Annular solutions to the partitioning problem in a ball

TL;DR

This paper constructs a family of constant mean curvature annuli with free boundary in the unit ball, demonstrating non-uniqueness and symmetry properties in the partitioning problem.

Contribution

It introduces a new family of embedded CMC annuli with prismatic symmetry, providing a negative answer to the uniqueness question for annular solutions.

Findings

Constructed a one-parameter family of CMC annuli in the ball.

Demonstrated non-uniqueness of annular solutions.

Showed these solutions have specific prismatic symmetry groups.

Abstract

For any $n\in \mathbb{N}$, $n\geq 2$, we construct a real analytic, one-parameter family of compact embedded CMC annuli with free boundary in the unit ball $\mathbb{B}^3$ of $\mathbb{R}^3$ with a prismatic symmetry group of order $4n$. These examples give a negative answer to the uniqueness problem by Nitsche and Wente of whether any annular solution to the partitioning problem in the ball should be rotational.