Two optimization problems for the Loewner energy

A Jordan curve on the Riemann sphere can be encoded by its conformal welding, a circle homeomorphism. The Loewner energy measures how far a Jordan curve is away from being a circle, or equivalently, how far its welding homeomorphism is away from being a Möbius transformation. We consider two optimizing problems for the Loewner energy, one under the constraint for the curves to pass through $n$ given points on the Riemann sphere, which is the conformal boundary of hyperbolic $3$-space ${\mathbb{H}}^3$; the other under the constraint for $n$ given points on the circle to be welded to another $n$ given points of the circle. The latter problem can be viewed as optimizing positive curves on the boundary of AdS${}^3$ space passing through $n$ prescribed points. We observe that the answers to the two problems exhibit interesting symmetries: optimizing the Jordan curve in ${\mathrm{\partial }}_{\mathrm{\infty }}{\mathbb{H}}^3$ gives rise to a welding homeomorphism that is the boundary of a pleated plane in AdS${}^3$, whereas optimizing the positive curve in ${\mathrm{\partial }}_{\mathrm{\infty }}\text{\hspace{0.17em}⁣}{\mathrm{AdS}}^3$ gives rise to a Jordan curve that is the boundary of a pleated plane in ${\mathbb{H}}^3$.