The boundary behavior of domains with complete translating, minimal and CMC graphs in N2×ℝ

Abstract

In this paper, we discuss graphs over a domain $\Omega\subset~N^2$ in the product manifold $N^2\PLH\R$. Here $N^2$ is a complete Riemannian surface and $\Omega$ has piecewise smooth boundary. Let $\gamma~\subset~\P\Omega$ be a smooth connected arc and $\Sigma$ be a complete graph in $N^2\PLH\R$ over $\Omega$. We show that if $\Sigma$ is a minimal or translating graph, then $\gamma$ is a geodesic in $N^2$. Moreover if $\Sigma$ is a CMC graph, then $\gamma$ has constant principal curvature in $N^2$. This explains the infinity value boundary condition upon domains having Jenkins-Serrin theorems on minimal and constant meancurvature (CMC) graphs in $N^2\PLH\R$.