Poisson integrals for homogeneous, rank 1 Koszul manifolds
Let \(X\) be a complex manifold which is homogeneous under the real analytic Lie group \(G\). From a \(G\)-invariant volume form \(Kdx\) on \(X\) is constructed a new form, the Koszul form, which is del-delbar of \(K\). If the Koszul form is non-degenerate then it gives \(X\) the structure of a pseudo-Kähler manifold. This then gives rise to a canonical second- order differential operator, the Laplace-Beltrami operator. An investigation of the properties of these forms leads to a complete solution of the eigenvalue problem for the Laplace-Beltrami operator for large classes of groups.
homogeneous under the real analytic Lie group
eigenvalue problem for the Laplace-Beltrami operator
scientific article; zbMATH DE number 704490