  In this note we show that if $g$ is a smooth Riemannian metric on
$\mathbb{S}^2$ such that the first eigenvalue of the operator $L_g:=-\Delta_g
+K_g$ satisfies $\lambda_1(L_g)=0$ then $(\mathbb{S}^2, g)$ arises as an
apparent horizon in an asymptotically flat initial data set with ADM mass
arbitrarily close to the associated Hawking mass
$\sqrt{\text{area}(\mathbb{S}^2, g)/16\pi}$. In particular, this determines the
Bartnik quasilocal mass (introduced by Bartnik \cite{Bartnik} in 1989)
associated with $(\mathbb{S}^2, g)$ in this setting. We prove these by
modifying the construction of Mantoulidis-Schoen \cite{MS} who proved the same
results in the case $\lambda_1(L_g)>0$. It follows that $\lambda_1(g)\geq 0$ is
necessary and sufficient for $(\mathbb{S}^2, g)$ to arise from an apparent
horizon in an asyptotically flat space-time under the dominant energy condition
and in the time symmetric setting, and that the Bartnik mass of the horizon is
$\sqrt{\text{area}(\mathbb{S}^2, g)/16\pi}$.
