Holomorphic invariant forms of a bounded domain

Given a complete ortho-normal system ? = (?₀,?₁,?₂,…) of L ² H(\(\mathcal{D}\)), the space of holomorphic and absolutely square integrable functions in the bounded domain \(\mathcal{D}\) of ? ⁿ , we construct a holomorphic imbedding \(\iota _\phi :\mathcal{D} \to \mathfrak{F}(n,\infty )\), the complex infinite dimensional Grassmann manifold of rank n. It is known that in \(\mathfrak{F}(n,\infty )\) there are n closed (μ, μ)-forms τ_μ (μ = 1,…,n) which are invariant under the holomorphic isometric automorphism of \(\mathfrak{F}(n,\infty )\) and generate algebraically all the harmonic differential forms of \(\mathfrak{F}(n,\infty )\). So we obtain in \(\mathcal{D}\) a set of (μ, μ)-forms ι _? ^* τ_μ (μ = 1,…, n), which are independent of the system ? chosen and are invariant under the bi-holomorphic transformations of \(\mathcal{D}\). Especially the differential metric ds ₂ ¹ associated to the Kähler form ι _? ^* τ₁ is a Kähler metric which differs from the Bergman metric ds ² of \(\mathcal{D}\) in general, but in case that the Bergman metric is an Einstein metric, ds ₁ ² differs from ds ² only by a positive constant factor.