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Minimal Surfaces in ℝ4 with Three Planar Ends II: Symmetry and Nonexistence
Abstract
In this paper, we study complete minimal surfaces in ℝ4 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≥ 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.
Related
- Free Boundary Minimal surfaces in the Euclidean Three-Ball close to the boundary
- Embedded minimal surfaces with three ends associated with the Hoffman-Osserman catenoids
- Topology, Morse Index, and Area of Minimal Surfaces in Three-Manifolds
- Entire area-minimizing surfaces in Rn of density two are quadratic or planar
- Inequivalent complexity criteria for free boundary minimal surfaces
BibTeX
@misc{lee2026minimal,
title = {{Minimal Surfaces in $\mathbbR^4$ with Three Planar Ends II: Symmetry and Nonexistence}},
author = {Jaehoon Lee and Eungbeom Yeon},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32111},
eprint = {2610.32111},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32111},
}
RIS
TY - UNPB
TI - Minimal Surfaces in ℝ^4 with Three Planar Ends II: Symmetry and Nonexistence
AU - Jaehoon Lee
AU - Eungbeom Yeon
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32111
UR - https://arxiv.org/abs/2610.32111
AB - In this paper, we study complete minimal surfaces in 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≥ 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.
ER -