Slab Theorem and Halfspace Theorem for constant mean curvature surfaces in $\mathbb H^2\times\mathbb R$

Laurent Hauswirth; Ana Menezes; Magdalena Rodriguez

Slab Theorem and Halfspace Theorem for constant mean curvature surfaces in $\mathbb H^2\times\mathbb R$

Differential Geometry 2022-07-28 v2

Authors: Laurent Hauswirth , Ana Menezes , Magdalena Rodriguez

Abstract

We prove that a properly embedded annular end of a surface in $\mathbb H^2\times\mathbb R$ with constant mean curvature $0<H\leq \frac{1}{2}$ can not be contained in any horizontal slab. Moreover, we show that a properly embedded surface with constant mean curvature $0<H\leq \frac{1}{2}$ contained in $\mathbb H^2\times[0,+\infty)$ and with finite topology is necessarily a graph over a simply connected domain of $\mathbb H^2$ . For the case $H=\frac{1}{2}$ , the graph is entire.

Keywords

constant mean curvature slab theorem halfspace theorem embedded surfaces hyperbolic space graphs

Cite

@article{arxiv.2105.11428,
  title  = {Slab Theorem and Halfspace Theorem for constant mean curvature surfaces in $\mathbb H^2\times\mathbb R$},
  author = {Laurent Hauswirth and Ana Menezes and Magdalena Rodriguez},
  journal= {arXiv preprint arXiv:2105.11428},
  year   = {2022}
}

Comments

Final version. Accepted for publication in Revista Matematica Iberoamericana

Slab Theorem and Halfspace Theorem for constant mean curvature surfaces in $\mathbb H^2\times\mathbb R$

Abstract

Keywords

Cite

Comments

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