A Counterexample to a Conjecture by Błocki–Zwonek

ABSTRACT

For a bounded pseudoconvex domain and pluricomplex Green function g_Ω(z, a) with pole at a ∈ Ω, it was conjectured by Błocki and Zwonek that β(t) = log λ_n({z ∈ Ω: g_Ω(z, a) < t}) is a convex function on ( − ∞, 0). With Ω the annulus the Green function g_Ω(z, a) with pole at a = 1 + 0i can be explicitly given in terms of Jacobi theta functions. We show numerically that in this case β is not convex.

KEYWORDS:

• Błocki–Zwonek conjectures
• Bergman kernel
• Green functions
• Jacobi functions

2000 AMS Subject Classification:

• 32F45
• 32U35
• 32A25

Acknowledgments

This work was inspired by Ian Kilmister and his Motörhead.

Additional information

Funding

The second-named author was partially supported by NCN grant DEC-2013/08/A/ST1/00312.