Half-space theorems for minimal surfaces in Nil_3 and Sol_3

TL;DR

This paper establishes half-space theorems for minimal surfaces in Nil_3 and Sol_3, showing that properly immersed minimal surfaces constrained by certain barriers must be translations or specific planes.

Contribution

It provides new half-space theorems for minimal surfaces in Nil_3 and Sol_3, characterizing their global behavior under geometric constraints.

Findings

Properly immersed minimal surfaces on one side of a minimal graph are vertical translations of it in Nil_3.

In Sol_3, such surfaces on one side of a special plane are themselves that plane.

Theorems extend understanding of minimal surface behavior in these Lie groups.

Abstract

We prove some half-space theorems for minimal surfaces in the Heisenberg group Nil_3 and the Lie group Sol_3 endowed with their left-invariant Riemannian metrics. If S is a properly immersed minimal surface in Nil_3 that lies on one side of some entire minimal graph G, then S is the image of G by a vertical translation. If S is a properly immersed minimal surface in Sol_3 that lies on one side of a special plane, then S is another special plane.