Properly immersed minimal surfaces in a slab of $${\mathbb {H} \times {\mathbb {R}}}$$ H × R , $${\mathbb {H}}$$ H the hyperbolic plane

Abstract

We prove that the ends of a properly immersed simply or one connected minimal surface in H × R contained in a slab of height less than π of H × R, are multi-graphs. When such a surface is embedded then the ends are graphs. When embedded and simply connected, it is an entire graph.

On doubly periodic minimal surfaces in $${\mathbb {H}}^2 \times {\mathbb {R}}$$ H 2 × R with finite total curvature in the

“…The end of E is contained in a slab of width 2 > 0 and by a result of Collin, Hauswirth and Rosenberg [3], E is a graph outside a compact domain of H 2 × R . This implies that E has bounded curvature.…”

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

“…Dragging Lemma. [5,6] Let Σ be a properly immersed minimal surface in a complete Riemannian 3-manifold M. Let S be a compact surface with boundary and f : S × [0, 1] → M a C 1 map such that for each 0 ≤ t ≤ 1,…”

On the characterization of minimal surfaces with finite total curvature in $\mathbb H^2\times\mathbb R$ and $\widetilde{\rm PSL}_2(\mathbb{R},τ)$

“…After that we will study the geometry of subdomains of ends which are contained in some vertical slab with small width. We will use what we call Dragging Lemma, a technique developped by Collin, Hauswirth and Rosenberg in [3,4], to prove that the end is a horizontal multigraph, hence has uniform bounded curvature at infinity using stability or Lemma 1. Then we will prove that the property on the boundary at infinity implies that the ends have finite total curvature.…”