Heinz mean curvature estimates in warped product spaces $M\times_{e^{\psi}}N$

TL;DR

This paper derives Heinz type estimates for the mean curvature of graphs in warped product spaces with parallel mean curvature, extending known results and providing conditions for minimality using weighted geometric analysis.

Contribution

It introduces new Heinz type estimates for graphs with parallel mean curvature in warped products, generalizing previous cases and connecting to weighted Cheeger constants and calibration methods.

Findings

Heinz type estimate: $m\|H ext{ extbar} ext{ extbar} extbar$ on compact domains.

Zero mean curvature when weighted Cheeger constant is zero.

Minimality conditions established via calibration and weighted Ricci bounds.

Abstract

If a graph submanifold $(x,f(x))$ of a Riemannian warped product space $(M^m\times_{e^{\psi}}N^n,\tilde{g}=g+e^{2\psi}h)$ 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 $M$, $m\|H\|\leq \frac{A_{\psi}(\partial D)}{V_{\psi}(D)}$ holds, where $A_{\psi}(\partial D)$ and $V_{\psi}(D)$ are the ${\psi}$-weighted area and volume, respectively. In particular, $H=0$ if $(M,g)$ has zero weighted Cheeger constant, a concept recently introduced by D.\ Impera et al.\ (\cite{[Im]}). This generalizes the known cases $n=1$ or $\psi=0$. We also conclude minimality using a closed calibration, assuming $(M,g_*)$ is complete where $g_*=g+e^{2\psi}f^*h$, and for some constants $\alpha\geq \delta\geq 0$, $C_1>0$ and $\beta\in [0,1)$, $\|\nabla^*\psi\|^2_{g_*}\leq \delta$, $\mathrm{Ricci}_{\psi,g_*}\geq \alpha$, and $det_g(g_*)\leq C_1 r^{2\beta}$ holds when $r\to +\infty$, where $r(x)$ is the distance function on $(M,g_*)$ 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.