Estimates for constant mean curvature graphs in MxR

Abstract

We will discuss some sharp estimates for CMC graphs in a Riemannian 3-manifold MxR whose boundary is contained in a slice. We will 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 that such a surface reaches in MxR. We will also give a lower 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.