On ideal lemniscates, butterflies, and circular waves

Abstract

We prove the existence of infinitely many lemniscate-like, butterfly-like, and circular-wave critical points for the length-penalised ideal energy. For the lemniscate and butterfly family, we use a staged direct minimisation process to establish existence. A boundary-layer analysis reveals critical points near two Fresnel phases: $3π/4$ modulo $2π$, corresponding to lemniscates, and $7π/4$ modulo $2π$, which are the butterflies. The family of circular waves bifurcates, in a sense, from multiply-covered circles. We use an adapted shooting method to establish their existence.