Half-space theorems for minimal surfaces in Nil3 and Sol 3

Abstract

We prove some half-space theorems for minimal surfaces in the Heisenberg group Nil3 and the Lie group Sol3 endowed with their standard left-invariant Riemannian metrics. If S is a properly immersed minimal surface in Nil3 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 Sol3 that lies on one side of a special plane εt (see the discussion just before Theorem 1.5 for the definition of a special plane in Sol3), then S is the special plane εu for some u ∈ ℝ.