Fix a bounded planar domain $\Omega.$ If an operator $T,$ in
theCowen-Douglas class $B_1(\Omega),$ admits the compact set $\bar{\Omega}$  as a
spectral set, then the curvature inequality $\mathcal K_T(w) \leq - 4 \pi^2
S_\Omega(w,w)^2,$ where $S_\Omega$ is the S\"{z}ego kernel of the domain $\Omega,$ is
evident. Except when $\Omega$ is simply connected, the existence of an operator for
which
$\mathcal K_T(w) = 4 \pi^2 S_\Omega(w,w)^2$ for all $w$ in $\Omega$ is not known.
However, one knows that if $w$ is a fixed but arbitrary point in $\Omega,$  then
there exists
a bundle shift of rank $1,$ say $S,$ depending on this $w,$ such that $\mathcal
K_{S^*}(w) =
4 \pi^2 S_\Omega(w,w)^2.$ We prove that these { extremal}  operators  are uniquely
determined: If $T_1$ and $T_2$ are two operators in $B_1(\Omega)$ each of which is the
adjoint of a rank $1$ bundle shift and $\mathcal{K}_{T_1}({w}) = -4\pi ^2
S(w,w)^2 = \mathcal{K}_{T_2}(w)$ for a fixed $w$ in $\Omega,$ then $T_1$ and $T_2$ are
unitarily equivalent. A surprising consequence is that the adjoint of only some of
the bundle
shifts of rank $1$ occur as extremal operators in domains of connectivity $> 1.$
These are
described
explicitly.