Estimates for constant mean curvature graphs in MxR

TL;DR

This paper derives sharp geometric estimates for constant mean curvature graphs in Riemannian 3-manifolds, focusing on boundary curvature, height restrictions, and interior-boundary distance bounds.

Contribution

It provides new sharp bounds for boundary geodesic curvature and interior-boundary distance in CMC graphs, with characterizations of equality cases.

Findings

Sharp lower bounds for boundary geodesic curvature.

Improved bounds under height restrictions.

Lower bounds for interior point to boundary distance.

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.