Rawnsley’s $$\varepsilon $$ -Function on a Class of Bounded Hartogs Domains and its Applications

In this paper, by using the hypergeometric functions, we obtain the formula for the Rawnsley’s \(\varepsilon \)-function of the Kähler manifold \((H^n_{\{k_i\},\gamma },g_{\mu ,\nu })\) with \(\mu \in ({\mathbb {R}}^+)^l\) and \(\nu \in ({\mathbb {R}}^+)^{n-k}\), where \(H^n_{\{k_i\},\gamma }\) is a class of bounded Hartogs domains defined by

$$\begin{aligned} H^n_{\{k_i\},\gamma }:=\big \{z\in {\mathbb {C}}^n:\max _{1\le i\le l}\Vert {\widetilde{z}}_i\Vert<|z_{k+1}|^\gamma<\ldots<|z_n|^\gamma <1\big \} \end{aligned}$$

and \(g_{\mu ,\nu }\) is a Kähler metric associated with the Kähler potential \(-\sum _{i=1}^l\mu _i\ln (|z_{k+1}|^{2\gamma }-\Vert {\widetilde{z}}_i\Vert ^2)-\sum _{j=k+1}^n\nu _j\ln (|z_{j+1}|^2-|z_j|^2)\). As applications of the main result, we obtain the existence of balanced metrics on \(H^n_{\{k_i\},\gamma }\) and prove that \(H^n_{\{k_i\},\gamma }\) admits a Berezin quantization.