The first two coefficients of the Bergman function expansions for Cartan–Hartogs domains

Abstract

Let [Formula: see text] be a globally defined real Kähler potential on a domain [Formula: see text], and [Formula: see text] be a Kähler metric on the Hartogs domain [Formula: see text] associated with the Kähler potential [Formula: see text]. First, we obtain explicit formulas of the coefficients [Formula: see text] of the Bergman function expansion for the Hartogs domain [Formula: see text] in a momentum profile [Formula: see text]. Second, using explicit expressions of [Formula: see text], we obtain necessary and sufficient conditions for the coefficients [Formula: see text] to be constants. Finally, we obtain all the invariant complete Kähler metrics on Cartan–Hartogs domains such that their the coefficients [Formula: see text] of the Bergman function expansions are constants.