May 31, 2026
In this note, we show that if the Bergman metric of a pseudoconvex domain in \(\mathbb{C}^n\)(\(n\geq 3\)) has constant scalar curvature, then every strongly pseudoconvex boundary point of the domain is spherical.
Let \(\Omega\subset\mathbb{C}^n\) (\(n\geq 1\)) be a possibly unbounded pseudoconvex domain containing a smooth strongly pseudoconvex boundary point \(p\in\partial\Omega\). Write \(\{\phi_j\}\) for an orthonormal basis of the Bergman space \(A^2(\Omega)\), the subspace of \(L^2(\Omega)\) consisting of \(L^2\)-integrable holomorphic functions on \(\Omega\). Denote by \(K_{\Omega}(z, z)=\sum_{j}|\phi_j(z)|^2\) the Bergman kernel function on \(\Omega\), and let \[g_{\Omega}=\sum_{i,j}g_{i\overline{j}}\,dz_i\otimes d\overline{z}_j, \;\;g_{i\overline{j}}=\frac{\partial^2}{\partial z_i\,\partial \overline{z}_j}\log K_{\Omega}(z,z)\] be the Bergman metric of \(\Omega\). The Bergman metric is well defined on a maximal open subset \(\Omega^*\subset \Omega\) that contains a one‑sided neighborhood of \(p\) (see [1]).
Write \(G_{\Omega}:=(g_{i\overline{j}})_{n\times n}\) and denote \(J_{\Omega}=\frac{\det G_{\Omega}}{K_{\Omega}},\) called the Bergman canonical invariant function of \(\Omega\). For the Bergman metric with metric tensor \(g_{i\bar{j}}\), the Ricci tensor is given by \(\mathrm{Ric}_{i\bar{j}} = -\,\frac{\partial^2}{\partial z^i \,\partial \bar{z}^j} \log \det\bigl(g_{k\bar{\ell}}\bigr).\) The scalar curvature is the trace of the Ricci tensor with respect to the metric: \(S_{\Omega}=g^{i\bar{j}} \,\mathrm{Ric}_{i\bar{j}}= -\,g^{i\bar{j}} \,\frac{\partial^2}{\partial z^i \,\partial \bar{z}^j} \log \det\bigl(g_{k\bar{\ell}}\bigr).\) It is known that near a strongly pseudoconvex boundary point [2], the scalar curvature \(S_{\Omega}\) and the invariant \(J_\Omega\) approach the limits \(-n\) and the constant \(c_n=\frac{(n+1)^n\pi^n}{n!}\), respectively. Moreover, the Bergman metric is asymptotically Kähler–Einstein with Ricci constant \(-1\) in the sense that \(\mathrm{Ric}_{i\bar{j}}+g_{i\bar{j}}\rightarrow 0\) as \(z\rightarrow p\).
Starting from the identity \(\log J_\Omega = \log \det G_\Omega - \log K_\Omega,\) and applying \(\partial_{z_k} \partial_{\overline{z_j}}\), then contracting with \(g^{k\bar{j}}\), we obtain that \(S_{\Omega}\) is constant if and only if \(\log J_\Omega\) is harmonic with respect to the Bergman metric, which was first derived by Sha in ([3]): \[\Delta_{g_\Omega} \log J_\Omega (z) = \sum_{j,k} g^{k\bar{j}} \frac{\partial^2}{\partial z^k \,\partial \overline{z^j}} \log J_\Omega=0 \quad \text{on } \Omega^*.\]
When the Bergman metric \(g_\Omega\) has constant scalar curvature on \(\Omega^*\), we say that \(\Omega\) admits a constant scalar curvature (csc) Bergman metric. Since this condition can be expressed as a real-analytic equation on \(K(z,z)\) in \(\Omega\), it follows from the uniqueness of real analytic functions that \(\Omega\) has a csc Bergman metric if and only if \(S_\Omega\) is constant on some nonempty open subset of \(\Omega^*\).
In this note, we present a proof of the following fact:
Theorem 1. Let \(\Omega \subset {\mathbb{C}}^n\) with \(n\ge 3\) be a pseudoconvex domain with \(p\in \partial\Omega\) a smooth strongly pseudoconvex boundary point of \(\Omega\). If \(\Omega\) has a csc Bergman metric then \(\partial\Omega\) is locally spherical near \(p\), namely, a small open piece of \(\partial\Omega\) is CR-diffeomorphic to an open piece of the boundary of the unit ball \(\mathbb{B}^n\).
The proof is similar to that of the corresponding known result (see, e.g., [4], [5]), where the csc condition is replaced by the slightly stronger Kähler–Einstein condition. A new observation here is the use of a recently obtained formula of Martin for the expansion of \(J_\Omega\) [6], in addition to Christoffers’s formula for the Bergman expansion [7]. We also use several asymptotic expansion formulas of Engliš [8] to simplify the computation.
Proof. Let \(G\subset\Omega\) be a small smoothly bounded strongly pseudoconvex domain with \(U\cap G=U\cap \Omega\) for some neighborhood \(U\) of the strongly pseudoconvex boundary point \(p\) in \(\mathbb{C}^n\). By localization of Bergman kernels (see [9]), we have \[\label{localization} K_\Omega(z, z)=K_G(z, z)+\varphi(z),\tag{1}\] where \(\varphi(z)\in C^\infty(U\cap\overline{G})\). Let \(\rho\in C^\infty(\overline{G})\) be a positively signed Fefferman defining function for \(G\), namely, \(J_{MA}(\rho)=1+O(\rho^{n+1})\) with the Monge-Ampere operator \(J_{MA}(\rho)\) defined in the following (4 ). With respect to such a \(\rho\), then we have the Fefferman expansion of \(K_G(z, z)\) as follows: \[\label{expansion} K_G(z, z)=\frac{\phi}{\rho^{n+1}}+\psi\log\rho\tag{2}\] with \(\phi, \psi\in C^\infty(\overline{G})\) and \(\phi=\frac{n!}{\pi^n}+O(\rho^2)\). It follows from (1 ) and (2 ) that \[\label{AP} K_{\Omega}(z, z)=\frac{\phi+\psi\rho^{n+1}\log\rho+\varphi\rho^{n+1}}{\rho^{n+1}}=\frac{\Phi}{\rho^{n+1}},\tag{3}\] where \(\Phi=\phi+\psi\rho^{n+1}\log\rho+\varphi\rho^{n+1}\in C^n(U\cap\overline{G})\). Moreover, \(\Phi=\phi+o(\rho^n)\) on \(U\cap\overline{G}\). Let \(a, b\) be smooth functions on \(\overline{G}\) such that \[\phi=\frac{n!}{\pi^n}(1+a\rho^2+b\rho^3+O(\rho^4)).\] Here, \(a\) is uniquely determined up to \(O(\rho^3)\) near \(p\). Then by a result of Christoffers [7], \(a(p)=0\) if and only if \(p\) is a CR umbilic point of \(\partial G\) in the sense of Chern-Moser [10]. Write \(u_{\Omega}=(\frac{\pi^n}{n!}K_{\Omega}(z, z))^{\frac{-1}{n+1}}, z\in\Omega^\ast\) and \(u_G=(\frac{\pi^n}{n!}K_{G}(z, z))^{\frac{-1}{n+1}}, z\in G\). For any real-valued \(C^2\)-smooth function \(u\), define \[\label{MA} J_{MA}(u)=(-1)^n\det\left[\begin{array}{cc} u & u_{\overline{\beta}} \\ u_{\alpha}& u_{\alpha\overline{\beta}} \end{array}\right]\tag{4}\] called the Fefferman-Monge-Ampere operator. By a result of Martin [6], we have \[J_{MA}(u_{G})=1-3\frac{n-1}{n+1}a\rho^2+o(\rho^2)~\text{on} ~U\cap \overline{G}.\] On the other hand, applying the following Fefferman formula [6] \[\label{Fefferman32formula} J_{G}=\frac{(n+1)^n\pi^n}{n!} J_{MA}(u_{G}):=c_n J_{MA}(u_{G}) \;\;wherec_n=\frac{(n+1)^n\pi^n}{n!}\tag{5}\] we have \[J_G(z)=c_n-3c_n\frac{n-1}{n+1}a\rho^2+o(\rho^2).\]
Again by the Fefferman formula, we verify that \(J_\Omega=J_G+O(\rho^3)\) near \(p\). Thus \[J_{\Omega}(z)=c_n-3c_n\frac{n-1}{n+1}a\rho^2+o(\rho^2)~\text{near }p.\]
Since \(J_\Omega(z)\) is independent of the chosen defining function and two defining functions differ by a positive smooth function in a neighborhood of \(p\), we may assume, without loss of generality, in the following computation that \(\rho\) is strongly plurisubharmonic near \(p\) to prove \(a(p) = 0\). By (3 ), we have the following expansion of the Bergman canonical invariant function from [8] (Engliš stated this result for strongly pseudoconvex domains, with (3 ), however, his proof carries over without any change to our local setting):
\[\begin{align} J_{\Omega}(z)&\sim\sum_{j=0}^\infty(\rho^{n+1}\log\rho)^j\eta_j, \eta_j\in C^\infty(U\cap\overline{G})\\ & \sim\eta_0+\eta_1\rho^{n+1}\log\rho+\cdots. \end{align}\] Here, the sum is in the asymptotic sense, that is, for any \(k\in\mathbb{N}\), the difference \[J_\Omega(z)-\sum_{j=0}^{k-1}(\rho^{n+1}\log\rho)^j\eta_j\in C^{k(n+1)-1}(U\cap\overline{G})\] and vanishes on \(U\cap\partial G\) with all its partial derivatives of orders \(\leq k(n+1)-1\). Then we can write \[\log( J_{\Omega}(z))=\log(c_n)-3\frac{n-1}{n+1}a(z)\rho^2+b(z)\rho^2\] where \(b(z)\in C^\infty(U\cap G)\cap C^{1,\frac{1}{2}}(U\cap\overline{G})\) and \[\label{5-24-a1} b(z)\sim a_1(z)\rho+\rho^{-2}\sum_{j=1}^\infty \tilde{\eta}_{j}(\rho^{n+1}\log\rho)^j,\tag{6}\] with \(\tilde{\eta}_k(z)\in C^\infty(U\cap\overline{G})\) for \(k\ge 1.\)
Since \(\Delta_{g_\Omega}\log J_{\Omega}=0\), we have \[\label{5-23-a3} \sum_{i, j=1}^ng^{\overline{j}i}\frac{\partial^2}{\partial z_i\partial\overline{z}_j}[3a\frac{n-1}{n+1}\rho^2- b\rho^2]\equiv 0~\text{on} ~U\cap G.\tag{7}\]
By direct calculation, \[\label{5-23-a1} \begin{align} g^{\overline{j}i}\frac{\partial^2}{\partial z_i\partial\overline{z}_j}(a\rho^2)=&\rho^2 g^{\overline{j} i}\frac{\partial^2 a}{\partial z_i\partial\overline{z}_j}+2\rho g^{\overline{j} i}\left(\frac{\partial a}{\partial z_i}\frac{\partial\rho}{\partial \overline{z}_j}+\frac{\partial a}{\partial \overline{z}_j}\frac{\partial \rho}{\partial z_i}\right)+\\ &2a g^{\overline{j} i}\frac{\partial \rho}{\partial\overline{z}_j}\frac{\partial\rho}{\partial z_i}+2a\rho g^{\overline{j}i}\frac{\partial^2 \rho}{\partial z_i\partial\overline{z}_j}. \end{align}\tag{8}\] Since we can write \(K_{\Omega}=\rho^{-(n+1)}[\widetilde{\eta}_0+(\rho^{n+1}\log\rho)\widetilde{\eta}_1],\) where \(\widetilde{\eta}_0=\phi+\varphi\rho^{n+1}\) and \(\widetilde{\eta}_1=\psi\), from [8], we have \[\rho^{-1} g^{\overline{j} i}\in C^n(U\cap\overline{G}).\] Moreover, from [8] we have in a small neighborhood \(U_p\subset {\mathbb{C}}^n\) of \(p\) \[\label{5-23-a2} \begin{align} &\rho^{-1}g^{\overline{j} i}=\rho^{-1}[\log \rho]^{\overline{j} l}H^i_l, H^i_l\in C^n(U_p\cap\overline{G}), H^i_l|_{\partial G\cap U_p}=-\frac{1}{n+1}\delta^i_l.\\ &\frac{1}{\rho^2}[\log\rho]^{\overline{j} i}\rho_i, ~~\frac{1}{\rho^2}[\log\rho]^{\overline{j} i}\rho_{\overline{j}}\in C^\infty(\overline{G}\cap U_p),~[\log \rho]^{\overline{j} i}\rho_{\overline{j}}\rho_i=\rho [\log \rho]^{\overline{j} i}\rho_{i\overline{j}}-n\rho^2~\text{on}~U_p\cap\overline{G}.\\ &\lim_{z\rightarrow p} \frac{1}{\rho^2}[\log \rho]^{\overline{j} i}\rho_{\overline{j}}\rho_i=-1. \end{align}\tag{9}\] Replacing \(a(z)\) by \(b(z)\) from (6 ) in (8 ) and from (9 ) we have that \[\lim_{z\rightarrow p}\frac{1}{\rho^2}\sum_{i, j=1}^n g^{\overline{j}i}\frac{\partial^2}{\partial z_i\partial\overline{z}_j}(b\rho^2)=0.\] It follows from (7 ), (8 ) and (9 ) that \[0=\lim_{z\rightarrow p}\frac{1}{\rho^2}\sum_{i, j=1}^n g^{\overline{j}i}\frac{\partial^2}{\partial z_i\partial\overline{z}_j}(a\rho^2)=2a(p)(n-2).\] Since \(n\geq 3\), we have \(a(p)=0\) and thus an arbitrary given strongly pseueoconvex point \(p\) is a CR umbilic point of \(\partial\Omega\). It follows immediately from the Chern-Moser Lemma [10] that \(\partial\Omega\) is spherical near \(p\). ◻
Remark 2. For \(m>1\) an integer and \(n\geq 2\), let \[\mathcal{E}_m:=\left\{z\in\mathbb{C}^{n-1}\times \mathbb{C}:\sum_{j=1}^{n-1}|z_j|^2+|z_n|^{2m}<1\right\}.\] Then we claim that the Bergman metric of \(\mathcal{E}_m\) can not have constant scaler curvature.
Proof of the statement in Remark 2. We denote by \(S\) the scalar curvature of the Bergman metric of \(\mathcal{E}_m\). If \(\mathcal{E}_m\) has a csc Bergman metric, then \(S\equiv -n\) on \(\mathcal{E}_m\) by [2] as it has limit \(-n\) at any strongly pseudoconvex point. From [2], \[\label{scaler} S(0)=n(n+1)-4a_0\sum_{j=1}^n\frac{b_{jj}}{a_j^2}-a_0\sum_{j\neq k}\frac{b_{jk}}{a_ja_k}.\tag{10}\] Here, \(a_0=\frac{1}{vol(\mathcal{E}_m)}\), \(a_j=\frac{1}{\|z_j\|_{\mathcal{E}_m}^2}\), \(b_{jk}=\frac{1}{\|z_jz_k\|^2_{\mathcal{E}_m}}\) with \(\|\cdot\|\) the \(L^2\)-norm on \(\mathcal{E}_m\). By direct calculations, \[\label{5-28-a1} \begin{align} a_0=\frac{m\Gamma\!\left(n + \frac{1}{m}\right)}{\pi^{n} \; \Gamma\!\left(\frac{1}{m}\right)},~ a_1=\dots=a_{n-1}=\frac{m\Gamma\!\left(n +1+ \frac{1}{m}\right)}{\pi^{n} \; \Gamma\!\left(\frac{1}{m}\right)},~ a_n=\frac{m\;\Gamma\!\left(n+\frac{2}{m}\right)}{\pi^{n} \; \Gamma\!\left(\frac{2}{m}\right)} \end{align}\tag{11}\] and \[\label{5-28-a2} \begin{align} &b_{jj}=\dfrac{m}{2\pi^n}\,\dfrac{\Gamma\!\left(n+2+\frac{1}{m}\right)}{\Gamma\!\left(\frac{1}{m}\right)},~ j=1,\dots,n-1; ~b_{nn}=\dfrac{m}{\pi^n}\,\dfrac{\Gamma\!\left(n+\frac{3}{m}\right)}{\Gamma\!\left(\frac{3}{m}\right)},\\ &b_{jk}=\dfrac{m}{\pi^n}\,\dfrac{\Gamma\!\left(n+2+\frac{1}{m}\right)}{\Gamma\!\left(\frac{1}{m}\right)},~ j\neq k,\; j,k=1,\dots,n-1,\\ &b_{jn}=b_{nj}=\dfrac{m}{\pi^n}\,\dfrac{\Gamma\!\left(n+1+\frac{2}{m}\right)}{\Gamma\!\left(\frac{2}{m}\right)}, j=1,\dots,n-1 \end{align}\tag{12}\] Substituting (11 ) and (12 ) to (10 ) we have \[S(0)=2-\frac{(n-1)(n+\frac{2}{m})}{n+\frac{1}{m}} -4\,\frac{\Gamma\!\left(n+\frac{1}{m}\right)\,\; \Gamma\!\left(n+\frac{3}{m}\right)\Gamma\!\left(\frac{2}{m}\right)^2\,}{\Gamma\!\left(\frac{1}{m}\right)\,\Gamma\!\left(\frac{3}{m}\right)\, \Gamma\!\left(n+\frac{2}{m}\right)^2}.\] Set \(a=\frac{1}{m}\in(0, 1)\), \(H_n(a) = \frac{ \Gamma(n+3a)\Gamma(n+a)\Gamma(2a)^2 }{ \Gamma(3a)\Gamma(a)\Gamma(n+2a)^2 }=\prod_{j=0}^{n-1} \frac{(j+a)(j+3a)}{(j+2a)^2}\) and \(T_n(a) =\frac{3n+(4-n)a}{4(n+a)}.\) Then \[\begin{align} S(0)+n=4(T_n(a)-H_n(a)). \end{align}\] By direct calculation \(T_2(a)<H_2(a)\). Since \(H_{n+1}(a)=H_n(a) \frac{(n+a)(n+3a)}{(n+2a)^2}\) and \[T_n(a)\frac{(n+a)(n+3a)}{(n+2a)^2}-T_{n+1}(a)=\frac{an(1-a)^2}{4(n+2a)^2(n+1+a)}.\] Since \(0<a<1\) and \(n\geq 2\), the right-hand side is strictly positive, hence \[T_n(a)\frac{(n+a)(n+3a)}{(n+2a)^2}>T_{n+1}(a).\] Assume inductively that \[H_n(a)>T_n(a).\] Then \[H_{n+1}(a)=H_n(a)\frac{(n+a)(n+3a)}{(n+2a)^2}>T_n(a)\frac{(n+a)(n+3a)}{(n+2a)^2}>T_{n+1}(a).\] Thus \[H_n(a)>T_n(a)\] for every \(n\ge2\). Consequently, \[S(0)+n=4(T_n(a)-H_n(a))<0.\] Therefore, \(S(0)<-n\) and thus we get a contradiction. ◻
Remark 3. We remark that Theorem 1, together with Remark 2, allows us to restate some of the results in [9] in a slightly different form. For instance, Theorem 1.1 and Corollary 1.4 of [9] can now be stated as: A bounded real analytic pseudoconvex domain or a smoothly bounded convex domain of finite D’Angelo type admits a csc Bergman metric if and only if it is biholomorphic to the ball. Indeed, the Kähler–Einstein condition in [9] is used only to ensure sphericity at strongly pseudoconvex points; and the proof of Theorem 3.3 in [9] carries over verbatim to the csc Bergman metric setting by using the formula \(N_\Omega\) in Proposition 2.1 (iv) of [2] in place of \(\lambda_\Omega\), and by observing that the same localization results in Proposition 2.4 of [2] hold for unbounded pseudoconvex domains as in [1].
Acknowledgement. In a personal communication, M. Xiao informed us that he and his coauthors have a work in progress in which, among other things, they also obtain results closely related to Theorem 1.1. The authors thank Song-Ying Li for several helpful discussions.