Minimal Surfaces in ${\mathbb{R}}^4$ with Three Planar Ends II: Symmetry and Nonexistence

Abstract

In this paper, we study complete minimal surfaces in ${\mathbb{R}}^4$ with three embedded planar ends parallel to those of the union of the Lagrangian catenoid and the plane containing its waist circle. We prove that no such complete embedded minimal surface of genus $g\ge 2$ with finite total curvature admits at least $4(g+1)$ symmetries. This contrasts sharply with the genus-one case, where highly symmetric examples do exist and are uniquely determined up to scaling and rigid motion.

arXiv comment43 pages. This paper is Part II of a manuscript originally posted as arXiv: 2504.02282, which has since been divided into two parts