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 M\"obius 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 $\partial_\infty \mathbb H^3$ gives rise to a welding homeomorphism that is the boundary of a pleated plane in AdS$^3$, whereas optimizing the positive curve in $\partial_\infty\!\operatorname{AdS}^3$ gives rise to a Jordan curve that is the boundary of a pleated plane in $\mathbb H^3$.