Complete Minimal Surfaces in $\mathbb{R}^4$ with Three Embedded Planar Ends
Jaehoon Lee, Eungbeom Yeon
TL;DR
This work classifies complete minimal surfaces in $\mathbb{R}^4$ with three embedded planar ends parallel to the Lagrangian catenoid planes, showing genus-1 surfaces must be $J$-holomorphic for some almost complex structure. Under embeddedness and with at least eight symmetries, the authors prove the symmetry count is restricted to $8$ or $12$ and that the surfaces are unique up to rigid motions and scaling; they also derive a nonexistence result for genus $g\ge 2$ with $4(g+1)$ or more symmetries. The analysis blends the generalized Weierstrass representation with Costa–Hoffman–Meeks–style cyclic-cover arguments, exploiting elliptic-function data and symmetry reduction to obtain tight constraints on possible surfaces. A surprising outcome is the appearance of holomorphicity in codimension two under planar-end constraints, contrasting with the $\mathbb{R}^3$ case and guiding future investigations into lower-symmetry regimes and higher-genus obstructions.
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 passing through its waist circle. We show that any complete, oriented, immersed minimal surface in $\mathbb{R}^4$ of finite total curvature with genus $1$ and three such ends must be $J$-holomorphic for some almost complex structure $J$. Under the additional assumptions of embeddedness and at least $8$ symmetries, we prove that the number of symmetries must be either $8$ or $12$, and in each case, the surface is uniquely determined up to rigid motions and scalings. Furthermore, we establish a nonexistence result for genus $g\geq2$ when the surface is embedded and has at least $4(g+1)$ symmetries. Our approach is based on a modification of the method of Costa and Hoffman-Meeks in the setting of $\mathbb{R}^4$, utilizing the generalized Weierstrass representation.