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 H³; 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 space-like curves on the boundary of AdS³ space passing through n prescribed points. We observe that the answers to the two problems exhibit interesting symmetries: optimizing the Jordan curve in ∂_∞ H³ gives rise to a welding homeomorphism that is the boundary of a pleated plane in AdS³, whereas optimizing the space-like curve in ∂_∞\!AdS³ gives rise to a Jordan curve that is the boundary of a pleated plane in H³.