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Canonical metrics on some domains of \(\mathbb{C}^n\). (English) Zbl 1222.32045

Actes de Séminaire de Théorie Spectrale et Géométrie. Année 2008–2009. St. Martin d’Hères: Université de Grenoble I, Institut Fourier. Séminaire de Théorie Spectrale et Géométrie 27, 143-156 (2009).
This paper is a survey on the study of the existence and uniqueness of preferred Kähler metrics on complex manifolds \(M\).
The author recalls the main results and open questions for the most important canonical metrics (Kähler-Einstein metrics, Kähler metrics with constant scalar curvature, extremal metrics, Kähler-Ricci solitons), in the compact and the non-compact case.
A particular class of complex domains \(D \subset\mathbb C^n\), the so-called Hartogs domains, is considered.
Let \(x_0\in\mathbb R^+\cup\{+\infty\}\) and let \(F : [0,x_0) \to (0, +\infty)\) be a decreasing continuous function, smooth on \((0, x_0)\). The Hartogs domain \(D_F\subset\mathbb C^n\) associated to the function \(F\) is defined by
\[ D_F = \Big\{(z_0, z_1,\dots, z_{n-1})\in \mathbb C^n \;\Big|\; |z_0|^2 < x_0,\, |z_1|^2 +\dots + |z_{n-1}|^2 < F\big(|z_0|^2\big)\Big\}. \]
The interest for Hartogs domains comes from the fact that they yield a lot of examples of noncompact Kähler manifolds enjoying a priori different geometric properties which can be controlled by integro-differential conditions on \(F\) itself. In the class of Hartogs domains, thought as domains endowed with the metric \(g_F\) associated to the natural Kähler form \(\omega_F\), there are no examples of canonical metrics but the complex hyperbolic space. More precisely, the following theorems hold true.
Theorem 1. Let \((D_F,g_F)\) be a Hartogs domain in \(\mathbb C^n\). Assume that one of the following assumptions is satisfied:
{(1)} \(g_F\) is a Kähler-Einstein metric;
{(2)} \(g_F\) is a Kähler metric with constant scalar curvature;
{(3)} \(g_F\) is an extremal metric;
{(4)} there exists a holomorphic field \(X\) such that \((g_F,X)\) is a Kähler-Ricci soliton.
Then \((D_F,g_F)\) is holomorphically isometric to an open domain of the hyperbolic space \(\mathbb C\mathbb H^n\).
Theorem 2. Let \((D^{}_F,g^{}_F)\) be a bounded Hartogs domain in \(\mathbb C^n\), and let \(g^{}_B\) denote the Bergman metric of \(D^{}_F\). If \(g^{}_B=\lambda g^{}_F\) for some \(\lambda > 0\), then \((D^{}_F,g^{}_F)\) is holomorphically isometric to an open domain of the hyperbolic space \(\mathbb C\mathbb H^n\).
The results presented by the author are proved in [A. J. Di Scala, A. Loi and F. Zuddas, Int. J. Math. 20, No. 2, 139–148 (2009; Zbl 1178.32017)] and [A. Loi and F. Zuddas, Osaka J. Math. 47, No. 2, 507–521 (2010; Zbl 1484.53105)].
For the entire collection see [Zbl 1206.35007].

MSC:

32Q15 Kähler manifolds
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