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 $\hat{I}»:M\to \mathbb{R}$, such that every unit speed solution of the ODE ${\ddot{\hat{I}}}^3(t)=\hat{I}»({\hat{I}}^3(t)){\hat{I}}^3(t{)}^{\perp }$ is periodic and the minimal period depends continuously on ${\hat{I}}^3$. The trivial example is given by $g$ with constant curvature $K$ and $\hat{I}»\equiv const.$ such that $\hat{I}{»}^2+K>0$. 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 $(g,\hat{I}»)$ one associates its transport twistor space $Z(g,\hat{I}»)$, 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 ${\hat{I}}^2:Z(g,\hat{I}»)\to W$ into certain ruled surfaces $W\to M$ are constructed, mapping $\partial Z(g,\hat{I}»)$ onto a Lagrangian $P\subset W$. For small Lagrangian perturbations $P^{\prime }\approx P$ the procedure can be reversed and this results in a large class of (non-trivial) nearby Zoll magnetic systems.