The half space property for cmc 1/2 graphs in $$\mathbb {E}(-1,\tau )$$ E ( - 1 , τ )

Abstract

In this paper, we prove a half-space theorem with respect to constant mean curvature 1/2 entire graphs in E(−1, τ ). If Σ is such an entire graph and Σ ′ is a properly immersed constant mean curvature 1/2 surface included in the mean convex side of Σ then Σ ′ is a vertical translate of Σ. We also have an equivalent statement for the non mean convex side of Σ.

A half-space theorem for graphs of constant mean curvature $0<H<\frac{1}{2}$ in $\mathbb{H}^{2}\times\mathbb{R}$

“…In that case, we prove a result similar to the one of Menezes. We notice that the value H = 1 2 is critical in this setting (see [12] for the H = 1 2 case). The graphs that we will work with are graphs of functions u defined in a domain D ⊂ H 2 whose boundary ∂D is composed of complete arcs {A i } and {B j }, such that the curvatures of the arcs with respect to the domain are κ(A i ) = 2H and κ(B j ) = −2H.…”

The slab theorem for minimal surfaces in $$\mathbb {E}(-1,\tau )$$ E ( - 1 , τ )

“…Good references for the geometry of E(−1, τ ) are the papers [9,12,17,19], and the results not proved here can be found in them.…”

The halfspace theorem for minimal hypersurfaces in regions bounded by minimal cones

“…It is then natural to investigate some sort of halfspace theorem in higher dimensions or in others contexts. In fact, there exits an active research on this topic and many results have been obtained (see, for instance, ).…”