Complex Gradient Systems

Abstract

Let $\widetilde{M}$ be a complex manifold of complex dimension n+k. We say that the functions u ₁,…,u _k and the vector fields ξ ₁,…,ξ _k on $\widetilde{M}$ form a complex gradient system if ξ ₁,…,ξ _k ,Jξ ₁,…,Jξ _k are linearly independent at each point $p\in \widetilde{M}$ and generate an integrable distribution of $T \widetilde{M}$ of dimension 2k and du _α (ξ _β )=0, d^c u _α (ξ _β )=δ _αβ for α,β=1,…,k. We prove a Cauchy theorem for such complex gradient systems with initial data along a CR-submanifold of type (n,k). We also give a complete local characterization for the complex gradient systems which are holomorphic and abelian, which means that the vector fields $\xi _{\alpha }^{c}=\xi _{\alpha }-iJ \xi _{\alpha }$, α=1,…,k, are holomorphic and satisfy $[\xi _{\alpha }^{c},\bar{\xi _{\beta }^{c}} ]=0$ for each α,β=1,…,k.