Variational character of the Cauchy-Riemann equations

Abstract

Cauchy-Riemann equations for smooth maps ϕ: ℂ → ℂk are proved to be of variational type if and only if k is even. This fact is seen to be related to a complex differential form of degree (3, 0) on ℂ × ℂk, which exists only for an even k. The Lie algebra of infinitesimal symmetries of the Hamiltonian structure associated with Cauchy-Riemann equations is also determined.