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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)

##### MSC:
 32U35 Plurisubharmonic extremal functions, pluricomplex Green functions
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