  given two minimal surfaces embedded in $\S3$ of genus $g$ we prove the
existence of a sequence of non-congruent compact minimal surfaces embedded in
$\S3$ of genus $g$ that converges in $C^{2,\alpha}$ to a compact embedded
minimal surface provided some conditions are satisfied. These conditions also
imply that, if any of these two surfaces is embedded by the first eigenvalue,
so is the other.
