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The pluricomplex Green function with two poles of the unit ball of \(\mathbb{C}^n\). (English) Zbl 1015.32029
In this paper the author finds the formula for the pluricomplex Green function of the unit ball \(B^n\) of \(\mathbb{C}^n\) with two poles at \(p\neq q\in B^n\) and with weight one at each pole. By using a suitable automorphism of \(B^n\) there might assume without loss of generality that \(p=-q=(\beta,0,\dots,0)\), for some \(\beta\in(0,1)\). The author considers the two dimensional case \(n=2\). The general case \(n\geq 2\) follows easily from this one. For the computing of the pluricomplex Green function, the unit ball \(B^2\) is divided into three regions: \(\Gamma_p\) and \(\Gamma_q\), which are intersections of \(B^2\) with two closed complex cones with vertex at \(p\) and \(q\) respectively, and the complement of their union, \(D\). The main result of this paper is the following:
Theorem. The pluricomplex Green function of the unit ball of \(\mathbb{C}^2\) with poles at \(p=-q=(\beta,0)\) is given by \[ g_2(z,p,q)=\begin{cases} g_2(z,p), z=(z_1,z_2)\in\Gamma_p,\\ \frac 12\log\frac{|\beta^2-z_1^2|^2+\beta^4|z_2|^4+2(1-\beta^4)|z_2|^2+ \sqrt{M(z)}}{2|1-\beta^2z_1^2|^2}, z\in D,\\ g_2(z,q), z\in\Gamma_q, \end{cases}. \] where \(g_2(\cdot,p)\) and \(g_2(\cdot,q)\) are the pluricomplex Green functions of \(B^2\) with poles at \(p\) and \(q\) respectively, \(M(z)=(\beta^4|z_2|^4-|\beta^2-z_1^2|^2)^2+4(1-\beta^4)|z_2|^2\left|\beta^2|z_2|^2-(\beta^2-z_1^2)\right|^2\). The function \( g_2(\cdot,p,q)\) is real analytic in int \(\Gamma_p\cup D\cup\) int \(\Gamma_q\), it is of class \(C^{1,1}\) on \(B^2\backslash\{p,q\}\), and its first order partial derivatives extend continuously to \(\partial B^2\). The domain \(D\) is foliated by a one parameter family of complex curves \(L_\gamma\), \(\gamma\in B^1\), which are given by the formula \[ L_\gamma=\{z\in B^2:\gamma z_1^2=\beta^2(\gamma-z_2)(1-\bar{\gamma}z_2)\} . \] The leaves \( L_\gamma\) are properly embedded submanifolds of \(B^2\) and the restriction of \( g_2(\cdot,p,q)\) to each \(L_\gamma\) is harmonic away from \(p\) and \(q\).
Similar results actually hold in the general case when the weights of the poles are arbitrary.
Reviewer: K.Malyutin (Sumy)

32U35 Plurisubharmonic extremal functions, pluricomplex Green functions
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