  We show under natural assumptions that stable solutions to the abelian
Yang--Mills--Higgs equations on Hermitian line bundles over the round
$2$-sphere actually satisfy the vortex equations, which are a first-order
reduction of the (second-order) abelian Yang--Mills--Higgs equations. We also
obtain a similar result for stable solutions on a flat $2$-torus. Our method of
proof comes from the work of Bourguignon--Lawson concerning stable $SU(2)$
Yang--Mills connections on compact homogeneous $4$-manifolds.
