Estimates for constant mean curvature graphs in $M\times \mathbb{R}$

Abstract

We discuss some sharp estimates for a constant mean curvature graph $\Sigma$ in a Riemannian 3-manifold $M\times \mathbb{R}$ whose boundary $\partial \Sigma$ is contained in a slice $M\times \{t_0\}$ and satisfies a capillarity condition. We start by giving sharp lower bounds for the geodesic curvature of the boundary and improve these bounds when assuming additional restrictions on the maximum height attained by the graph in $M\times \mathbb{R}$. We also give a bound for the distance from an interior point to the boundary in terms of the height at that point, and characterize when these bounds are attained.