A half-space theorem for ideal Scherk graphs in $M\times\mathbb R$

TL;DR

This paper establishes a half-space theorem for ideal Scherk graphs in product manifolds with negatively curved Hadamard surfaces, showing that certain minimal surfaces must be translates of these graphs.

Contribution

It proves a new half-space theorem for minimal surfaces over polygonal domains in negatively curved Hadamard surfaces, extending classical results to this geometric setting.

Findings

Properly immersed minimal surfaces disjoint from the Scherk graph are translates of it.

The theorem applies to surfaces in $M\times\mathbb R$ where $M$ has bounded negative curvature.

The result generalizes classical half-space theorems to a broader geometric context.

Abstract

We prove a half-space theorem for an ideal Scherk graph $\Sigma\subset M\times\mathbb R$ over a polygonal domain $D\subset M,$ where $M$ is a Hadamard surface whose curvature is bounded above by a negative constant. More precisely, we show that a properly immersed minimal surface contained in $D\times\mathbb R$ and disjoint from $\Sigma$ is a translate of $\Sigma.$