The pluricomplex Green function with two poles of the unit ball of \(\mathbb{C}^n\).

*(English)*Zbl 1015.32029In 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.

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)

##### MSC:

32U35 | Plurisubharmonic extremal functions, pluricomplex Green functions |