Holomorphic Dirichlet forms on complex manifolds - Gross, L; Qian, Z (2004)
A complete description of all holomorphic Dirichlet forms on complex hyperbolic space is given. They are classified in terms of the Lie algebra of the automorphism group, which we take to act in the unit ball of ℂ n. We show that the measure defining the holomorphic Dirichlet form is finite if and only if the real part of the associated holomorphic vector field points "outward" at the boundary of the unit ball. A logarithmic Sobolev inequality is proven to hold whenever this measure is finite. All holomorphic Dirichlet forms over complex hyperbolic space, complex projective space and Cn are now known.