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We present some geometric applications, of global character, of the bubblinganalysis developed by Buzano and Sharp for closed minimal surfaces, obtainingsmooth multiplicity one convergence results under upper bounds on the Morseindex 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 thethree-dimensional sphere the class of embedded minimal surfaces of index oneand genus $\gamma$ is sequentially compact for any $\gamma\geq 1$. Furthemore,we give a quantitative description of how the genus drops as a sequence ofminimal surfaces converges smoothly, with mutiplicity $m\geq 1$, away fromfinitely many points where curvature concentration may happen. This resultexploits a sharp estimate on the multiplicity of convergence in terms of thenumber of ends of the bubbles that appear in the process.

Geometric convergence results for closed minimal surfaces via bubbling analysis