Remarks on the balanced metric on Hartogs triangles with integral exponent

In this paper we study the balanced metrics on some Hartogs triangles of exponent γ ∈ ℤ + , i.e. equipped with a natural Kähler form with where μ = ( μ 1 , …, μ n ), μ i > 0, depending on n parameters. The purpose of this paper is threefold. First, we compute the explicit expression for the weighted Bergman kernel function for (Ω n (γ), g ( μ )) and we prove that g ( μ ) is balanced if and only if μ 1 > 1 and γμ 1 is an integer, μ i are integers such that μ i ≽ 2 for all i = 2, …, n − 1, and μ n > 1. Second, we prove that g ( μ ) is Kähler-Einstein if and only if μ 1 = μ 2 = … = μ n = 2λ, where λ is a nonzero constant. Finally, we show that if g ( μ ) is balanced then (Ω n (γ), g ( μ )) admits a Berezin-Engliš quantization.