  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 space-like 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 space-like curve in $\partial_\infty\!\operatorname{AdS}^3$
gives rise to a Jordan curve that is the boundary of a pleated plane in
$\mathbb H^3$.
