Bulletin of the
Korean Mathematical Society BKMS

We give area and height estimates for cmc-graphs over a bounded planar $C^{2, \alpha}$ domain $\Omega\subset \mathbb R^3$. For a constant $H$ satisfying $H^2 |\Omega| \le 9\pi /16$, we show that the height $h$ of $H$-graphs over $\Omega$ with vanishing boundary satisfies $|h| < (\tilde{r}/2\pi) H |\Omega|$, where $\tilde{r}$ is the middle zero of $(x - 1) ( H^2 |\Omega| ( x + 2)^2 -9 \pi (x - 1))$. We use this height estimate to prove the following existence result for cmc $H$-graphs: for a constant $H$ satisfying ${H}^2 |\Omega| < {(\sqrt{297}- 13) \pi/ 8} $, there exists an $H$-graph with vanishing boundary.

Keywords: constant mean curvature, height estimate, Dirichlet problem

MSC numbers: 53A10, 35J25