Constant mean curvature $1/2$ surfaces in $\mathbb{H}^2\times\mathbb{R}$ asymptotic to the ends of horizontal catenoids

Abstract

This thesis lies in the field of constant mean curvature (cmc) hypersurfaces and specifically cmc $1/2$ surfaces in the three-manifold $\mathbb{H}^2\times\mathbb{R}$. The value $1/2$ is the critical mean curvature for $\mathbb{H}^2\times\mathbb{R}$, in that there do no exist closed cmc surfaces with mean curvature $1/2$ or less. Daniel and Hauswirth have constructed a one-parameter family of complete, cmc $1/2$ annuli that are symmetric about a reflection in the horizontal plane $\mathbb{H}^2 \times \{0\}$, the horizontal catenoids. In this thesis we prove that these catenoids converge to a singular limit of two tangent horocylinders as the neck size tends to zero. We discuss the analytic gluing construction that this fact suggests, which would create a multitude of cmc $1/2$ surfaces with positive genus. The main result of the thesis concerns a key step in such an analytic gluing construction. We construct families of cmc $1/2$ annuli with boundary, whose single end is asymptotic to an end of a horizontal catenoid. We produce these families by solving the mean curvature equation for normal graphs off the end of a horizontal catenoid. This is a non-linear boundary value problem, which we solve by perturbative methods. To do so we analyse the linearised mean curvature operator, known as the Jacobi operator. We show that on carefully chosen weighted H\"older spaces the Jacobi operator can be inverted, modulo a finite-dimensional subspace, and provided the neck size of the horizontal catenoid is sufficiently small. Using these linear results we solve the boundary value problem for the mean curvature equation by a contraction mapping argument.