Existence of barriers for surfaces with prescribed curvatures in M2×RM2×R

We present a deformation process of surfaces from a product space M1×RM1×R into another product space M2×RM2×R such that the relation of the principal curvatures of the deformed surfaces can be controlled in terms of the curvatures of M1M1 and M2M2. Thus, we obtain subsolutions for the existence or barriers for the non-existence of surfaces with prescribed curvatures in a general product space M×RM×R.