1$ and then $R_1$ small so that $1+\tau ^{-n}R^\alpha <2$ for all $R<R_1$, we have

$$\begin{aligned} \rho ^{2-2n}\int _{B_\rho (x_0)} \left| Dg\right| ^2 \le \tau ^{2\gamma } R^{2-2n}\int _{B_R(x_0)} \left| Dg\right| ^2 + C(\gamma ) R^2 \end{aligned}$$

for $R<R_1$. From here, a routine iteration shows that

$$\begin{aligned} \rho ^{2-2n}\int _{B_\rho (x_0)} \left| Dg\right| ^2 \le C \rho ^{2\gamma }. \end{aligned}$$

Hence, by the Hölder inequality and the equivalence between the Companato space and the Hölder space, we have

$$\begin{aligned} \left\| g_{i\bar{j}}\right\| _{C^\gamma (B_{3/4})}\le C\left( \sigma ,\Lambda ,\gamma , \left\| g_{i\bar{j}}\right\| _{C^\alpha (B_1)}\right) . \end{aligned}$$

$\square$

Finally, we prove the $C^{1,\alpha }$ estimate.

Lemma A.3

For $g_{i\bar{j}}$ as in Lemma A.1, there exists some $\alpha '\in (0,1)$ such that

$$\begin{aligned} \left\| g_{i\bar{j}}\right\| _{C^{1,\alpha '}(B_{1/2})} \le C\left( \sigma ,\Lambda ,\left\| g_{i\bar{j}}\right\| _{C^\alpha (B_1)}\right) . \end{aligned}$$

Proof

For any $x_0\in B_{1/2}$ and $R\le 1/4$, as in the proof of Lemma A.2, let $v_{i\bar{j}}$ be the solution of

$$\begin{aligned} \left\{ \begin{array}[]{l@{\quad }l} g^{k\bar{l}}(x_0) \partial _k \bar{\partial }_l v_{i\bar{j}} =0 &{} \quad \text{ on } \quad B_R(x_0) \\ v_{i\bar{j}}= g_{i\bar{j}} &{}\quad \text{ on } \quad \partial B_R(x_0). \end{array} \right. \end{aligned}$$

Again, applying Theorem 2.1 on page 78 of [4] to $Dv_{i\bar{j}}$, we get a constant depending only on $\sigma$ such that

$$\begin{aligned} \int _{B_{\rho }(x_0)} \left| Dv_{i\bar{j}}- (Dv_{i\bar{j}})_{x_0,\rho }\right| ^2 \le C\left( \frac{\rho }{R}\right) ^{2n+2} \int _{B_R(x_0)} \left| Dv_{i\bar{j}}-(Dv_{i\bar{j}})_{x_0,R}\right| ^2. \end{aligned}$$

(A.10)

Here $(u)_{x,r}$ means the average of u in the ball $B_r(x)$.

Next, we set $w_{i\bar{j}}=g_{i\bar{j}}-v_{i\bar{j}}$ on $B_R(x_0)$. Triangle inequalities and (A.10) imply that

$$\begin{aligned}&\int _{B_{\rho }(x_0)} \left| Dg_{i\bar{j}}- (Dg_{i\bar{j}})_{x_0,\rho }\right| ^2 \nonumber \\&\quad \le \int _{B_{\rho }(x_0)} \left| Dv_{i\bar{j}}- (Dv_{i\bar{j}})_{x_0,\rho }\right| ^2+ \int _{B_{\rho }(x_0)} \left| Dw_{i\bar{j}}- (Dw_{i\bar{j}})_{x_0,\rho }\right| ^2\nonumber \\&\quad \le C\left( \frac{\rho }{R}\right) ^{2n+2} \int _{B_{R}(x_0)} \left| Dv_{i\bar{j}}- (Dv_{i\bar{j}})_{x_0,R}\right| ^2 + C \int _{B_{\rho }(x_0)} \left| Dw_{i\bar{j}}\right| ^2 \nonumber \\&\quad \le C\left( \frac{\rho }{R}\right) ^{2n+2} \int _{B_{R}(x_0)} \left| Dg_{i\bar{j}}- (Dg_{i\bar{j}})_{x_0,R}\right| ^2 + C \int _{B_{R}(x_0)} \left| Dw_{i\bar{j}}\right| ^2. \end{aligned}$$

(A.11)

Using $w_{i\bar{j}}$ as the test function of (A.1) as in the proof of Lemma A.2 gives

$$\begin{aligned} \int _{B_R(x_0)} \left| Dw_{i\bar{j}}\right| ^2\le & {} C \int _{B_R(x_0)} \partial _k \bar{\partial }_l g_{i\bar{j}} \left( w (g^{k\bar{l}}-g^{k\bar{l}}(x_0)) \right) + \left| w\right| \left| D g\right| ^2 + \left| w\right| \\\le & {} C \int _{B_R(x_0)} \left| D g\right| \left| \nabla w\right| \left| g^{k\bar{l}}-g^{k\bar{l}}(x_0)\right| + \left| w\right| \left| D g\right| ^2 + \left| w\right| . \end{aligned}$$

The Young’s inequality and the fact that $g_{i\bar{j}}$ lies in $C^\gamma (B_{3/4})$ imply that

$$\begin{aligned} \int _{B_R(x_0)} \left| Dw\right| ^2\le & {} C R^{\gamma } \int _{B_R(x_0)} \left| D g\right| ^2 + CR^{2n+\gamma }. \end{aligned}$$

Notice that we have proved (see (A.5))

$$\begin{aligned} \int _{B_R(x_0)} \left| Dg\right| ^2 \le C R^{2n-2+2\gamma } \end{aligned}$$

in the proof of Lemma A.2. By Lemma A.2, we can choose and fix $\gamma$ so that

$$\begin{aligned} \int _{B_R(x_0)}\left| Dw\right| ^2 \le C R^{2n+\gamma '} \end{aligned}$$

(A.12)

for some $\gamma '>0$ (small). Combining (A.11) and (A.12), we get

$$\begin{aligned}&\rho ^{-2n}\int _{B_{\rho }(x_0)} \left| Dg_{i\bar{j}}- (Dg_{i\bar{j}})_{x_0,\rho }\right| ^2 \le C\left( \frac{\rho }{R}\right) ^2 R^{-2n} \int _{B_R(x_0)}\left| Dg_{i\bar{j}}- (Dg_{i\bar{j}})_{x_0,\rho }\right| ^2 \\&\quad + C(R/\rho )^{2n} R^{\gamma '}. \end{aligned}$$

As before, picking $\tau \in (0,1)$ with $C \tau ^{2-\gamma '}=1$ and setting $\rho =\tau R$, we get

$$\begin{aligned} \rho ^{-2n}\int _{B_{\rho }(x_0)} \left| Dg_{i\bar{j}}- (Dg_{i\bar{j}})_{x_0,\rho }\right| ^2 \le \tau ^{\gamma '} R^{-2n} \int _{B_R(x_0)}\left| Dg_{i\bar{j}}- (Dg_{i\bar{j}})_{x_0,\rho }\right| ^2 + C(\tau ) R^{\gamma '}. \end{aligned}$$

Iteration again implies that

$$\begin{aligned} \int _{B_{\rho }(x_0)} \left| Dg_{i\bar{j}}- (Dg_{i\bar{j}})_{x_0,\rho }\right| ^2 \le C \rho ^{2n+\gamma '}, \end{aligned}$$

which concludes the proof of the lemma by Theorem 1.2 in Chapter III of [4]. $\square$