A variational construction for Einstein metrics on connected sums of $S^2\times S^2$

For each positive integer $n$, we give a variational construction of a positive Einstein metric on the connected sum of $n$ copies of $S^2\times S^2$, with coupled toric and dihedral symmetries. More precisely, we obtain these metrics via a min-max construction for the total scalar curvature, which is in some sense dual to the usual max-min formulation of the smooth Yamabe invariant within this symmetry class, with more favorable analytic properties.