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

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 $M$, $m\Vert H\Vert \le \frac{A_{\psi }(\partial D)}{V_{\psi }(D)}$ 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 $(M,g_{\ast })$ is complete where $g_{\ast }=g+e^{2\psi }{f}^{\ast }h$, and for some constants $\alpha \ge \delta \ge 0$, $C_1>0$ and $\beta \in [0,1)$, $\Vert {\nabla }^{\ast }\psi {\Vert }_{g_{\ast }}^2\le \delta$, ${\mathrm{Ricci}}_{\psi ,g_{\ast }}\ge \alpha$, and $de{t}_g(g_{\ast })\le C_1{r}^{2\beta }$ holds when $r\to +\infty$, where $r(x)$ is the distance function on $(M,g_{\ast })$ 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.

arXiv commentv4. 20 Pages, we add examples, and correct some constants. v2. 12 pages. We obtain a substancial improvement of the estimation of the mean curvature by using the weighted Cheeger constant recently introduced by Impera at all. in Arxiv 1612.01257

References in corpus (1)

• Dirichlet principal eigenvalue comparison theorems in geometry with torsion