Localization of invariant metrics (Q699773)

The author generalizes certain localization results for invariant functions from the case of bounded domains to the general case. The main results are the following two theorems. (1) Let \(D\) be an open set in \(\mathbb C^n\) and let \(z_0\in\partial D\) be a plurisubharmonic antipeak point. Let \(U\) be a neighborhood of \(z_0\) such that the set \(G=D\cap U\) satisfies the following condition (P): \(\lim_{z\to z_0} \inf\{g_G(z,w): w\in G\setminus W\}=0\), for any open neighborhood \(W\) of \(z_0\) with \(G\setminus W\neq\varnothing\), where \(g_G\) is the pluricomplex Green function. Then \(D\) satisfies (P) and there exists a neighborhood \(V\subset U\) of \(z_0\) such that \(\lim_{z\to z_0} \inf\{g_D(z,w)-g_{D\cap V}(z,w): w\in D\cap V\setminus\{z\}\}=0\). In the case where \(D\) is bounded the result was proved by \textit{D. Coman} [Ark. Mat. 36, 341-353 (1998)]. (2) Let \(D\subset\mathbb C^n\) be an open pseudoconvex set, let \(z_0\in\partial D\) be a local peak point, and let \(U\) be a neighborhood of \(z_0\). Then \[ \lim_{z\to z_0}\frac{K_D(z)}{K_{D\cap U}(z)}= \lim_{z\to z_0}\frac{I_D^j(z,X)}{I_{D\cap U}^j(z,X)}= \lim_{z\to z_0}\frac{B_D(z,X)}{B_{D\cap U}(z,X)}= \lim_{z\to z_0}\frac{2-R_D(z,X)}{2-R_{D\cap U}(z,X)}=1 \] locally uniformly in \(\mathbb C^n\setminus\{0\}\), where: \(K_D\) denotes the Bergman kernel, \[ I_D^j(z,X):=\sup\{|\frac 1{j!}f^{(j)}(z)(X)|^2: f\in L^2_h(D),\;\|f\|_{L^2(D)} \leq 1,\;\text{ord}_zf\geq j\}, \] \(B_D(z,X):=(\frac{I_D^1(z,X)}{K_D(z)})^{1/2}\) is the Bergman metric, and \(R_D(z,X):=2-4K_D(z)\frac{I_D^2(z,X)}{(I_D^1(z,X))^2}\) stands for the sectional curvature. The case where \(D\) is additionally bounded was proved by \textit{K. T. Kim} and \textit{J. Yu} [Pac. J. Math. 176, 141-163 (1996; Zbl 0886.32020)].