  We present some geometric applications, of global character, of the bubbling
analysis developed by Buzano and Sharp for closed minimal surfaces, obtaining
smooth multiplicity one convergence results under upper bounds on the Morse
index and suitable lower bounds on either the genus or the area. For instance,
we show that given any Riemannian metric of positive scalar curvature on the
three-dimensional sphere the class of embedded minimal surfaces of index one
and genus $\gamma$ is sequentially compact for any $\gamma\geq 1$. Furthemore,
we give a quantitative description of how the genus drops as a sequence of
minimal surfaces converges smoothly, with mutiplicity $m\geq 1$, away from
finitely many points where curvature concentration may happen. This result
exploits a sharp estimate on the multiplicity of convergence in terms of the
number of ends of the bubbles that appear in the process.
