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).
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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
We say that the metric g is also Einstein if its Ricci curvature
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:
has a unique strictly plurisubharmonic solution \(u\in C^\infty (D)\). Moreover, the Kähler metric
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
where
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
then (1.3) is the same as (1.5). Moreover, one can prove (see [14] and references therein) that
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]:
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
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]:
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
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
with
and
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
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
It is easy to verify that
In fact, since
Remark 1
When H(r) is not positive definite on \(\partial D\), we can replace r by
Then r[a] is positive definite with a large a and
From now on, we will always assume that \(r(z)\in C^\infty (\overline{D})\) is a negative defining function for D such that
is strictly plurisubharmonic in D. It is known from [5, 14–16] that the following identity holds:
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
with
Then
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
by choosing \(a\ge 0\) so that r[a] is strictly plurisubharmonic. Therefore, we can write
with \(B_0\in C^\infty (\overline{D})\). Since
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
Since
Notice that \(\exp ((n+1)\ell (\rho _1))=\exp ((n+1)B) J(r) \exp ((n+1)\ell (r))\), we have
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
on \(\partial D\), where \(B(z)=B[r](z)\) is given by (2.10).
Proof
Let
Theorem 2.1 implies that \(\rho (z)=\rho _1(z)+O(r(z)^3)\). A simple calculation shows that
By (2.13) (\(B=(-r) B_0\)), one can easily see that
and
For any \(z\in \partial D\), by (2.20), one has
This proves Proposition 2.2. \(\square \)
Let \(u^{D_j}\) be the potential functions for the Kähler–Einstein metric of \(D_j\) and let
Proposition 2.3
Let \(\phi : D_1\rightarrow D_2\) be a smooth biholomorphic mapping. Then
In particular, if \(\det \phi '(z)\) is constant c then
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:
Then
and
In particular, when \(\det \phi '(z)=c\), one has
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
Moreover, if \(|\det \phi '(z)|^2\) is a constant on \(B(0,\delta _0)\) for some \(\delta _0>0\)
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
where
According to Theorem 2.1, one has
Let \(\rho =\rho ^D\) be the solution of (1.5) such that \(\ell (\rho )\) is strictly plurisubharmonic in D. Then
By Proposition 2.2 and
where
Thus for \(z_0\in \partial D\), one has
Let
and
Then it is easy to see that
Therefore, by (2.21) and Lemma 3.1 in [14], at \(z=z_0\in \partial D\), one has
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
and
we have
Thus,
where
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
on \(\partial D\). For ant \(z_0\in \partial D\), by rotation, we may assume that \( r_n(z_0)>0\). Thus
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
and
Proof
The following two identities will be used later.
and
By (4.3) and (4.2), for \(z\in \partial D\), one has
Then for \(z\in \partial D\), we have
and
Moreover,
Therefore,
Therefore,
and
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
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
Since
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
Let
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
and since
we have
and
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
Since, at \(z=0\), we have
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
Thus,
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
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
one can easily see that \(\partial D\) not convex at \(z=0\). Thus, \(\partial D\) is not convex. However,
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
For each i
and, on \(\partial D\), we have
Notice that if \(z\in D\), then
This implies that
Thus
We claim that
Otherwise, \(4C|z_k|^2\ge 1/8\). Therefore, \( C|z_k|^4-(\alpha -1)|z_k|^2<{1\over 1+\alpha } \) implies
This is a contradiction with \(4C|z_k|\ge 1/8\). Therefore, the claim is true. Notice
we have
and
Thus by (5.3)
and
Therefore, since (5.1), we have
Therefore,
and
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. \(\square \)
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Acknowledgments
The author would like to thank Professor Fefferman and Xiaodong Wang for some useful conversations he has had with them. The author is greatly appreciative and thank Professor R. Graham who pointed out that there is a mistake in computation at (3.21): \(r_{nn}^b=r_{nn}+b r_n^2\) at \(z_0\) in the the previous version of the paper (it should be \(r_{nn}^b=r_{nn}+2b r_n^2\)), as well as his valuable suggestions for the current revision.
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Communicated by Neil Trudinger.
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Li, SY. Plurisubharmonicity for the solution of the Fefferman equation and applications. Bull. Math. Sci. 6, 287–309 (2016). https://doi.org/10.1007/s13373-015-0078-6
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DOI: https://doi.org/10.1007/s13373-015-0078-6