  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. 
Convergent sequences of closed minimal surfaces embedded in $\S3$