ℝ-equivariant minimal Lagrangian surfaces in the complex quadric Q2 and the complex hyperbolic quadric Q2*
Abstract
We study global and variational properties of ℝ-equivariant minimal Lagrangian surfaces in the complex quadric Q2 and the complex hyperbolic quadric Q2* arising from loop group potentials. In the Q2* case, we work under the timelike-axis condition throughout the associated 𝕊1-family. Excluding the flat case in Q2, we give a necessary and sufficient criterion for closing in the y-direction: a member fλ0 closes if and only if the spectral ratio μ2/μ1 is a positive rational number. The closing parameters form a dense countable subset of 𝕊1, while the non-closing parameters are dense and have full arc-length measure. The closing members yield cylindrical quotients in Q2 and annular quotients in Q2*. Apart from the flat case in Q2, the surfaces under consideration have infinite absolute total curvature. The equivariant surfaces considered here in Q2 are unstable, whereas minimal Lagrangian surfaces in Q2* are stable. Finally, for compact cylindrical domains of the y-closing surfaces in Q2, we obtain explicit sufficient width conditions for Hamiltonian stability and instability under variations fixed near the boundary. We also establish the existence of a finite geometric critical width and determine it explicitly in the flat case.
Related
- Minimal Lagrangian surfaces in the two-dimensional complex hyperbolic quadric via the loop group method
- Minimal surfaces: A Lagrangian derivation of first and second variations
- Ramification and Finiteness of Equivariant Minimal Surfaces
- Equivariant isometric immersions of surfaces in hyperbolic space
- Hamiltonian stationary Lagrangian surfaces of constant Gaussian curvature in complex space forms
BibTeX
@misc{zeng2026ℝequivariant,
title = {{$\mathbbR$-equivariant minimal Lagrangian surfaces in the complex quadric $Q_2$ and the complex hyperbolic quadric $Q_2^*$}},
author = {Sihao Zeng},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09555},
eprint = {2610.09555},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09555},
}
RIS
TY - UNPB
TI - ℝ-equivariant minimal Lagrangian surfaces in the complex quadric Q_2 and the complex hyperbolic quadric Q_2^*
AU - Sihao Zeng
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.09555
UR - https://arxiv.org/abs/2610.09555
AB - We study global and variational properties of ℝ-equivariant minimal Lagrangian surfaces in the complex quadric Q_2 and the complex hyperbolic quadric Q_2^* arising from loop group potentials. In the Q_2^* case, we work under the timelike-axis condition throughout the associated 𝕊^1-family. Excluding the flat case in Q_2, we give a necessary and sufficient criterion for closing in the y-direction: a member f^λ_0 closes if and only if the spectral ratio μ_2/μ_1 is a positive rational number. The closing parameters form a dense countable subset of 𝕊^1, while the non-closing parameters are dense and have full arc-length measure. The closing members yield cylindrical quotients in Q_2 and annular quotients in Q_2^*. Apart from the flat case in Q_2, the surfaces under consideration have infinite absolute total curvature. The equivariant surfaces considered here in Q_2 are unstable, whereas minimal Lagrangian surfaces in Q_2^* are stable. Finally, for compact cylindrical domains of the y-closing surfaces in Q_2, we obtain explicit sufficient width conditions for Hamiltonian stability and instability under variations fixed near the boundary. We also establish the existence of a finite geometric critical width and determine it explicitly in the flat case.
ER -