Estimates of invariant metrics on pseudoconvex domains with comparable Levi form. (Q1411728)

Let \(\varOmega=\{z\in\mathbb C^n: r(z)<0\}\) be a smooth bounded pseudoconvex domain with defining function \(r\). Let \(z_0\in\partial\varOmega\) be a point of finite type \(m\) for which there exists a \(c>0\) such that \(\lambda_k(z)\geq c\sum_{j=1}^{n-1}\lambda_j(z)\) for \(z\) in a neighborhood of \(z_0\), \(k=1,\dots,n-1\), where \(\lambda_1(z),\dots,\lambda_{n-1}(z)\) are the eigenvalues of the Levi form \(\partial\overline\partial r\) of \(\partial\varOmega\) at \(z\). The main result of the paper says that there exist a neighborhood \(U\) of \(z_0\) and constants \(c, C>0\) such that \(cM(z;\cdot)\leq\delta_\varOmega(z;\cdot)\leq CM(z;\cdot)\), \(z\in U\cap\varOmega\), where \(\delta_G\) is the Carathéodory--Reiffen, Kobayashi-Royden, or Bergman metric for \(\varOmega\), and \(M\) is a metric defined as follows. Suppose that \(r'_{z_n}(z)\neq0\), \(z\in U\). Then \[ M(z;X)=(| b_1| +\dots+| b_{n-1}| )\sum_{j=2}^m\Big| \frac{C_j(z)}{r(z)}\Big| ^{1/j}+ \Big| \frac{b_n}{r(z)}\Big| ,\quad z\in U, \] \[ X=\sum_{j=1}^{n-1}b_j\Big(e_j-\frac{r'_{z_j} (z)}{r'_{z_n}(z)}e_n\Big)+b_ne_n, \] where the functions \(C_j\), \(j=2,\dots,m\), are also effectively given.