Opening nodes on horosphere packings (Q2790710)

In this paper, using different methods, the author provides another proof of a result due to \textit{F. Pacard} and \textit{F. A. A. Pimentel} [J. Inst. Math. Jussieu 3, No. 3, 421--459 (2004; Zbl 1059.53049)]. More precisely, using Bryant's representation of CMC-1 surfaces in the hyperbolic space \(\mathbb{H}^3\), the author shows the following: NEWLINENEWLINENEWLINETheorem. Given a packing of \(n\) horospheres with \(m\) tangency points, there exists a smooth family \((M_s)_{0<s<\varepsilon}\) of complete, embedded CMC-1 surfaces in hyperbolic space \(\mathbb{H}^3\) such that \(M_s\) converges when \(s\to 0\) to the given horosphere packing. The surfaces \(M_s\) have genus \(m-n + 1\) and \(n\) catenoid-cousin-type ends.