Piecewise geodesic Jordan curves on the sphere

In this talk I will discuss Jordan curves on the Riemann sphere passing through finitely many given points with the property that every arc between two consecutive points is a hyperbolic geodesic in the simply connected region bounded by the rest of the arcs. During the first half of the talk I will present interesting alternative characterisations of these curves and discuss their existence, mainly following earlier works by Marshall, Rohde and Wang. The second half will be focusing on showing that every isotopy class of curves passing through at least three fixed points contains a unique differentiable piecewise geodesic curve, and is based on upcoming joint work with Bonk, Marshall, Rohde and Wang.