Plurisubharmonicity for the solution of the Fefferman equation and applications

Abstract

In this paper, the author introduces a concept of the super-pseudoconvex domain. He proves that the solution of the Fefferman equation on a smoothly bounded strictly pseudoconvex domain D in ${\mathbb {C}}^n$ is plurisubharmonic in D if and only if D is super-pseudoconvex. As an application, when D is super-pseudoconvex, he gives the sharp lower bound for the bottom of the spectrum of the Laplace-Beltrami operators by using the result of Li and Wang (Int. Math. Res. Not. 4351–4371, 2012).

1 Introduction

Let D be a smoothly bounded pseudoconvex domain D in ${\mathbb {C}}^n$. Let $u\in C^2(D)$ be a real-valued function and let H(u) denote the $n\times n$ complex Hessian matrix of u. We say that u is strictly plurisubharmonic in D if H(u) is positive definite on D. When u is strictly plurisubharmonic in D, u induces a Kähler metric

$$\begin{aligned} g=g[u] =\sum _{i,j=1}^n {\partial ^2 u\over \partial z_i \partial \overline{z}_j} dz^i \otimes d \overline{z}^j. \end{aligned}$$

(1.1)

We say that the metric g is also Einstein if its Ricci curvature

$$\begin{aligned} R_{k\overline{\ell }}=-{\partial ^2 \log \det [g_{i\overline{j}}] \over \partial z_k \partial \overline{z}_{\ell }} \end{aligned}$$

(1.2)

satisfies the equation: $R_{k\overline{\ell }}=c g_{k\overline{\ell }}$ for some constant c.

When $c<0$, after a normalization, we may assume $c=-(n+1)$. It was proved by Cheng and Yau [5] that the following Monge–Ampère equation:

$$\begin{aligned} \left\{ \begin{array}{c@{\quad }l} \det H(u)=e^{(n+1) u}, &{} z\in D\\ u=+\infty , &{} z\in \partial D\\ \end{array}\right. \end{aligned}$$

(1.3)

has a unique strictly plurisubharmonic solution $u\in C^\infty (D)$. Moreover, the Kähler metric

$$\begin{aligned} g[u]=\sum _{i,j=1}^n {\partial ^2 u\over \partial z_i \partial \overline{z}_j} dz_i\otimes d\overline{z}_j \end{aligned}$$

(1.4)

induced by u is a complete Kähler–Einstein metric on D.

When D is also strictly pseudoconvex, the existence and uniqueness problem was studied by Fefferman [6] earlier. He considered the following Fefferman equation

$$\begin{aligned} \left\{ \begin{array}{r@{\quad }l} \det J(\rho )=1, &{} z\in D,\\ \rho =0, &{} z\in \partial D,\\ \end{array}\right. \end{aligned}$$

(1.5)

where

$$\begin{aligned} J(\rho )= & {} -\det \left[ \begin{matrix} \rho &{}\overline{\partial }\rho \\ (\overline{\partial }\rho )^* &{} H(\rho )\\ \end{matrix}\right] ,\quad \overline{\partial }\rho =\left( {\partial \rho \over \partial \overline{z}_1},\ldots , {\partial \rho \over \partial \overline{z}_n}\right) \hbox { and } \nonumber \\ (\overline{\partial }\rho )^*= & {} \left( {\partial \rho \over \partial z_1},\ldots , {\partial \rho \over \partial z_n}\right) ^t. \end{aligned}$$

(1.6)

Fefferman searched for a solution $\rho <0$ on D such that $u=-\log (-\rho )$ is strictly plurisubharmonic in D. He proved the uniqueness and gave a formal or approximation solution for (1.5).

If the relation between $\rho$ and u is given by

$$\begin{aligned} \rho (z)=-e^{-u(z)},\qquad z\in D, \end{aligned}$$

(1.7)

then (1.3) is the same as (1.5). Moreover, one can prove (see [14] and references therein) that

$$\begin{aligned} \det H(u)= J(\rho ) e^{(n+1) u}. \end{aligned}$$

(1.8)

When D is smoothly bounded strictly pseudoconvex, it was proved by Cheng and Yau [5] that $\rho \in C^{n+3/2}(\overline{D})$. In fact, $\rho \in C^{n+2-\epsilon }(\overline{D})$ for any small $\epsilon >0$. This follows from an asymptotic expansion formula for $\rho$ obtained by Lee and Melrose [10]:

$$\begin{aligned} \rho (z)=r(z)\Big (a_0(z)+\sum _{j=1}^\infty a_j (r^{n+1} \log (-r))^j\Big ), \end{aligned}$$

(1.9)

where $r\in C^\infty (\overline{D})$ is any defining function for D and $a_j\in C^\infty (\overline{D})$ and $a_0(z)>0$ on $\partial D$.

When D is a bounded strictly pseudoconvex domain in ${\mathbb {C}}^n$ with smooth defining function r, one can view $(\partial D, \theta )$ as a pseudo-Hermitian CR manifold with the contact/pseudo Hermitian form

$$\begin{aligned} \theta ={1\over 2 i} (\partial r-\overline{\partial }r). \end{aligned}$$

(1.10)

An interesting and useful question is: How to find a defining function r such that $(\partial D, \theta )$ has positive the Webster-Tanaka pseudo Ricci curvature or pseudo scalar curvature? Under the assumption $u=-\log (-r)$ is strictly plurisubharmonic near and on $\partial D$, the following formula for the pseudo-Ricci curvature was discovered by Li and Luk [18]:

$$\begin{aligned} { Ric}_z(w, \overline{v})=-\sum _{k,\ell =1}^n {\partial ^2\log J(r)(z)\over \partial z_k\partial \overline{z}_{\ell }}w_k \overline{v}_{\ell }+ n{\det H(r)\over J(r)} \sum _{j,k=1}^n {\partial ^2 r(z) \over \partial z_k\partial \overline{z}_{\ell }}w_k \overline{v}_{\ell }\nonumber \\ \end{aligned}$$

(1.11)

for $w, v\in H_z=\{ v=(v_1,\ldots , v_n)\in {\mathbb {C}}^n: \sum _{j=1}^n {\partial r(z)\over \partial z_j} v_j=0\}$.

When g[u] is asymptotic Einstein (i.e. $J(r)=1+O(r^2))$, one has that

$$\begin{aligned} { Ric}_z(w, \overline{v})= n {\det H(r)\over J(r)} \sum _{j,k=1}^n {\partial ^2 r(z) \over \partial z_k\partial \overline{z}_{\ell }}w_k \overline{v}_{\ell } \end{aligned}$$

(1.12)

for $w, v\in H_z=\{ v=(v_1,\ldots , v_n)\in {\mathbb {C}}^n : \sum _{j=1}^n {\partial r(z)\over \partial z_j} v_j=0\}$. In this case, the Webster-Tanaka pseudo-Hermitian metric is a pseudo Einstein metric. Moreover, the pseudo Ricci curvature is positive on $\partial D$ if and only if $\det H(r)>0$ on $\partial D$.

Many researches [14, 15, 19, 20] indicate that the following problem is very interesting and very important.

Problem 1

Assume that D is a smoothly bounded strictly pseudoconvex domain in ${\mathbb {C}}^n$. Let $\rho$ be the solution of the Fefferman equation (1.5) such that $u=-\log (-\rho )$ is strictly plurisubharmonic in D. For what extra condition on D, one has that $\rho$ is strictly plurisubharmonic in $\overline{D}$.

It is well known that $\rho (z)=|z|^2-1$ is strictly plurisubharmonic when $D=B_n$, the unit ball in ${\mathbb {C}}^n$. It was proved by the Li [14] that $\rho$ is strictly plurisubharmonic when D is a bounded domain in ${\mathbb {C}}^n$ whose boundary is a real ellipsoid. In particular, when $n=2$ case, this result was also proved by Chanillo, Chiu and Yang [2] later.

One of the main purposes of this paper is to give a characterization for domains D in ${\mathbb {C}}^n$ where the answer of Problem 1 is affirmatively true. We first introduce the following definition.

Definition 1.1

Let D be a smoothly bounded domain in ${\mathbb {C}}^n$. We say that D is strictly super-pseudoconvex (super-superconvex) if there is a strictly plurisubharmonic defining function $r\in C^4(\overline{D})$ such that ${{\mathcal {L}}}_2[r]>0$ (${{\mathcal {L}}}_2[r]\ge 0$) on $\partial D$, respectively. Here

$$\begin{aligned} {{\mathcal {L}}}_2[r]\!=:\!1\!+\!{|\partial r|^2_r\over n(n+1)} {\tilde{\Delta }} \log J(r)-{2 \mathrm{Re}\,R \log J(r)\over n+1} \!-\!|\partial r|^2_r |\tilde{\nabla }\log J(r)|^2,\qquad \end{aligned}$$

(1.13)

with

$$\begin{aligned} \tilde{\Delta }=\sum _{i,j=1}^n a^{i\overline{j}}[r]{\partial ^2\over \partial z_i \partial \overline{z}_j},\quad R=\sum _{j=1}^nr^j {\partial \over \partial z_j}, \quad |\tilde{\nabla }f|^2= \sum _{i,j=1}^n a^{i\overline{j}}[r]{\partial f \over \partial z_i } {\partial f\over \partial \overline{z}_j}\nonumber \\ \end{aligned}$$

(1.14)

and

$$\begin{aligned} r^i \!=\!\sum _{j=1}^n r^{i\overline{j}} r_{\overline{j}},\quad \Big [r^{i\overline{j}}\Big ]^t=H(r)^{-1},\quad a^{i\overline{j}}[r]=: r^{i\overline{j}}-{r^i r^{\overline{j}}\over -r+|\partial r|_r^2},\quad 1\le i,j \le n.\nonumber \\ \end{aligned}$$

(1.15)

Another motivation of this paper is to apply the result (the solution of Problem 1) to estimate the lower bound of the bottom of the spectrum of Laplace-Beltrami operator $\Delta _{g[u]}$.

Definition 1.2

Let D be a smoothly bounded strictly pseudoconvex domain in ${\mathbb {C}}^n$. Let $r\in C^\infty (\overline{D})$ be a defining function for D such that $u=-\log (-r)$ is strictly plurisubharmonic. We say that the Kähler metric g[u] induced by u is super asymptotic Einstein if

• (i) the Ricci curvature $R_{i\overline{j}}\ge -(n+1) g_{i\overline{j}}$ on D; and

• (ii) $J(r)=1+O(r^2)$.

Let $(M^n, g)$ be a Kähler manifold with the Kähler metric g. Let $\Delta _g$ be the Laplcae-Beltrami operator associated to g. Let $\lambda _1$ denote the bottom of the spectrum of $\Delta _g$. Then the problem of estimating the upper bound and lower bound for $\lambda _1$ have studied by many authors, including Cheng [4], Lee [9], Li and Wang [12, 13], Munteanu [22], Li and Tran [19] and Li and Wang [20], Wang [24], etc... When the Ricci curvature is super Einstein: $R_{i\overline{j}}\ge -(n+1) g_{i\overline{j}}$, Munteanu [22] proves that $\lambda _1\le n^2$. For the lower bound estimate of $\lambda _1$, Li and Tran [19] and Li and Wang [20] consider a smoothly bounded pseudoconvex domain in ${\mathbb {C}}^n$ with defining function $r\in C^4(\overline{D})$ such that $u=:-\log (-r)$ is strictly plurisubharmonic in D. When r is plurisubharmonic in D, Li and Tran [19] prove that $\lambda _1=n^2$. When g[u] is super asymptotic Einstein and $\det H(r)\ge 0$ on $\partial D$, Li and Wang [20] prove $\lambda _1=n^2$. We will show that $\det H(r)\ge 0$ on $\partial D$ when D is super-pseudoconvex.

The first result of the paper is the following theorems.

Theorem 1.3

Let D be a smoothly bounded strictly pseudoconvex domain in ${\mathbb {C}}^n$. Let $\tilde{\rho }\in C^4(\overline{D})$ be a defining function for D such that $\tilde{u}=-\log (-\tilde{\rho })$ is strictly plurisubharmonic. If the Kähler metric $g[\tilde{u}]$ induced by $\tilde{u}$ is the super asymptotic Einstein, then the following two statements hold:

• (i) $\tilde{\rho }$ is strictly plurisubharmonic on $\overline{D}$ if and only if D is strictly super-pseudoconvex. In particular if $\tilde{\rho }=\rho (z)$ is the solution of (1.5) then $\rho$ is strictly plurisubharmonic in $\overline{D}$ when D is strictly super-pseudoconvex;

• (ii) If D is also super-pseudoconvex then $\lambda _1(\Delta _{g[\tilde{u}]})=n^2$, where $\Delta _g=-4\sum _{i,j=1}^n g^{i\overline{j}}{\partial ^2\over \partial z_i \partial \overline{z}_j}$.

It is interesting to bridge the relation between convex and super-pseudoconvex. The second result of the paper is:

Theorem 1.4

Let D be a smoothly bounded domain in ${\mathbb {C}}^n$. Then

• (i) When $n=1$, D is strictly super-pseudoconvex (super-pseudoconvex) if and only if D is strictly convex (convex);

• (ii) When $n>1$, if D is convex and if there is a strictly plurisubharmonic defining function $r \in C^4(\overline{D})$ such that

  $$\begin{aligned} n-1 \!+\! { |\partial r|^2\over n } a^{k\overline{\ell }}[r]\Big [\tilde{\Delta }r_{k\overline{\ell }}\!-\!a^{i\overline{q}}[r] r^{p\overline{j}} r_{i\overline{j}k} r_{p\overline{q}\overline{\ell }} \!-\! (\tilde{\Delta }r_k) (\tilde{\Delta }r_{\overline{\ell }}) \Big ] \!-\!2\mathrm{Re}\,r^k \tilde{\Delta }r_k > 0,\nonumber \\ \end{aligned}$$

  (1.16)

  then D is strictly super-pseudoconvex;

• (iii) The convexity does not imply super-pseudoconvexity and the super-pseudoconvexity does not imply the convexity either.

The paper is organized as follows: Sect. 2, we give an approximation formula. Theorem 1.3 will be proved in Sect. 3; Part (i) and Part (ii) of Theorem 1.4 will be proved in Sect. 4. Finally, in Sect. 5, we provide two domains in ${\mathbb {C}}^2$; one is strictly convex but not super-pseudoconvex and the other is super-pseudoconvex but not convex. These prove Part (iii) of Theorem 1.4.

2 An approximation formula

Let D be a bounded domain in ${\mathbb {C}}^n$ with smooth boundary. Let $r\in C^2(\overline{D})$ be a real-valued, negative defining function for D. Then the Fefferman operator [5, 6] acting on r is defined by

$$\begin{aligned} J(r)=-\det \left[ \begin{matrix} r &{} \overline{\partial }r\\ (\overline{\partial }r)^* &{} H(r)\\ \end{matrix}\right] , \end{aligned}$$

(2.1)

where $\overline{\partial }r=({\partial r\over \partial \overline{z}_1},\ldots , {\partial r\over \partial \overline{z}_n})=(r_{\overline{1}},\ldots , r_{\overline{n}})$ is a row vector in ${\mathbb {C}}^n$ and $(\overline{\partial }r)^*$ is its adjoint vector, which is column vector in ${\mathbb {C}}^n$ and $H(r)=[{\partial ^2 r\over \partial z_i \partial \overline{z}_j}]$ is the $n\times n$ complex Hessian matrix of r.

If $H(r)=[r_{i\overline{j}}]$ is invertible, in particular it is positive definite, then we use the notation $[r^{i\overline{j}}]^t=:H(r)^{-1}$ and

$$\begin{aligned} |\partial r|_r^2=\sum _{i,j=1}^n r^{i\overline{j}} r_i r_{\overline{j}}. \end{aligned}$$

(2.2)

It is easy to verify that

$$\begin{aligned} J(r)=\det H(r) (-r+|\partial r|_r^2). \end{aligned}$$

(2.3)

In fact, since

$$\begin{aligned} J(r )= & {} (-r) \det \left[ H(r)- {(\overline{\partial }r)^* (\overline{\partial }r) \over r}\right] \nonumber \\= & {} (-r) \det H(r) \left( 1-{|\partial r|_r^2 \over r}\right) \nonumber \\= & {} \det H(r) \left( -r+|\partial r|_r^2\right) . \end{aligned}$$

(2.4)

Remark 1

When H(r) is not positive definite on $\partial D$, we can replace r by

$$\begin{aligned} r[a]=:r(z)+{a\over 2} r^2. \end{aligned}$$

(2.5)

Then r[a] is positive definite with a large a and

$$\begin{aligned} J(r)={1\over (1+a r)^n}\det H(r[a]) (-r +(1+2a\, r) |\partial r|_{r[a]}). \end{aligned}$$

(2.6)

From now on, we will always assume that $r(z)\in C^\infty (\overline{D})$ is a negative defining function for D such that

$$\begin{aligned} \ell (r)=-\log (-r) \end{aligned}$$

(2.7)

is strictly plurisubharmonic in D. It is known from [5, 14–16] that the following identity holds:

$$\begin{aligned} \det H(\ell (r))=J(r) e^{(n+1)\ell (r)}. \end{aligned}$$

(2.8)

This implies that

• (i) $u=:\ell (r)$ is strictly plurisubharmonic on D if and only if $J(r)>0$ on D;

• (ii) $J(r)=1$ if and only if $\det H(u)=e^{(n+1) u}$ with $u=:\ell (r)$.

Fefferman [6] gave a formula to approximate the potential function $\rho$ [for Eq. (1.5)]. He proved that $J(r\, J(r)^{-1/(n+1)})=1+O(r)$ near $\partial D$. Higher order approximation can be iterated through the previous steps. Based on the Fefferman’s idea, the iteration formula of the approximation was given in more detail by Graham in [7]. The author [14] gave another modification. For convenience of readers and further argument for the current paper, we will state and prove a second order approximation formula here.

Theorem 2.1

Let D be a smoothly bounded pseudoconvex domain in ${\mathbb {C}}^n$. Let r(z) be a smooth negative defining function for D such that $\ell (r)$ is strictly plurisubharmonic in D. Let

$$\begin{aligned} \rho _{1}(z) =r(z) J(r)^{-1/(n+1)} e^{-B(z)} \end{aligned}$$

(2.9)

with

$$\begin{aligned} B(z)=B[r](z)={ \hbox {tr}(H(\ell ( r ))^{-1} H(\log J(r)) \over 2n (n+1)}. \end{aligned}$$

(2.10)

Then

$$\begin{aligned} J(\rho _{1})(z)=1+O(r^2). \end{aligned}$$

(2.11)

Moreover, if $J(r)=1+O(r^2)$ then $\rho _1=r+O(r^3)$ and $J(\rho _1)=1+O(r^3)$.

Proof

Since

$$\begin{aligned} H(\ell (r))={1\over (-r)(1+ar)} \left[ H(r_a)+{1+2a\, r \over (-r)} (\overline{\partial }r)^*(\overline{\partial }r)\right] \end{aligned}$$

(2.12)

by choosing $a\ge 0$ so that r[a] is strictly plurisubharmonic. Therefore, we can write

$$\begin{aligned} B(z)=(-r) B_0(z), \end{aligned}$$

(2.13)

with $B_0\in C^\infty (\overline{D})$. Since

$$\begin{aligned} H(B)= & {} (-r) H(B_0)-B_0 \left( H(r)+{(\overline{\partial }r)^* \overline{\partial }r\over -r}\right) +B_0{(\overline{\partial }r)^* \overline{\partial }r\over -r}-(\overline{\partial }r)^* (\overline{\partial }B_0)\nonumber \\&-(\overline{\partial }B)^* (\overline{\partial }r). \end{aligned}$$

(2.14)

By complex rotation, one may assume that ${\partial r\over \partial z_j}(z_0)=0$ for $1\le j\le n-1$ and $H(r)(z_0)$ is diagonal, it is easy to verify that

$$\begin{aligned} \hbox {tr}(H(\ell (r)^{-1}H(B)) =-n B(z)+(-r)B_0+O(r^2)=-(n-1)B +O(r^2).\nonumber \\ \end{aligned}$$

(2.15)

Since

$$\begin{aligned} J(\rho _1)(z) e^{(n+1)\ell (\rho _1)}= & {} \det H(\ell (\rho _1)) \\= & {} \det \Big ( H(\ell (r)) +{1\over n+1} H(\log J) +H(B)\Big )\\ \!= & {} \!\det H(\ell (r)) \det \Bigg ( I_n\!+\! H(\ell (r))^{-1}\left[ {1\over n\!+\!1} H(\log J) \!+\!H(B)\right] \Bigg )\\ \!= & {} \!J(r)e^{(n+1)\ell (r)} \det \Bigg ( I_n\!+\! H(\ell (r))^{-1}\left[ {1\over n\!+\!1} H(\log J) \!+\!H(B)\!\right] \Bigg ) \end{aligned}$$

Notice that $\exp ((n+1)\ell (\rho _1))=\exp ((n+1)B) J(r) \exp ((n+1)\ell (r))$, we have

$$\begin{aligned} J(\rho _1)(z)= & {} e^{-(n+1) B} \det \Bigg ( I_n+ H(\ell (r))^{-1}\left[ {1\over n+1} H(\log J) +H(B)\right] \Bigg )\\= & {} e^{-(n+1) B}\left[ 1+\hbox {tr}\left[ H(\ell (r))^{-1}\left[ {1\over n+1} H(\log J) +H(B)\right] \right] \right] +O(r^2)\\= & {} e^{-(n+1) B}[1+2n B +\hbox {tr}( H(\ell (r))^{-1} H(B)] +O(r^2)\\= & {} e^{-(n+1)B}[1+2n B -(n-1) B+O(r^2)]+O(r^2)\\= & {} 1+{(n+1)^2\over 2} B^2+O(r^2)\\= & {} 1+O(r^2). \end{aligned}$$

When $J(r)=1+A r^2$ with A is smooth on $\overline{D}$, it is easy to prove $B=B_1 r^2$ with $B_1$ smooth in $\overline{D}$ near $\partial D$. It is also easy to verify that $\rho _1[r]=r+O(r^3)$ and $J(\rho _1[r])=1+O(r^3)$. This proves Theorem 2.1.$\square$

Proposition 2.2

Let D be a smoothly bounded strictly pseudoconvex domain in ${\mathbb {C}}^n$. Let u be the plurisubharmonic solution of (1.3) and $\rho (z)=-e^{-u}$. Then for any smooth defining function r of D with $\ell (r)$ being strictly plurisubharmonic in D, we have

$$\begin{aligned} \det H(\rho )= & {} J(r)^{-n\over n+1} \det \Big ( H(r) -{ [\partial _i r \partial _{\overline{j}} \log J+ \partial _i \log J(r)\, \partial _{\overline{j}} r]\over n+1}\nonumber \\&-[\partial _i r \partial _{\overline{j}} B(z)+\partial _i B \partial _{\overline{j}} r] \Big ) \end{aligned}$$

(2.16)

on $\partial D$, where $B(z)=B[r](z)$ is given by (2.10).

Proof

Let

$$\begin{aligned} \rho _1(z)=\rho _1[r]=:r(z) J(r)^{-1/(n+1)} e^{-B}. \end{aligned}$$

(2.17)

Theorem 2.1 implies that $\rho (z)=\rho _1(z)+O(r(z)^3)$. A simple calculation shows that

$$\begin{aligned} \det H(\rho )=\det H(\rho _1),\quad z\in \partial D. \end{aligned}$$

(2.18)

By (2.13) ($B=(-r) B_0$), one can easily see that

$$\begin{aligned} \rho _1(z)=r(z) J(r)^{-1/(n+1)} -r(z) J(r)^{-1/(n+1)} B(z)+O(r(z)^3) \end{aligned}$$

(2.19)

and

$$\begin{aligned} \det H(\rho _1)=\det H\Big (r(z) J(r)^{-1/(n+1)} -r(z) J(r)^{-1/(n+1)} B(z)\Big ),\quad z\in \partial D.\nonumber \\ \end{aligned}$$

(2.20)

For any $z\in \partial D$, by (2.20), one has

$$\begin{aligned} \det H(\rho _1)(z)= & {} \det \Big ( H(r J(r)^{-1/(n+1)} ) -J(r)^{-1/(n+1)} [\partial _i r \partial _{\overline{j}} B+\partial _i B \partial _{\overline{j}} r] \Big )\nonumber \\= & {} \det \Big ( J(r)^{-1\over (n+1)} H(r) -{ J^{-(n+2)\over (n+1)}\over n+1} [\partial _i r \partial _{\overline{j}} J+ \partial _i J(r) \partial _{\overline{j}} r]\nonumber \\&-J(r)^{-1\over (n+1)} [\partial _i r \partial _{\overline{j}} B+\partial _i B \partial _{\overline{j}} r] \Big )\nonumber \\= & {} J(r)^{-n\over n+1} \det \Big ( H(r) -{ 1\over n+1} [\partial _i r \partial _{\overline{j}} \log J+ \partial _i \log J(r)\, \partial _{\overline{j}} r]\nonumber \\&-[\partial _i r \partial _{\overline{j}} B+\partial _i B \partial _{\overline{j}} r] \Big ). \end{aligned}$$

(2.21)

This proves Proposition 2.2. $\square$

Let $u^{D_j}$ be the potential functions for the Kähler–Einstein metric of $D_j$ and let

$$\begin{aligned} \rho ^{D_j}(z)=-e^{-u^{D_j}(z)},\quad j=1, 2. \end{aligned}$$

(2.22)

Proposition 2.3

Let $\phi : D_1\rightarrow D_2$ be a smooth biholomorphic mapping. Then

$$\begin{aligned} \rho ^{D_1}(z)=\rho ^{D_2}(\phi (z)) |\det \phi '(z)|^{-2/(n+1)} \end{aligned}$$

(2.23)

In particular, if $\det \phi '(z)$ is constant c then

$$\begin{aligned} \det H(\rho ^{D_1})(z)=|c|^{2/(n+1)} \det H(\rho ^{D_2})(\phi (z)). \end{aligned}$$

(2.24)

Proof

Since $\phi : D_1\rightarrow D_2$ is biholomorphic, one has that if $u^{D_j}$ is the unique plurisubharmonic solutions for the Monge–Ampère equation:

$$\begin{aligned} \left\{ \begin{array}{c@{\quad }l} \det H(u)=e^{(n+1) u}, &{} z\in D_j\\ u=\infty ,&{} z\in \partial D_j\\ \end{array}\right. \end{aligned}$$

(2.25)

Then

$$\begin{aligned} u^{D_1}(z)=u^{D_2}(\phi (z))+{1\over n+1}\log |\det \phi '(z)|^2,\quad z\in D_1 \end{aligned}$$

(2.26)

and

$$\begin{aligned} \rho ^{D_1}(z)=\rho ^{D_2}(\phi (z))|\det \phi '(z)|^{-2/(n+1)}. \end{aligned}$$

(2.27)

In particular, when $\det \phi '(z)=c$, one has

$$\begin{aligned} \det H(\rho ^{D_1})(z)=|c|^{-2n/(n+1)} \det H(\rho ^{D_2})(\phi (z)) |c|^2=|c|^{2/(n+1)} \det H(\rho ^{D_2})(\phi (z)) \end{aligned}$$

and the proof of Proposition 2.3 is complete. $\square$

We also need the following holomorphic change of variables formula.

Lemma 2.4

For $z_0\in \partial D$, if $z=\phi (w): B(0,\delta _0) \rightarrow B(z_0, 1)$ be a one-to-one holomorphic map with $\phi (0)=z_0$ and $r(z)=\tilde{r}(w)$, then

$$\begin{aligned} \rho _1(\phi (w))=|\det \phi '(w)|^{2/(n+1)} {\tilde{r}(w)\over J(\tilde{r}(w))^{1/(n+1)}}e^{-B(\tilde{r}(w))}. \end{aligned}$$

(2.28)

Moreover, if $|\det \phi '(z)|^2$ is a constant on $B(0,\delta _0)$ for some $\delta _0>0$

$$\begin{aligned} \det H(\rho _1)(z_0) |\det \phi '(0)|^{2\over n+1}= \det H\Big ( {\tilde{r}\over J(\tilde{r})^{1/(n+1)}}e^{-B(\tilde{r})}\Big )(0). \end{aligned}$$

(2.29)

Proof

Since $|\det \phi '(z)|^2$ is constant, by the definitions for B[r] and J(r) from Theorem 2.1, one can easily prove (2.27) and (2.29), and the proposition is proved. $\square$

3 Proof of Theorem 1.3

Let D be a smoothly bounded strictly pseudoconvex domain in ${\mathbb {C}}^n$. Let $r\in C^\infty (\overline{D})$ be any strictly plurisubharmonic defining function for D. Let

$$\begin{aligned} \rho _1(z)= r(z) J(r)^{-1/(n+1)} \exp (-B(z)) \end{aligned}$$

(3.1)

where

$$\begin{aligned} B(z)={ \hbox {tr}(H(\ell (r))^{-1} H(\log J(r)) \over 2n (n+1)}, \end{aligned}$$

(3.2)

According to Theorem 2.1, one has

$$\begin{aligned} J(\rho _1)=1+O(r(z)^2). \end{aligned}$$

(3.3)

Let $\rho =\rho ^D$ be the solution of (1.5) such that $\ell (\rho )$ is strictly plurisubharmonic in D. Then

$$\begin{aligned} \det H(\rho )(z)=\det H(\rho _1)(z)\quad \hbox {on }\partial D. \end{aligned}$$

(3.4)

By Proposition 2.2 and

$$\begin{aligned} B(z)= & {} {(-r)\over 2n (n+1)} \hbox {tr}\left[ \left( H(r)+{r_i r_{\overline{j}}\over -r}\right) ^{-1} H(\log J(r)) \right] (z)\nonumber \\= & {} {(-r)\over 2n (n+1)}\sum _{j,k=1}^n \left( r^{i\overline{j}} -{r^i r^{\overline{j}} \over -r+|\partial r|_r^2}\right) {\partial ^2 \log J(r)\over \partial z_i\partial \overline{z}_j} =-B^0(z) r, \end{aligned}$$

(3.5)

where

$$\begin{aligned} B^0(z)={1\over 2n (n+1)} \sum _{j,k=1}^n a^{i\overline{j}}[r] {\partial ^2 \log J(r)\over \partial z_i\partial \overline{z}_j}={1\over 2n(n+1)}{\tilde{\Delta }}_r \log J(r). \end{aligned}$$

(3.6)

Thus for $z_0\in \partial D$, one has

$$\begin{aligned} \partial _j B (z_0)=- B^0(z_0) \partial _j r (z_0), \quad \partial _{\overline{j}} B (z_0)=- B^0(z_0) \partial _{\overline{j}} r (z_0), \quad \hbox { for } 1\le j\le n.\nonumber \\ \end{aligned}$$

(3.7)

Let

$$\begin{aligned} R=\sum _{j=1}^n r^{j} {\partial \over \partial z_j},\quad \overline{R}=\sum _{j=1}^n r^{\overline{j}}{\partial \over \partial \overline{z}_j},\quad r^i=r^{i\overline{j}} r_{\overline{j}}, \ \ r^{\overline{j}}=r^{i\overline{j}} r_i. \end{aligned}$$

(3.8)

and

$$\begin{aligned} \Big | \tilde{\nabla _r} f \Big |^2=:\sum _{i,j=1}^n \left( r^{i\overline{j}}-{r^i r^{\overline{j}}\over -r +|\partial r|_r^2}\right) \partial _i f\partial _{\overline{j}} f=\sum _{i,j=1}^n r^{i\overline{j}} \partial _i f\partial _{\overline{j}} f-{|R f|^2\over -r +|\partial r|^2_r }.\nonumber \\ \end{aligned}$$

(3.9)

Then it is easy to see that

$$\begin{aligned} |\tilde{\nabla }_r r|^2=0\quad \hbox { on }\ \partial D. \end{aligned}$$

(3.10)

Therefore, by (2.21) and Lemma 3.1 in [14], at $z=z_0\in \partial D$, one has

$$\begin{aligned}&\det H(\rho )(z^0)\,J(r)^{n/(n+1)}(z^0)\nonumber \\&\quad =\det H(r) \Bigg (\Big | 1- r^{i\overline{j}} \left( \partial _i r\, \Bigg ({\partial _{\overline{j}}\log J(r) \over n+1}-B^0 \partial _{\overline{j}} r\Bigg )\right) \Big |^2\Bigg )\nonumber \\&\qquad - |\partial r|_r^2 \sum _{i,j=1}^n r^{i\overline{j}} \left( {\partial _{i}\log J(r) \over n+1}-B^0 \partial _i r\right) \left( {\partial _{\overline{j}}\log J(r) \over n+1}-B^0 \partial _{\overline{j}} r\right) \Bigg )\nonumber \\&\quad =\det H(r) \Big (\Big | 1- { \overline{R}\log J(r) \over n+1}+B^0 |\partial r|^2_r\Big |^2\nonumber \\&\qquad - |\partial r|_r^2 \sum _{i,j=1}^n r^{i\overline{j}} {\partial _{i}\log J(r) \partial _{\overline{j}} \log J(r)\over (n+1)^2} +|\partial r|_r^2 2\mathrm{Re}\,B^0 {\overline{R}\log J(r)\over n+1} - |\partial r|_r^4 |B^0|^2 \Big )\nonumber \\&\quad =\det H(r) \Big (1+2 B^0 |\partial r|^2 -2\mathrm{Re}\,{\overline{R}\log J(r)\over n+1} -{|\partial r|_r ^2\over (n+1)^2} |\tilde{\nabla }_r \log J(r)|^2 \Big )\nonumber \\&\quad \!=\!\det H(r) \Big (1\!+\!{|\partial r|^2\over n(n+1)} \tilde{\Delta }\log J(r) \!-\!2\mathrm{Re}\,{\overline{R}\log J(r)\over n+1} \!-\!{|\partial r|_r ^2\over (n+1)^2} |\tilde{\nabla }_r \log J(r)|^2 \Big )\nonumber \\&\quad > 0 \end{aligned}$$

(3.11)

since D is strictly super-pseudoconvex, there is a strictly plurisubharmonic function $r\in C^4(\overline{D})$ such that the above inequality holds on $\partial D$. If $\tilde{\rho }$ is smooth defining function for D such that the Kähler metric induced by $\tilde{u}=-\log (-\tilde{\rho })$ is super asymptotic Enistein, then $\det H(\tilde{\rho })=\det H(\rho ) >0$ on $\partial D$ by (3.11). By Lemma 2 in [20], one has that $\det H(\tilde{\rho })$ attains its minimum over $\overline{D}$ at some ponit in $\partial D$. Therefore, $\det H(\tilde{\rho })>0$ on $\overline{D}$ and the proof of Part (i) of Theorem 1.3 is complete. Part (ii) of Theorem  1.3 is a corollary of Part (i) and the result in [19] and [20]. Therefore, the proof of Theorem 1.3 is complete. $\square$

4 Super-pseudoconvex domains

In this section, we will study the relation between super-pseudoconvex domains and convex domains when $n=1$. We will also study and simplify some quantities in the definition of the super-pseudoconvex domain in ${\mathbb {C}}^n$. Since

$$\begin{aligned} \log J(r)= & {} \log \det H(r)+\log (-r+|\partial r|_r^2), \end{aligned}$$

(4.1)

$$\begin{aligned} {\partial (-r +|\partial r|^2_r)\over \partial z_k}= & {} -r_k+ \partial _k(r^{i\overline{j}}) r_ i r_{\overline{j}} +r^{i\overline{j}} r_{ik} r_{\overline{j}}+r^{i\overline{j}} r_i r_{k\overline{j}}\nonumber \\= & {} -r ^{i \overline{q}} r^{p\overline{j}} r_{p\overline{q}k} r_ i r_{\overline{j}} +r^{i\overline{j}} r_{ik} r_{\overline{j}} \nonumber \\= & {} -r ^{ \overline{q}} r^{p} r_{p\overline{q}k} +r^{i} r_{ik} \end{aligned}$$

(4.2)

and

$$\begin{aligned} {\partial \log J\over \partial z_k}= & {} {\partial \log \det H(r)+\log (-r+|\partial r|^2_r)\over \partial z_k}\nonumber \\= & {} \left( r^{i\overline{j}} -{r^i r^{\overline{j}}\over -r+|\partial r|_r^2} \right) r_{i\overline{j}k} +{r^i r_{ik}\over -r+|\partial r|_r^2}, \end{aligned}$$

(4.3)

we have

$$\begin{aligned} R \log J(r)(z_0)= r^k {\tilde{\Delta }} r_k + {r^i r^k\over |\partial r|_r^2} r_{ik}. \end{aligned}$$

(4.4)

Thus,

$$\begin{aligned} \det H(\rho )(z^0)\,J(r)^{n/(n+1)}(z^0) =\det H(r) \Big (1-{2 \mathrm{Re}\,r^k r^i r_{ik}\over (n+1)|\partial r|^2}+ \tilde{E}(r)\Big ), \end{aligned}$$

(4.5)

where

$$\begin{aligned} \tilde{E}(r)=: { |\partial r|^2 \over n(n+1)}\Big [ {\tilde{\Delta }} \log J(r) -{n |{\tilde{\nabla }}\log J(r)|^2\over (n+1)} -2n \mathrm{Re}\,\left( {r^k \tilde{\Delta }r_k\over |\partial r|_r^2} \right) \Big ]. \end{aligned}$$

(4.6)

Proposition 4.1

Let D be a smoothly bounded domain in the complex plane ${\mathbb {C}}$. Then D is (strictly) super-pseudoconvex if and only if D is (strictly) convex.

Proof

Let r be any smooth strictly subharmonic defining function on $D\subset {\mathbb {C}}$. By (4.5) and (4.6), we have $a^{1\overline{1}}[r]=0$ and $\tilde{E}(r)=0$ on $\partial D$. Therefore, D is strictly super-pseudoconvex if and only if

$$\begin{aligned} S_r(z)=:\det H(r) \Big (1-{2\over n+1}\mathrm{Re}\,{r^k r^i r_{ik}\over |\partial r|_r^2}\Big ) >0 \end{aligned}$$

(4.7)

on $\partial D$. For ant $z_0\in \partial D$, by rotation, we may assume that $r_n(z_0)>0$. Thus

$$\begin{aligned} S_r(z_0) =r_{1\overline{1}} -\mathrm{Re}\,r_{11}(z_0) \end{aligned}$$

(4.8)

is positive for all $z_0 \in \partial D$ if and only if $\partial D$ is strictly convex; and is non-negative for all $z_0\in \partial D$ if and only if $\partial D$ is convex, respectively. Therefore, the proof of the proposition is complete. $\square$

Next we estimate $\tilde{E}(r)$.

Proposition 4.2

With the notation above, for $z\in \partial D$, we have

$$\begin{aligned} \tilde{E}(r)\ge & {} { |\partial r|^2 a^{k\overline{\ell }}[r]\over n (n+1)} \Big [\tilde{\Delta }r_{k\overline{\ell }}-a^{i\overline{q}}[r] r^{p\overline{j}} r_{i\overline{j}k} r_{p\overline{q}\overline{\ell }} - (\tilde{\Delta }r_k) (\tilde{\Delta }r_{\overline{\ell }}) - n {r^i r_{ik} r^{\overline{j}} r_{\overline{j}\overline{\ell }}\over |\partial r|_r^4}\Big ]\nonumber \\&-{2\mathrm{Re}\,r^k \tilde{\Delta }r_k \over (n+1) } \end{aligned}$$

(4.9)

and

$$\begin{aligned} \tilde{E}(r)\le {|\partial r|^2 a^{k\overline{\ell }} \over n (n+1)}\Big [\tilde{\Delta }r_{k\overline{\ell }}+a^{i\overline{q}}[r] r^p r^{\overline{j}}r_{i\overline{j}k}r_{p\overline{q}\overline{\ell }}+2 a^{i\overline{q}}[r] \, {r_{i k} r_{\overline{q}\overline{\ell }} \over |\partial r|^2} \Big ]\!-\!{2\mathrm{Re}\,r^k \tilde{\Delta }r_k \over (n+1) }.\nonumber \\ \end{aligned}$$

(4.10)

Proof

The following two identities will be used later.

$$\begin{aligned} (r^i)_{\overline{\ell }}= & {} (r^{i\overline{q}} r_{\overline{q}})_{\overline{\ell }}\!=\!r_{\overline{q}} (r^{i\overline{q}})_{\overline{\ell }}+r^{i\overline{q}} r_{\overline{q}\overline{\ell }}\!=\!-r^{i \overline{t}} r^{s\overline{q}} r_{s \overline{t}\overline{\ell }} r_{\overline{q}}+r^{i \overline{q}} r_{\overline{q}\overline{\ell }}\\= & {} -r^{i\overline{t}} r^s r_{s\overline{t}\overline{\ell }}+r^{i\overline{q}} r_{\overline{q}\overline{\ell }} \end{aligned}$$

and

$$\begin{aligned} (r^{\overline{j}})_{\overline{\ell }}=(r^{p \overline{j}} r_p)_{\overline{\ell }}=-r^{\overline{q}} r^{i\overline{j}} r_{i \overline{q}\overline{\ell }}+\delta _{j\ell }. \end{aligned}$$

By (4.3) and (4.2), for $z\in \partial D$, one has

$$\begin{aligned} {\partial ^2 \log J(r)\over \partial z_k \partial \overline{z}_{\ell }}= & {} \left( r^{i\overline{j}} -{r^i r^{\overline{j}}\over |\partial r|_r^2} \right) r_{i\overline{j}k\overline{\ell }} +r_{i\overline{j}k}{\partial \over \partial \overline{z}_\ell } \left( r^{i\overline{j}}-{r^i r^{\overline{j}}\over -r +|\partial r|^2_r}\right) \\&+{\partial \over \partial \overline{z}_\ell }{r^i r_{ik}\over (-r+|\partial r|_r^2)}\\= & {} \tilde{\Delta }r_{ k\overline{\ell }} -r_{i\overline{j}k} r^{i\overline{q}} r^{p\overline{j}} r_{p\overline{q}\overline{\ell }}\\&+{1\over (|\partial r|_r^2)^2} (r_{i\overline{j}k}r^i r^{\overline{j}} -r^i r_{ik})\left( {\partial (-r +|\partial r|_r^2) \over \partial \overline{z}_\ell }\right) \\&-{r_{i\overline{j}k}\over |\partial r|^2_r} (r^i (r^{\overline{j}})_{\overline{\ell }} +r^{\overline{j}} (r^i)_{\overline{\ell }})+ {1\over |\partial r|^2} (r^i r_{ik\overline{\ell }}+ r_{ik} (r^i)_{\overline{\ell }})\\= & {} \tilde{\Delta }r_{ k\overline{\ell }} -r_{i\overline{j}k} r^{i\overline{q}} r^{p\overline{j}} r_{p\overline{q}\overline{\ell }}\\&+{1\over (|\partial r|_r^2)^2} (r_{i\overline{j}k}r^i r^{\overline{j}} -r^i r_{ik})( -r ^{ \overline{q}} r^{p} r_{p\overline{q}\overline{\ell }} + r^{\overline{q}} r_{\overline{q}\overline{\ell }} )\\&-{r_{i\overline{j}k}\over |\partial r|^2_r} \Big ( r^{\overline{j}}( -r^{i\overline{t}} r^s r_{s\overline{t}\overline{\ell }}+r^{i\overline{q}} r_{\overline{q}\overline{\ell }}) +r^i (-r^{\overline{q}} r^{p\overline{j}} r_{p \overline{q}\overline{\ell }}+\delta _{j\ell })\Big )\\&+ {1\over |\partial r|^2} \Big (r^i r_{ik\overline{\ell }}+ r_{ik} (-r^{i\overline{t}} r^s r_{s\overline{t}\overline{\ell }}+r^{i\overline{q}} r_{\overline{q}\overline{\ell }})\Big )\\ \!= & {} \!\tilde{\Delta }r_{k\overline{\ell }}\!-\!r^{i\overline{q}} r^{p\overline{j}} r_{i\overline{j}k} r_{p\overline{q}\overline{\ell }} \!-\!{1\over (|\partial r|_r^2)^2} (r_{i\overline{j}k}r^i r^{\overline{j}} -r^i r_{ik})( r ^{ \overline{q}} r^{p} r_{p\overline{q}\overline{\ell }} - r^{\overline{q}} r_{\overline{q}\overline{\ell }} )\\&+{1 \over |\partial r|^2_r} (r^p r^{\overline{j}} r^{i\overline{q}} +r^i r^{\overline{q}} r^{p\overline{j}}) r_{p\overline{q}\overline{\ell }}r_{i\overline{j}k}-{1\over |\partial r|_r^2} r^{\overline{j}} r^{i\overline{q}} r_{\overline{q}\overline{\ell }}r_{i\overline{j}k} -{r_{i\overline{\ell }k}\over |\partial r|_r^2} r^i\\&+ {1\over |\partial r|^2} \Big (r^i r_{ik\overline{\ell }}-r^{i\overline{t}} r^s r_{s\overline{t}\overline{\ell }} r_{ik} +r^{i\overline{q}} r_{\overline{q}\overline{\ell }} r_{ik}\Big )\\= & {} \tilde{\Delta }r_{k\overline{\ell }}-r^{i\overline{q}} r^{p\overline{j}} r_{i\overline{j}k} r_{p\overline{q}\overline{\ell }} -{r^i r^{\overline{j}} r^p r^{\overline{q}} \over |\partial r|_r^4} r_{i\overline{j}k} r_{p\overline{q}\overline{\ell }} +{1\over |\partial r|_r^4}(r^i r^{\overline{j}} r_{i\overline{j}k} r^{\overline{q}} r_{\overline{q}\overline{\ell }} \nonumber \\&+r^p r^{\overline{q}} r_{p\overline{q}\overline{\ell }} r^i r_{ik}) \\&+{1 \over |\partial r|^2_r} (r^p r^{\overline{j}} r^{i\overline{q}} +r^i r^{\overline{q}} r^{p\overline{j}}) r_{p\overline{q}\overline{\ell }}r_{i\overline{j}k}\\&- {1\over |\partial r_r^2} ( r^i r^{p\overline{j}} r_{pk} r_{i\overline{j}\overline{\ell }} +r^{\overline{j}} r^{i\overline{q}} r_{\overline{q}\overline{\ell }} r_{i\overline{j}k} ) +{1\over |\partial r|_r^2} \left( r^{i\overline{q}} -{r^i r^{\overline{q}}\over |\partial r|_r^2}\right) r_{\overline{q}\overline{\ell }} r_{ik} \\= & {} \tilde{\Delta }r_{k\overline{\ell }}-\left( r^{i\overline{q}}-{r^i r^{\overline{q}}\over |\partial r|_r^2}\right) \left( r^{p\overline{j}} -{r^p r^{\overline{j}}\over |\partial r|_r^2}\right) r_{i\overline{j}k} r_{p\overline{q}\overline{\ell }} \\&- {1\over |\partial r|_r^2} \left( r^i \left( r^{p\overline{j}} -{r^p r^{\overline{j}}\over |\partial r|_r^2} \right) r_{pk} r_{i\overline{j}\overline{\ell }} +r^{\overline{j}} \left( r^{i\overline{q}} -{r^i r^{\overline{q}}\over |\partial r|_r^2}\right) r_{\overline{q}\overline{\ell }} r_{i\overline{j}k} \right) \nonumber \\&+{1\over |\partial r|_r^2} \left( r^{i\overline{q}} -{r^i r^{\overline{q}}\over |\partial r|_r^2}\right) r_{\overline{q}\overline{\ell }} r_{ik}. \end{aligned}$$

Then for $z\in \partial D$, we have

$$\begin{aligned} \tilde{\Delta }\log J(r)(z)\ge & {} a^{k\overline{\ell }}[r]\tilde{\Delta }r_{k\overline{\ell }}-a^{k\overline{\ell }}[r] a^{i\overline{q}}[r] a^{p\overline{j}}[r] r_{i\overline{j}k} r_{p\overline{q}\overline{\ell }} \\&-a^{k\overline{\ell }}[r] {a^{i\overline{q}}[r] \over |\partial r|^2_r} \Big ( r^{\overline{j}} r_{i\overline{j}k} r^p r_{p\overline{q}\overline{\ell }} +r_{ki} r_{\overline{q}\overline{\ell }}\Big ) +{1\over |\partial r|_r^2} a^{k\overline{\ell }}[r] a^{i\overline{q}}[r] r_{\overline{q}\overline{\ell }} r_{ik} \\= & {} a^{k\overline{\ell }}\tilde{\Delta }r_{k\overline{\ell }}-a^{k \overline{\ell }}[r] a^{i\overline{q}}[r] \, r^{p\overline{j}} r_{i\overline{j}k} r_{p\overline{q}\overline{\ell }} \end{aligned}$$

and

$$\begin{aligned} \tilde{\Delta }\log J(r)(z)\le & {} a^{k\overline{\ell }}\tilde{\Delta }r_{k\overline{\ell }}+2a^{k \overline{\ell }}[r] a^{i\overline{q}}[r] \, {r_{i k} r_{\overline{q}\overline{\ell }} \over |\partial r|^2} +a^{k\overline{\ell }}[r] a^{i\overline{q}}[r] r^p r^{\overline{j}} r_{i\overline{j}k} r_{p\overline{q}\overline{\ell }}. \end{aligned}$$

Moreover,

$$\begin{aligned} |\tilde{\nabla }\log J(r)|^2= & {} a^{k\overline{\ell }}[r]\, \left( \tilde{\Delta }r_k +{r^i r_{ik}\over |\partial r|_r^2}\right) \left( \tilde{\Delta }r_{\overline{\ell }}+{r^{\overline{j}} r_{\overline{j}\overline{\ell }}\over |\partial r|_r^2}\right) \\= & {} a^{k\overline{\ell }}[r]\, \Big [(\tilde{\Delta }r_k) (\tilde{\Delta }r_{\overline{\ell }}) +(\tilde{\Delta }r_k)\left( {r^{\overline{j}} r_{\overline{j}\overline{\ell }}\over |\partial r|_r^2}\right) +{r^i r_{ik}\over |\partial r|_r^2} \tilde{\Delta }r_{\overline{\ell }}+{r^i r_{ik}\over |\partial r|_r^2} {r^{\overline{j}} r_{\overline{j}\overline{\ell }}\over |\partial r|_r^2}\Big ]\\\le & {} a^{k\overline{\ell }}[r]\, \Big [{n+1\over n} (\tilde{\Delta }r_k) (\tilde{\Delta }r_{\overline{\ell }}) +(n+1){r^i r_{ik}\over |\partial r|_r^2} {r^{\overline{j}} r_{\overline{j}\overline{\ell }}\over |\partial r|_r^2}\Big ]. \end{aligned}$$

Therefore,

$$\begin{aligned} \tilde{\Delta }\log J(r)- & {} {n\over n+1} |\tilde{\nabla }\log J|^2 \ge a^{k\overline{\ell }}[r]\Big (\tilde{\Delta }r_{k\overline{\ell }}-a^{i\overline{q}}[r] r^{p\overline{j}} r_{i\overline{j}k} r_{p\overline{q}\overline{\ell }} \Big ) \nonumber \\- & {} a^{k\overline{\ell }}[r]\Big ( (\tilde{\Delta }r_k) (\tilde{\Delta }r_{\overline{\ell }}) +n {r^i r_{ik}\over |\partial r|_r^2} {r^{\overline{j}} r_{\overline{j}\overline{\ell }}\over |\partial r|_r^2}\Big ). \end{aligned}$$

Therefore,

$$\begin{aligned} \tilde{E}(r)\ge & {} { |\partial r|^2 a^{k\overline{\ell }}[r]\over n (n+1)}\Big (\tilde{\Delta }r_{k\overline{\ell }}-a^{i\overline{q}}[r] r^{p\overline{j}} r_{i\overline{j}k} r_{p\overline{q}\overline{\ell }} - (\tilde{\Delta }r_k) (\tilde{\Delta }r_{\overline{\ell }}) -n {r^i r_{ik}\over |\partial r|_r^2} {r^{\overline{j}} r_{\overline{j}\overline{\ell }}\over |\partial r|_r^2}\Big )\\&-{2\mathrm{Re}\,r^k \tilde{\Delta }r_k \over (n+1) } \end{aligned}$$

and

$$\begin{aligned} \tilde{E}(r)\le {|\partial r|^2 a^{k\overline{\ell }}[r] \over n (n+1)}\Big [\tilde{\Delta }r_{k\overline{\ell }}+ a^{i\overline{q}} r^p r^{\overline{j}} r_{i\overline{j}k} r_{p\overline{q}\overline{\ell }}+ 2a^{i\overline{q}}[r] \, {r_{i k} r_{\overline{q}\overline{\ell }} \over |\partial r|^2} \Big ]-{2\mathrm{Re}\,r^k \tilde{\Delta }r_k \over (n+1) } \end{aligned}$$

Therefore, the proof of the proposition is complete. $\square$

Corollary 4.3

Let D be smoothly bounded convex domain in ${\mathbb {C}}^n$. If there is a strictly plurisubharmonic defining function $r\in C^4(\overline{D})$ such that

$$\begin{aligned}&{n-1\over n+1}+ { |\partial r|^2 a^{k\overline{\ell }}[r]\over n (n+1)}\Big (\tilde{\Delta }r_{k\overline{\ell }}-a^{i\overline{q}}[r] r^{p\overline{j}} r_{i\overline{j}k} r_{p\overline{q}\overline{\ell }} - (\tilde{\Delta }r_k) (\tilde{\Delta }r_{\overline{\ell }}) \Big )\nonumber \\&\quad -{2\mathrm{Re}\,r^k \tilde{\Delta }r_k \over (n+1) } >0 \hbox { on }\ \partial D, \end{aligned}$$

(4.11)

then D is strictly super-pseudoconvex.

Proof

If $\partial D$ is convex then for any strictly plurisubharmonic defining function $r\in C^4(\overline{D})$, we have

$$\begin{aligned} {2\over n+1}-{2\over n+1}\mathrm{Re}\,{ r^k r^i r_{ik}\over |\partial r|^2} - {a^{k\overline{\ell }}[r] r^i r_{ik} r^{\overline{j}} r_{\overline{j}\overline{\ell }}\over (n+1) |\partial r|_r^2}\ge 0 \quad \hbox {on }\ \partial D. \end{aligned}$$

(4.12)

Since

$$\begin{aligned} \tilde{E}(r)+ & {} {1\over n+1} a^{k\overline{\ell }}[r] r^i r_{ik} r^{\overline{j}} r_{\overline{j}\overline{\ell }} = { |\partial r|^2 a^{k\overline{\ell }}[r] \over n (n+1)}\nonumber \\\times & {} \Big (\tilde{\Delta }r_{k\overline{\ell }}-a^{i\overline{q}}[r] r^{p\overline{j}} r_{i\overline{j}k} r_{p\overline{q}\overline{\ell }} - (\tilde{\Delta }r_k) (\tilde{\Delta }r_{\overline{\ell }}) \Big ) \quad -{2\mathrm{Re}\,r^k \tilde{\Delta }r_k \over (n+1) } \end{aligned}$$

and $1-{2\over n+1}={n-1\over n+1}$, by (4.5), (4.11) and (4.12), we have $\det H(\rho )>0$ on $\partial D$. This implies $\rho$ is strictly plurisubharmonic on $\overline{D}$ by Lemma 2 in [20]. This proves Parts (i) and (ii) in Theorem 1.4. $\square$

5 Examples

In this section, we will provide two examples in ${\mathbb {C}}^2$ which give the proof of Part (iii) of Theorem 1.4.

For $\delta =4^{-12}$, we let

$$\begin{aligned} g(t)=:g_{\delta } (t)=: \left\{ \begin{array}{l@{\quad }l} e^{-{\delta \over \delta -t}}, &{} \hbox { if } t<\delta ,\\ 0,&{} \hbox { if } t \ge \delta .\\ \end{array}\right. \end{aligned}$$

(5.1)

Let

$$\begin{aligned} r(z)=-2\mathrm{Re}\,z_2+|z|^2-8 |z_1|^4 g(|z_1|^2),\quad z=(z_1, z_2)\in {\mathbb {C}}^2. \end{aligned}$$

(5.2)

Example 1

Let $D=\{z\in {\mathbb {C}}^2: r(z)<0\}$. Then

• (i) D is strictly convex.

• (ii) If $\rho _D$ the solution of Fefferman equation (2), then $\rho _D$ is not plurisubharmonic in D.

Proof

Since

$$\begin{aligned} {\partial |z_1|^4 g(|z_1|^2)\over \partial x_1}= & {} 4|z_1|^2 x_1 g(|z_1|^2)+|z_1|^4 g'(|z_1|^2) 2x_1,\\ {\partial |z_1|^4 g(|z_1|^2)\over \partial y_1}= & {} 4|z_1|^2 y_1 g(|z_1|^2)+|z_1|^4 g'(|z_1|^2) 2y_1,\\ {\partial ^2 \, |z_1|^4 g(|z_1|^2)\over \partial x_1^2 }= & {} 16 |z_1|^2 x_1^2 g'(|z_1|^2) +2|z_1|^4 g'(|z_1|^2)+ 4(|z_1|^2 +2 x_1^2) g(|z_1|^2) \\&+ 4|z_1|^4 g''(|z_1|^2) x_1^2,\\ {\partial |z_1|^4 g(|z_1|)\over \partial y_1^2}= & {} 16 |z_1|^2 y_1^2 g'(|z_1|^2) +2|z_1|^4 g'(|z_1|^2)+ 4(|z_1|^2 +2 y_1^2) g(|z_1|^2)\\&+ 4|z_1|^4 g''(|z_1|^2) y_1^2,\\ {\partial ^2 (|z_1|^4 g(|z_1|^2)\over \partial x_1 \partial y_1}= & {} {\partial (4|z_1|^2 x_1 g(|z_1|^2)+|z_1|^4 g'(|z_1|^2) 2x_1)\over \partial y_1}\\= & {} 8 x_1y_1 g(|z_1|^2)+16|z_1|^2 x_1 y_1 g'(|z_1|^2)+4|z_1|^4x_1 y_1 g''(|z_1|^2); \end{aligned}$$

and since

$$\begin{aligned} 20 t^2 |g'(t)|+12 t g(t)+4t^3 |g''(t)|= & {} 4t g(t) \left[ 3 +5 {t \delta \over (\delta -t)^2}+{t^2 (\delta ^2+2\delta (\delta -t))\over (\delta -t)^4}\right] \\\le & {} 4t g(t) \left[ {11\delta ^4\over (\delta -t)^4}\right] \\\le & {} 4^7 \delta \\\le & {} 4^{-5}, \end{aligned}$$

we have

$$\begin{aligned}&18|z_1|^4|g'(|z_1|^2)|+12|z_1|^2 g(|z_1|^2)+4|z_1|^6 |g''(|z_1|^2)|\le {1/4},\\&\Big |{\partial (|z_1|^4 g(|z_1|^2)\over \partial x_1^2}\Big |<1/4,\\&\Big |{\partial (|z_1|^4 g(|z_1|^2)\over \partial y_1^2}\Big |<1/4 \end{aligned}$$

and

$$\begin{aligned} \Big |{\partial (|z_1|^4 g(|z_1|^2)\over \partial x_1\partial y_1}\Big |<1/2. \end{aligned}$$

Therefore, $D^2 r(z)=2I_n+D^2 (|z_1|^4 g(|z_1|^2))$ is positive definite in ${\mathbb {R}}^4$. Therefore, D is strictly convex. Moreover, $H(r)(0)=I_2$. We claim that

$$\begin{aligned} \det H(\rho _D)(0)<0. \end{aligned}$$

Since, at $z=0$, we have

$$\begin{aligned} {\partial r\over \partial z_2}=-1,\quad r_{kj}(0)=r_{i\overline{j}k}(0)=0,\quad 1\le i,j,k\le 2 \end{aligned}$$

By (4.3). This implies ${\partial \log J(r )\over \partial z_j}(0)=0$ for all $1\le j\le 2$. By (4.6) and (4.10), we have

$$\begin{aligned} r_{1\overline{1}1\overline{1}}(0)=-32e^{-1},\quad \tilde{E}(r)(0)={|\partial r|^2 \over 6 } r_{1\overline{1} 1\overline{1}}(0)=-{32\over 6} e^{-1} \end{aligned}$$

Thus,

$$\begin{aligned} \det H(\rho _D) J(r)^{2/3} =1-{2\over 3}-{32\over 6 e}<0. \end{aligned}$$

This completes the proof of the statement in the example. $\square$

Example 2

For $n\ge 2, \alpha =21/20$ and $0<C\le (9-8\alpha )(1+\alpha )/256$, we let $r(z)=|z|^2+2\mathrm{Re}\,z_n +\alpha \mathrm{Re}\,\sum _{j=1}^n z_j^2 +C\sum _{j=1}^n |z_j|^4$ and let

$$\begin{aligned} D=\{z\in {\mathbb {C}}^n: r(z)<0\} \end{aligned}$$

Then D is super-pseudoconvex, but D is not convex.

Proof

At $z=(0,0,\ldots , 0)\in \partial D$, we have that ${\partial \over \partial x_j}, {\partial \over \partial y_j}$ and ${\partial \over \partial y_n}$ are tangent vectors to $\partial D$ for $1\le j\le n-1$. Notice that

$$\begin{aligned} {\partial ^2 r\over \partial y_n^2}=2-2\alpha =-2(\alpha -1)<0, \end{aligned}$$

one can easily see that $\partial D$ not convex at $z=0$. Thus, $\partial D$ is not convex. However,

$$\begin{aligned} H(r)=I_n+4C\hbox {Diag}(|z_1|^2,\ldots , |z_n|^2), \end{aligned}$$

where $\hbox {Diag}(|z_1|^2,\ldots , |z_n|^2)$ is a diagonal matrix with diagonal entries $|z_1|^2,\ldots , |z_n|^2$, respectively. Then

$$\begin{aligned} {\partial ^2 r \over \partial z_i \partial \overline{z}_j \partial z_k \partial \overline{z}_\ell }(z)= & {} 4C \delta _{ij} \delta _{k\ell }\delta _{ik},\quad {\partial ^3 r\over \partial z_k \partial \overline{z}_\ell \partial z_j}=4C\delta _{k\ell } \delta _{kj} \overline{z}_j,\\&{\partial ^2 r\over \partial z_i \partial z_j}=( \alpha +2C \overline{z}_j^2) \delta _{ij}. \end{aligned}$$

For each i

$$\begin{aligned} r^i={r_{\overline{i}}\over 1+4C|z_i|^2},\quad |\partial r|_r^2=r^i r_i=\sum _{i=1}^n {|r_i|^2\over 1+4C|z_i|^2} \end{aligned}$$

and, on $\partial D$, we have

$$\begin{aligned} \tilde{\Delta }=\sum _{i, j=1}^n \left( {\delta _{ij}\over 1+4C|z_j|^2}-{r_{\overline{i}} r_j\over (1+4C|z_i|^2)(1+4C|z_j|^2) |\partial r|_r^2}\right) {\partial ^2 \over \partial z_i \partial \overline{z}_j}. \end{aligned}$$

Notice that if $z\in D$, then

$$\begin{aligned} 2x_n+(1+\alpha )\sum _{j=1}^n x_j^2+(1-\alpha )\sum _{j=1}^n y_j^2+C\sum (x_j^2+y_j^2)^2<0. \end{aligned}$$

This implies that

$$\begin{aligned} 2x_n+(1+\alpha ) x_n^2<0\iff -{2\over 1+\alpha } <x_n<0. \end{aligned}$$

(5.3)

Thus

$$\begin{aligned} 2x_n+(1+\alpha )x_n^2>{-1\over 1+\alpha } \quad \hbox {and } \ C|z_k|^4-(\alpha -1)|z_k|^2<{1\over 1+\alpha }. \end{aligned}$$

(5.4)

We claim that

$$\begin{aligned} 4C|z_k|^2 \le 1/8 \quad \hbox {if }\ 0<C\le {(9-8\alpha )(1+\alpha )\over 256},\quad 1<\alpha <9/8. \end{aligned}$$

(5.5)

Otherwise, $4C|z_k|^2\ge 1/8$. Therefore, $C|z_k|^4-(\alpha -1)|z_k|^2<{1\over 1+\alpha }$ implies

$$\begin{aligned} |z_k|^2<{8 \over (1+\alpha )(9-8\alpha )}. \end{aligned}$$

This is a contradiction with $4C|z_k|\ge 1/8$. Therefore, the claim is true. Notice

$$\begin{aligned} a^{k\overline{\ell }}[r] r_{\overline{\ell }}=0, \quad \hbox { for all } 1\le k\le n, \end{aligned}$$

we have

$$\begin{aligned} \left( r^{k\overline{\ell }}-{r^k r^{\overline{\ell }}\over |\partial r|^2}\right) \left( r^i r_{ik} r^{\overline{j}} r_{\overline{j}\overline{\ell }}\right)= & {} \left( r^{k\overline{\ell }}-{r^k r^{\overline{\ell }}\over |\partial r|^2}\right) r^k r^{\overline{\ell }} (\alpha +2C\overline{z}_k^2)(\alpha +2Cz_{\ell }^2)\\= & {} \left( r^{k\overline{\ell }}-{r^k r^{\overline{\ell }}\over |\partial r|^2}\right) r_k r_{\overline{\ell }} \left( \alpha +2C{\overline{z}_k^2-2\alpha |z_k|^2\over 1+4C|z_k|^2}\right) \\&\times \left( \alpha +2C{z_{\ell }^2-2\alpha |z_\ell |^2\over 1+4C|z_\ell |^2}\right) \\= & {} \left( r^{k\overline{\ell }}-{r^k r^{\overline{\ell }}\over |\partial r|^2}\right) r_k r_{\overline{\ell }}\alpha ^2 \\&+\,4C\alpha \mathrm{Re}\,\left( r^{k\overline{\ell }}-{r^k r^{\overline{\ell }}\over |\partial r|^2}\right) r_k r_{\overline{\ell }} {\overline{z}_k^2-2\alpha |z_k|^2\over 1+4C|z_k|^2}) \\&+\,4C^2 \left( r^{k\overline{\ell }}\!-\!{r^k r^{\overline{\ell }}\over |\partial r|^2}\right) r_k r_{\overline{\ell }} {(\overline{z}_k^2\!-\!2\alpha |z_k|^2) (z_{\ell }^2\!-\!2\alpha |z_{\ell }|^2)\over (1\!+\!4C|z_k|^2)(1\!+\!4C|z_{\ell }|^2)}\\\le & {} {4C^2 (2\alpha +1)^2 |z_k|^4\over (1+4C|z_k|^2)^2} r^{k\overline{k}} |r_k|^2\\\le & {} {(2\alpha +1)^2 \over 256}|\partial r|^2,\\ \left( r^{k\overline{\ell }} -{r^k r^{\overline{\ell }}\over |\partial r|_r^2}\right) \tilde{\Delta }r_{k\overline{\ell }}= & {} 4C\left( r^{k\overline{k}} -{r^k r^{\overline{k}}\over |\partial r|_r^2}\right) \tilde{\Delta }|z_k|^2=4C\left( r^{k\overline{k}} -{r^k r^{\overline{k}}\over |\partial r|_r^2}\right) ^2,\\ \tilde{\Delta }r_k= & {} 4C\left( r^{k\overline{k}}-{r^k r^{\overline{k}}\over |\partial r|^2}\right) \overline{z}_k \end{aligned}$$

and

$$\begin{aligned} r_{\overline{k}}=(1+2C|z_k|^2) z_k +2\alpha \overline{z}_k. \end{aligned}$$

Thus by (5.3)

$$\begin{aligned} \mathrm{Re}\,r^k \tilde{\Delta }r_k= & {} 4C\mathrm{Re}\,\left( r^{k\overline{k}}-{r^k r^{\overline{k}}\over |\partial r|^2}\right) r^k \overline{z}_k\\\le & {} 4C\left( r^{k\overline{k}}-{r^k r^{\overline{k}}\over |\partial r|^2}\right) r^{k\overline{k}} (1+2\alpha +2C|z_k|^2)|z_k|^2\\= & {} {4C|z_k|^2 (1+2\alpha +2C|z_k|^2) \over (1+4C|z_k|^2)^2} \le {2\alpha +1\over 8},\\ \left( r^{k\overline{\ell }}-{r^k r^{\overline{\ell }}\over |\partial r|^2}\right) \tilde{\Delta }r_k \tilde{\Delta }r_{\overline{\ell }}= & {} 16 C^2 \left( r^{k\overline{\ell }}-{r^k r^{\overline{\ell }}\over |\partial r|^2}\right) \overline{z}_k z_{\ell } \left( r^{k\overline{k}}-{r^k r^{\overline{k}}\over |\partial r|^2}\right) \left( r^{\ell \overline{\ell }}-{r^\ell r^{\overline{\ell }}\over |\partial r|^2}\right) \\\le & {} 16 C^2 r^{k\overline{\ell }}\overline{z}_k z_{\ell } \left( r^{k\overline{k}}-{r^k r^{\overline{k}}\over |\partial r|^2}\right) \left( r^{\ell \overline{\ell }}-{r^\ell r^{\overline{\ell }}\over |\partial r|^2}\right) \\\le & {} 4C {4C|z_k|^2 \over 1+4C|z_k|^2}\left( r^{k\overline{k}}-{r^k r^{\overline{k}}\over |\partial r|^2}\right) ^2 \end{aligned}$$

and

$$\begin{aligned} \left( r^{k\overline{\ell }}\!-\!{r^k r^{\overline{\ell }}\over |\partial r|^2}\right) \left( r^{i\overline{q}}\!-\!{r^i r^{\overline{q}}\over |\partial r|^2}\right) r^{p\overline{j}} r_{i\overline{j}k} r_{p\overline{q}\overline{\ell }} \!= & {} \!16C^2 \left( r^{k\overline{\ell }}\!-\!{r^k r^{\overline{\ell }}\over |\partial r|^2}\!\right) ^2 r^{\ell \overline{k}} \overline{z}_k \delta _{ik} \delta _{jk} z_{\ell } \delta _{p\ell } \delta _{q\ell }\\= & {} 16C^2 |z_k|^2 \left( r^{k\overline{k}}-{r^k r^{\overline{k}}\over |\partial r|^2}\right) ^2 r^{k \overline{k}} \\= & {} 4C {4C |z_k|^2 \over (1+4C|z_k|^2)} \left( r^{k\overline{k}}-{r^{kk} |r_k|^2\over |\partial r|^2}\right) ^2. \end{aligned}$$

Therefore, since (5.1), we have

$$\begin{aligned}&\left( r^{k\overline{\ell }} -{r^k r^{\overline{\ell }}\over |\partial r|_r^2}\right) \tilde{\Delta }r_{k\overline{\ell }} -\left( r^{k\overline{\ell }}-{r^k r^{\overline{\ell }}\over |\partial r|^2}\right) \left( r^{i\overline{q}}-{r^i r^{\overline{q}}\over |\partial r|^2}\right) r^{p\overline{j}} r_{i\overline{j}k} r_{p\overline{q}\overline{\ell }} \\&\qquad - 4C {4C|z_k|^2 \over 1+4C|z_k|^2}\left( r^{k\overline{k}}-{r^k r^{\overline{k}}\over |\partial r|^2}\right) ^2\\&\quad =4C\left( r^{k\overline{k}} -{r^k r^{\overline{k}}\over |\partial r|_r^2}\right) ^2-4C {4C |z_k|^2 \over (1+4C|z_k|^2)} \left( r^{k\overline{k}}-{r^{kk} |r_k|^2\over |\partial r|^2}\right) ^2\\&\qquad - 4C {4C|z_k|^2 \over 1+4C|z_k|^2}\left( r^{k\overline{k}}-{r^k r^{\overline{k}}\over |\partial r|^2}\right) ^2\\&\quad = 4C(1-2{4C|z_k|^2\over 1+4C|z_k|^2})\left( r^{k\overline{k}}-{r^k r^{\overline{k}} \over |\partial r|^2}\right) ^2\\&\quad \ge 0. \end{aligned}$$

Therefore,

$$\begin{aligned} \tilde{E}(r)\ge - {2\mathrm{Re}\,r^k \tilde{\Delta }r_k\over n+1}-a^{k\overline{\ell }}[r] {r^i r_{ik} r^{\overline{j}} r_{\overline{j}\overline{\ell }}\over (n+1)|\partial r|^2} \ge -{(1+2\alpha )\over 4(n+1)}-{(2\alpha +1)^2\over 256(n+1)} \end{aligned}$$

and

$$\begin{aligned} 1-{2\over n+1} \mathrm{Re}\,{r^i r^k r_{ik} \over |\partial r|^2} +\tilde{E}(r)\ge & {} 1-{2\over n+1}\mathrm{Re}\,{r^i r^i (\alpha +2C\overline{z}_i^2 )\over |\partial r|^2} -{(1+2\alpha )\over 4(n+1)}\\&-{(2\alpha +1)^2\over 256(n+1)} \\= & {} 1-{2 \over n+1}\mathrm{Re}\,{ r^ {i\overline{i}} r_i^2 r^{i\overline{i}} (\alpha +2C\overline{z}_i^2 )\over |\partial r|^2} -{(1+2\alpha )\over 4(n+1)}\\&-{(2\alpha +1)^2\over 256 (n+1)}\\\ge & {} 1-{2\alpha \over n+1} -{(1+2\alpha )\over 4(n+1)}-{(2\alpha +1)^2 \over 256(n+1)}\\> & {} 1-{10\alpha +1\over 4(n+1)}-{10 \over 256(n+1)}\\\ge & {} 1-{23\over 24}-{1\over 25}\\> & {} 0 \end{aligned}$$

if $n\ge 2$ and $\alpha \le 21/20$. Therefore, by (1.13) in Definition 1.1 and (4.5) and (4.6), D is strictly super-pseudoconvex and the proof is complete. $\square$