In this paper, we use the the well-known Calabi ansatz, further generalized by Hwang–Singer and Apostolov–Calderbank–Gauduchon, to study the existence of constant scalar curvature Kähler (cscK for short) metrics and balanced metrics on certain holomorphic polyball bundles M which are locally expressed as \(M=\Big \{(z_1,\ldots ,z_{{\mathcal {N}}}, u_1,\ldots ,u_{\ell })\in \prod _{j=1}^{{\mathcal {N}}}\Omega _j \times \prod _{i=1}^{\ell }{\mathbb {C}}^{r_i}: e^{\sum _{j=1}^{{\mathcal {N}}}\lambda _{ij}\phi _j(z_j)}\Vert u_i\Vert ^2<1,1\le i\le \ell \Big \}.\) Let \(g_F\) be the natural Kähler metrics on M associated with the Kähler forms locally expressed as \(\omega _F=\sqrt{-1}\partial \overline{\partial } \Big (\sum _{j=1}^{{\mathcal {N}}}\nu _j\phi _j(z_j) +\sum _{i=1}^{\ell }f_i(\sum _{j=1}^{{\mathcal {N}}} \lambda _{ij}\phi _j(z_j)+\log \Vert u_i\Vert ^2)\Big )\). Firstly, we obtain sufficient and necessary conditions for \(g_F\) to be cscK metrics. Secondly, using this result, we obtain necessary and sufficient conditions for \(mg_F\) to be balanced metrics for all sufficiently large positive integer numbers m. Finally, we obtain complete cscK metrics and balanced metrics on the polyball bundles over simply connected Riemann surfaces, or products of simply connected Riemannian surfaces and unit balls. The main contribution of this paper is the explicit construction of complete, non-compact cscK metrics and balanced metrics.

在本文中，我们使用著名的 Calabi ansatz（由 Hwang-Singer 和 Apostolov-Calderbank-Gauduchon 进一步推广）来研究某些全纯多球丛上常标量曲率 Kähler（简称 cscK）度量和平衡度量的存在性M局部表示为\(M=\Big \{(z_1,\ldots ,z_{{\mathcal {N}}}, u_1,\ldots ,u_{\ell })\in \prod _{j= 1}^{{\mathcal {N}}}\Omega _j \times \prod _{i=1}^{\ell }{\mathbb {C}}^{r_i}: e^{\sum _{j =1}^{{\mathcal {N}}}\lambda _{ij}\phi _j(z_j)}\Vert u_i\Vert ^2<1,1\le i\le \ell \Big \}.\ )设\(g_F\)为M上的自然凯勒度量，与局部表示为\(\omega _F=\sqrt{-1}\partial \overline{\partial } \Big (\sum _{j =1}^{{\mathcal {N}}}\nu _j\phi _j(z_j) +\sum _{i=1}^{\ell }f_i(\sum _{j=1}^{{\ mathcal {N}}} \lambda _{ij}\phi _j(z_j)+\log \Vert u_i\Vert ^2)\Big )\)。首先，我们获得\(g_F\)作为cscK度量的充分必要条件。其次，利用这个结果，我们获得了\(mg_F\)成为所有足够大的正整数m的平衡度量的充分必要条件。最后，我们获得简单连接黎曼曲面上的多球丛的完整 cscK 度量和平衡度量，或者简单连接黎曼曲面和单位球的乘积。本文的主要贡献是完整的、非紧凑的 cscK 度量和平衡度量的显式构建。