Half-space Theorems for Minimal Surfaces in Nil$_3$ and Sol$_3$

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 standard left-invariant Riemannian metrics. If $\mathcal{S}$ is a properly immersed minimal surface in Nil$_3$ that lies on one side of some entire minimal graph $\mathcal{G}$, then $\mathcal{S}$ is the image of $\mathcal{G}$ by a vertical translation. If $\mathcal{S}$ is a properly immersed minimal surface in Sol$_3$ that lies on one side of a special plane $\mathcal{E}^t$ (see the discussion just before Theorem 1.5 for the definition of a special plane inSol$_3$), then $\mathcal{S}$ is the special plane $\mathcal{E}^u$ for some $u\in \mathbb{R}$.