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

TL;DR

This paper investigates the boundary behavior of complete graphs with minimal, translating, and constant mean curvature in product manifolds, revealing geometric conditions on boundary arcs related to their curvature and geodesic properties.

Contribution

It establishes geometric boundary conditions for complete minimal, translating, and CMC graphs over domains in product manifolds, linking boundary arc properties to curvature and geodesic conditions.

Findings

Boundary arcs are geodesics for minimal and translating graphs.

Boundary arcs have constant principal curvature for CMC graphs.

Results connect boundary behavior to Jenkins-Serrin type theorems.

Abstract

In this note we discuss graphs over a domain $\Omega\subset N^2$ in the product manifold $N^2\times \mathbb{R}$. Here $N^2$ is a complete Riemannian surface and $\Omega$ has peice-wise smooth boundary. Let $\gamma \subset\partial\Omega$ be a smooth connected arc and $\Sigma$ be a complete graph in $N^2\times \mathbb{R}$ 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 principle curvature in $N^2$. This explains the infinity value boundary condition upon domains having Jenkins-Serrin theorems on minimal and CMC graphs in $N^2\times \mathbb{R}$.