Deformations of constant mean curvature 1/2 surfaces in H2xR with vertical ends at infinity

TL;DR

This paper investigates deformations of constant mean curvature 1/2 surfaces in H2xR with vertical ends, establishing a manifold structure, uniqueness at infinity, and constructing new examples of annuli with specific asymptotic properties.

Contribution

It introduces a framework for deforming CMC 1/2 surfaces in H2xR, including a manifold structure and new examples with non-aligned ends, expanding understanding of their geometric properties.

Findings

Complete entire graphs form an infinite dimensional manifold.

Uniqueness results at infinity for these surfaces.

Construction of new non-rotational annuli with vertical ends.

Abstract

We study constant mean curvature 1/2 surfaces in H2xR that admit a compactification of the mean curvature operator. We show that a particular family of complete entire graphs over H2 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 constant mean curvature 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.