Piecewise geodesic Jordan curves II: Loewner energy, projective structures, and accessory parameters

In this paper we consider Jordan curves on the Riemann sphere passing through $n \ge 3$ given points. We show that in each relative isotopy class of such curves, there exists a unique curve that minimizes the Loewner energy. These curves have the property that each arc between two consecutive points is a hyperbolic geodesic in the domain bounded by the other arcs. This geodesic property lets us define a complex projective structure whose holonomy lies in $\mathrm{PSL}(2,\mathbb{R})$. We show that the quadratic differential comparing this projective structure to the trivial projective structure on the sphere has simple poles whose residues (accessory parameters) are given by the Wirtinger derivatives of the minimal Loewner energy. This is reminiscent of Polyakov's conjecture for Fuchsian projective structures, proven by Takhtajan and Zograf. Finally, we show that the projective structures we obtain are related to Fuchsian projective structures through $\pi$-grafting.