Estimates for the Constant Mean Curvature Dirichlet Problem on Catenoids

In this article, we solve the constant mean curvature dirichlet problem on catenoidal necks with small scale in $\mb{R}^3$. The solutions are found in exponentially weighted H\"older spaces with non-integer weight and are a-priori bounded by a uniform constant times $r^{1 + \gamma}$, where $r$ denotes the distance to the axis of the neck and where $\gamma$ belongs to the interval $(0, 1)$. By comparing the solutions with their limits on the disk, we improve the estimate to $\gamma =1$. As a corollary, we prove differentiability of solutions in $\tau$ down to $\tau = 0$. The surfaces we construct have applications to gluing constructions.