On the Existence of Stable Unduloids of Dimension Eight

TL;DR

This paper investigates the stability of n-dimensional unduloids with constant mean curvature, establishing conditions for stability and proving the existence of stable unduloids specifically in dimension eight.

Contribution

It provides a stability criterion based on volume behavior and demonstrates the existence of stable unduloids in eight dimensions for the first time.

Findings

Stability is linked to volume increase or decrease along unduloid families.

Existence of stable unduloids is proven specifically for dimension eight.

Conditions on volume functions determine stability in higher dimensions.

Abstract

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.