LAUSR

If a graph submanifold (x, f(x)) of a Riemannian warped product space $$(M^m\times _{e^{\psi }}N^n,\tilde{g}=g+ e^{2\psi }h)$$(Mm×eψNn,g~=g+e2ψh) 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, $$m\Vert H\Vert \le \frac{A_{\psi }(\partial D)}{V_{\psi }(D)}$$m‖H‖≤Aψ(∂D)Vψ(D) holds, where $$A_{\psi }(\partial D)$$Aψ(∂D) and $$V_{\psi }(D)$$Vψ(D) are the $${\psi }$$ψ-weighted area and volume, respectively. In particular, $$H=0$$H=0 if (M, g) has zero-weighted Cheeger constant, a concept recently introduced by Impera et al. (Height estimates for killing graphs. arXiv:1612.01257, 2016). This generalizes the known cases $$n=1$$n=1 or $$\psi =0$$ψ=0. We also conclude minimality using a closed calibration, assuming $$(M,g_*)$$(M,g∗) is complete where $$g_*=g+e^{2\psi }f^*h$$g∗=g+e2ψf∗h, and for some constants $$\alpha \ge \delta \ge 0$$α≥δ≥0, $$C_1>0$$C1>0 and $$\beta \in [0,1)$$β∈[0,1), $$\Vert \nabla ^*\psi \Vert ^2_{g_*}\le \delta $$‖∇∗ψ‖g∗2≤δ, $$\mathrm {Ricci}_{\psi ,g_*}\ge \alpha $$Ricciψ,g∗≥α, and $${\mathrm{det}}_g(g_*)\le C_1 r^{2\beta }$$detg(g∗)≤C1r2β holds when $$r\rightarrow +\infty $$r→+∞, where r(x) is the distance function on $$(M,g_*)$$(M,g∗) 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.