# zbMATH — the first resource for mathematics

Le cône des fonctions plurisousharmoniques négatives et une conjecture de Coman. (A cone of plurisubharmonic negative functions and a conjecture of Coman). (French) Zbl 1026.32066
The 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.

##### MSC:
 32U35 Plurisubharmonic extremal functions, pluricomplex Green functions 32U05 Plurisubharmonic functions and generalizations
Full Text: