Inserted: 6 may 2011
Journal: J. Reine Angew. Math.
For the update version and eventual errata see the webpage http:/www.math.uzh.chdelellis
In this paper we prove genus bounds for closed embedded minimal surfaces in a closed $3$-dimensional manifold constructed via min-max arguments. A stronger estimate was announced by Pitts and Rubistein but to our knowledge its proof has never been published. Our proof follows ideas of Simon and uses an extension of a famous result of Meeks, Simon and Yau on the convergence of minimizing sequences of isotopic surfaces. This result is proved in the second part of the paper.