An estimate for the Bergman distance on pseudoconvex domains. (English) Zbl 0828.32002

Let \(D\) be a bounded pseudoconvex domain in \(\mathbb{C}^n\). Let \(\delta_D (z)\) denote the euclidean distance from \(z\) to the boundary of \(D\), and let \(\text{dist}_D (z,w)\) denote the Bergman distance between \(z\) and \(w\) with respect to \(D\). The authors’ main theorem states that if there are a \({\mathcal C}^\infty\) plurisubharmonic exhaustion function \(\rho : D \to [- 1,0)\) and positive constants \(c_1, c_2\) such that \[ {1 \over c_1} \delta^{c_2}_D < - \rho < c_1 \delta_D^{1/c_2}, \] then for any given \(z_0 \in D\) there are positive constants \(c_3\), \(c_4\) such that \[ \text{dist}_D (z_0, z) > c_3 \log \biggl |\log \bigl( c_4 \delta_D (z) \bigr) \biggr |- 1 \] for all \(z \in D\); in particular, the Bergman metric of \(D\) is complete. It should be noted that no smoothness is assumed for the boundary of \(D\), but the hypothesis will be satisfied if the boundary has class \({\mathcal C}^2\). The proof employs an \({\mathcal L}^2\) estimate for the \(\overline \partial\)-equation, with weight function \(e^{- \varphi}\), where \(\varphi\) is \({\mathcal C}^\infty\) but not necessarily plurisubharmonic.


32T99 Pseudoconvex domains
32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)
32Q15 Kähler manifolds
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