Minimal genus problem for $T^2$-bundles over surfaces

For any positive integer $g$, we completely determine the minimal genus function for ${\mathrm{\Sigma }}_g\times T^2$. We show that the lower bound given by the adjunction inequality is not sharp for some class in $H_2({\mathrm{\Sigma }}_g\times T^2)$. However, we construct a suitable embedded surface for each class and we have exact values of minimal genus functions.