  This note expands on the recent proof \cite{ABKT} that the extremal domains
for analytic content in two dimensions can only be disks and annuli. This
result's unexpected implication for theoretical physics is that, for extremal
domains, the analytic content is a measure of non-commutativity of the
(multiplicative) adjoint operators $T, T^{\dag}$, where $T^{\dag} = \bar z$,
and therefore of the quantum deformation parameter (``Planck's constant"). The
annular solution (which includes the disk as a special case) is, in fact, a
continuous family of solutions, corresponding to all possible positive values
of the deformation parameter, consistent with the physical requirement that
conformal invariance in two dimensions forbids the existence of a special
length scale.
