Periodic solutions of holomorphic differential equations

Abstract

Publisher Summary This chapter describes the periodic solutions of holomorphic differential equations. It presents an assumption in which M is a complex manifold and F is the collection of holomorphic time-dependent vector fields of (fixed) period ω . Therefore, ƒ ∈ ɛ when f maps M x ℝ continuously into TM , the tangent bundle of M , in such a way that (1) ƒ ( x, t + ω ) = ƒ ( x,t ), (2) f is holomorphic in x , and (3) f (·, t ) is a section of TM for each t . ɛ is given the topology of uniform convergence on compact sets of M × ℝ . The orbits of ƒ are given locally by the solutions of an ordinary differential equation ɛ = ƒ * ( t, z ).