On the Existence of Stable Unduloids of Dimension Eight

In this paper we study the stability of $n$-dimensional constant mean curvature unduloids embedded in slabs in $\mathbb{R}^{n+1}$. We prove that among the family of half period unduloids stability is determined by whether the volume is increasing or decreasing along this family provided some conditions on the volume function are met. We then use this theorem to prove the existence of stable unduloids of dimension eight.