Two optimization problems for the Loewner energy
Abstract: 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 $\mathbb H3$; 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 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 $\partial_\infty \mathbb H3$ 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 $\mathbb H3$.
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