The boundary behavior of domains with complete translating, minimal and CMC graphs in $N^2\times \mathbb{R}$

Abstract

In this note we discuss graphs over a domain $Ω\subset N^2$ in the product manifold $N^2\times \mathbb{R}$. Here $N^2$ is a complete Riemannian surface and $Ω$ has peice-wise smooth boundary. Let $γ\subset\partialΩ$ be a smooth connected arc and $Σ$ be a complete graph in $N^2\times \mathbb{R}$ over $Ω$. We show that if $Σ$ is a minimal or translating graph, then $γ$ is a geodesic in $N^2$. Moreover if $Σ$ is a CMC graph, then $γ$ has constant principle curvature in $N^2$. This explains the infinity value boundary condition upon domains having Jenkins-Serrin theorems on minimal and CMC graphs in $N^2\times \mathbb{R}$.