Abstract:
We show the convexity properties and similarity classification with homogeneous operators on Hilbert spaces of holomorphic mappings, curves and functions. The sharp estimates of all homogeneous expansions for a class and subclass of quasi-convex mappings on the unit polydisk in the unitary space and of type B and order Α with the weak version of the Bieberbach conjecture in several complex variables are given. The centers of the quasi-homogeneous polynomial differential equations of degree three and global behaviour of the period of the sum of two quasi-homogeneous vector fields are determined. We obtain the first derivative of the period function for Hamiltonian systems with homogeneous non linearities and applications. The multiplicity-free and rigidity of the flag structure with classification of homogeneous and quasi-homogeneous operators and holomorphic curves in the Cowen-Douglas class are discussed.
