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For a Riemannian manifold $M$, possibly with boundary, we consider theRiemannian product $M\times\mathbb{R}^k$ with a smooth positive function thatweights the Riemannian measures. In this work we characterize parabolichypersurfaces with non-empty boundary and contained within certain regions of$M\times\mathbb{R}^k$ with suitable weights. Our results include half-space andBernstein-type theorems in weighted cylinders. We also generalize to thissetting some classical properties about the confinement of a compact minimalhypersurface to certain regions of Euclidean space according to the position ofits boundary. Finally, we show interesting situations where the statements areapplied, some of them in relation to the singularities of the mean curvatureflow.

Uniqueness results and enclosure properties for hypersurfaces with boundary in weighted cylinders