Communications in Analysis and Geometry

Volume 22 (2014)

Number 1

Deformations of constant mean curvature-1/2 surfaces in $\mathbb{H}^2 \times \mathbb{R}$ with vertical ends at infinity

Pages: 109 – 148

DOI: http://dx.doi.org/10.4310/CAG.2014.v22.n1.a2

Authors

Sébastien Cartier (Laboratoire d’Analyse et de Mathématiques Appliquées (UMR 8050), Université Paris-Est, Créteil, France)

Laurent Hauswirth (Laboratoire d’Analyse et de Mathématiques Appliquées (UMR 8050), Université Paris-Est, Créteil, France)

Abstract

We study constant mean curvature (CMC)-1/2 surfaces in $\mathbb{H}^2 \times \mathbb{R}$ that admit a compactification of the mean curvature operator. We show that a particular family of complete entire graphs over $\mathbb{H}^2$ admits a structure of infinite dimensional manifold with local control on the behaviors at infinity. These graphs also appear to have a half-space property and we deduce a uniqueness result at infinity. Deforming non-degenerate CMC-1/2 annuli, we provide a large class of (non-rotational) examples and construct (possibly embedded) annuli without axis, i.e., with two vertical, asymptotically rotational, non-aligned ends.

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