Geometric properties of pluricomplex Green functions with one or several poles in \({\mathbb{C}}^n\) (Q1591620)

For a domain \(D\subset\mathbb{C}^n\) let \(A_D\) be the Azukawa pseudometric and let \[ \gamma_p(w,X):=\sup\{|\sum_{|\nu|=p}\frac 1{\nu!}D^{\nu}f(w)X^\nu|^{1/p}: f\in\mathcal O(D,\Delta),\forall_{|\nu|\leq p-1}:D^\nu f(w)=0\}, \] \[ \Gamma_p(w,X):=\inf\{\frac 1{|t|}|\prod_{\lambda\in\varphi^{-1}(w) \cap(\Delta\setminus\{0\})}\lambda|:\varphi\in\mathcal O(\Delta,D), \] \(t\in\mathbb{C}: \varphi(0)=w\), \(\varphi'(0)=0,\dots,\varphi^{(p-1)}(0)=0\), \(\varphi^{(p)}(0)=p!tX\}\), \((w,X)\in D\times\mathbb{C}^n\), \(p\in\mathbb{N}\), where \(\Delta\) denotes the unit disc. The author proves that if \(D\subset\mathbb{C}^n\) is strictly hyperconvex, then \(A_D=\sup_{p\in\mathbb{N}}\gamma_p=\lim_{p\to+\infty}\gamma_p\) except for a pluripolar set and \(A_D\equiv\Gamma_p\) for any \(p\in\mathbb{N}\). She obtains moreover an effective formula for the pluricomplex Green function with two poles \(g_{\mathbb{B}_2}(p,q;\cdot)\), where \(\mathbb{B}_2\subset\mathbb{C}^2\) is the unit Euclidean ball, \(p:=(\beta,0)\), \(q:=(-\beta,0)\) (\(0<\beta<1\)). The formula has been first proved by \textit{D. Coman} [Pac. J. Math. 194, 257-283 (2000; Zbl 1015.32029)] and next, by a simpler method, by \textit{A. Edigarian} and \textit{W. Zwonek} [Complex Variables, Theory Appl. 35, No. 4, 367-380 (1998; Zbl 0910.32030)]. The author's method is the same as the one used by Edigarian and Zwonek. Using the formula the author presents a counterexample to Proposition 3.7 from the paper by \textit{S. M. Einstein-Matthews} [Nagoya Math. J. 138, 65-112 (1995; Zbl 0828.32001)], showing that for \(2\log\beta<c<0\) the level set \(B_c:=\{z\in\mathbb{B}_2:g_{\mathbb{B}_2}(p,q;z)<c\}\) is connected but not lineally convex.