  In this paper, we build up a min-max theory for minimal surfaces using
sweepouts of surfaces of genus $g\geq 2$. We develop a direct variational
methods similar to the proof of the famous Plateau problem by J. Douglas and T.
Rado. As a result, we show that the min-max value for the area functional can
be achieved by a bubble tree limit consisting of branched genus-$g$ minimal
surfaces with nodes, and possibly finitely many branched minimal spheres. We
also prove a Colding-Minicozzi type strong convergence theorem similar to the
classical mountain pass lemma. Our results extend the min-max theory developed
by Colding-Minicozzi and the author to all genera.
