The Poincar\'e-Bendixson Theorem and the Non-linear Cauchy-Riemann Equations

Fiedler and Mallet-Paret prove a version of the classical Poincar'e-Bendixson Theorem for scalar parabolic equations. We prove that a similar result holds for bounded solutions of the non-linear Cauchy-Riemann equations. The latter is an application of an abstract theorem for flows with a(n) (unbounded) discrete Lyapunov function.