Constant Mean Curvature surfaces with prescribed finite topologies

Abstract

In this article, we construct complete embedded constant mean curvature surfaces in $\mb{R}^3$ with freely prescribed genus and any number of ends greater than or equal to four. Heuristically, the surfaces are obtained by resolving finitely many points of tangency between collections of spheres. The construction relies a family of constant mean curvature surfaces constructed in \cite{Kleene}, constructed as graphs over catenoidal necks of small scale.