Le cône des fonctions plurisousharmoniques négatives et une conjecture de Coman. (A cone of plurisubharmonic negative functions and a conjecture of Coman).

*(French)*Zbl 1026.32066The negative plurisubharmonic functions on a domain \(\Omega\) in \(\mathbb C^n\) form a convex cone in the space of locally integrable functions on \(\Omega\). The authors give a few new examples of functions that are extremal points in this cone. These examples are pluricomplex Green functions with two poles. If \(w_j\), \(j=1,\dots,k\) are finitely many points in \(\Omega\), \(\nu_j\) are positive numbers, and \(A=\{(w_j,\nu_j)\); \(j=1,\dots,k\}\), then the pluricomplex Green function \(g(\cdot,A)\) with weights \(\nu_j\) at \(w_j\) is defined as the supremum of all negative plurisubharmonic functions \(u\) such that \(u-\nu_j\log|\cdot-w_j|\) is bounded near \(w_j\) for each \(j\). The points \(w_j\) are called the poles of the function \(g(\cdot, A)\).

The authors prove that the Green function with two poles of equal weight \(1\) in the unit ball are extremal as well as the Green function with the poles \((a,0)\) and \((b,0)\) of equal weight \(1\) in the bidisc. The Lempert function \(\delta(\cdot,A)\) with poles of weight \(\nu_j\) at \(w_j\) is defined so that \(\delta(z,A)\) is the infimum of sums of the form \(\sum_{j=1}^k\nu_j\log|\zeta_j|\) taken over all analytic discs \(f:D\to \Omega\) such that \(f(0)=z\) and \(f(\zeta_j)=w_j\). D. Coman studied these functions in his paper [Pac. J. Math. 194, 257-283 (2000; Zbl 1015.32029)] and conjectured that \(g(\cdot,A)=\delta(\cdot,A)\) for every \(A\) on any bounded convex domain. The authors show that for the bidisc with the points \((a,0)\) and \((b,0)\), \(0<|a|,|b|<1\), with the weights \(1\) and \(2\), these functions are not equal. They even prove the existence of strictly pseudoconvex domains in the bidisc for which the functions are different.

The authors prove that the Green function with two poles of equal weight \(1\) in the unit ball are extremal as well as the Green function with the poles \((a,0)\) and \((b,0)\) of equal weight \(1\) in the bidisc. The Lempert function \(\delta(\cdot,A)\) with poles of weight \(\nu_j\) at \(w_j\) is defined so that \(\delta(z,A)\) is the infimum of sums of the form \(\sum_{j=1}^k\nu_j\log|\zeta_j|\) taken over all analytic discs \(f:D\to \Omega\) such that \(f(0)=z\) and \(f(\zeta_j)=w_j\). D. Coman studied these functions in his paper [Pac. J. Math. 194, 257-283 (2000; Zbl 1015.32029)] and conjectured that \(g(\cdot,A)=\delta(\cdot,A)\) for every \(A\) on any bounded convex domain. The authors show that for the bidisc with the points \((a,0)\) and \((b,0)\), \(0<|a|,|b|<1\), with the weights \(1\) and \(2\), these functions are not equal. They even prove the existence of strictly pseudoconvex domains in the bidisc for which the functions are different.

Reviewer: Ragnar Sigurdsson (Reykjavik)

##### MSC:

32U35 | Plurisubharmonic extremal functions, pluricomplex Green functions |

32U05 | Plurisubharmonic functions and generalizations |