Zoll magnetic structures and ruled surfaces

Abstract

A Zoll magnetic system on an oriented closed surface $M$ is a Riemannian metric $g$ together with a function $λ\colon M\to \mathbb{R}$, such that every unit speed solution of the ODE $\ddot γ(t)=λ(γ(t))γ(t)^\perp$ is periodic and the minimal period depends continuously on $γ$. The trivial example is given by $g$ with constant curvature $K$ and $λ\equiv {\rm const.}$ such that $λ^2+K>0$. This article exhibits non-trivial Zoll magnetic systems for every genus-for genus $\ge 2$ these are the first such examples. The approach is twistor theoretic: To a general magnetic system $(g,λ)$ one associates its transport twistor space $Z(g,λ)$, which is the unit disk bundle $DM$, equipped with a degenerate complex structure that encodes the magnetic flow. For the trivial Zoll magnetic systems explicit holomorphic blow-down maps $β\colon Z(g,λ)\to W$ into certain ruled surfaces $W\to M$ are constructed, mapping $\partial Z(g,λ)$ onto a Lagrangian $P\subset W$. For small Lagrangian perturbations $P'\approx P$ the procedure can be reversed and this results in a large class of (non-trivial) nearby Zoll magnetic systems.