Heinz mean curvature estimates in warped product spaces M x e N

If a graph submanifold (x, f(x)) of a Riemannian warped product space is immersed with parallel mean curvature H, then we obtain a Heinz-type estimation of the mean curvature. Namely, on each compact domain D of M, holds, where and are the -weighted area and volume, respectively. In particular, if (M, g) has zero-weighted Cheeger constant, a concept recently introduced by Impera et al. (Height estimates for killing graphs. arXiv:1612.012571612.01257" TargetType=, 2016). This generalizes the known cases or . We also conclude minimality using a closed calibration, assuming is complete where , and for some constants , and , , , and holds when , where r(x) is the distance function on from some fixed point. Both results rely on expressing the squared norm of the mean curvature as a weighted divergence of a suitable vector field.