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 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 positive 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 positive curve in ∂∞ AdS3 gives rise to a Jordan curve that is the boundary of a pleated plane in H3.