Minimal Surfaces in ${\mathbb{R}}^4$ with Three Planar Ends I: Holomorphicity and Uniqueness

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 show that any complete, oriented, immersed minimal surface in ${\mathbb{R}}^4$ with finite total curvature and genus $1$, and having these three ends, must be $J$-holomorphic for some almost complex structure $J$. If the surface is also embedded and has at least $8$ symmetries, we prove that its symmetry group has order $8$ or $12$. In each case the surface is uniquely determined up to scaling and rigid motion.

arXiv comment41 pages; The original manuscript has been divided into two separate papers. This version contains Part I