Timelike surfaces of constant mean and constant Gaussian curvature in anti-de Sitter 3-space via loop groups
Abstract
The loop group description of surfaces whose Gauss map is Lorentz harmonic is known for timelike surfaces of constant mean curvature in the flat Lorentzian 3-space, and for surfaces of constant Gaussian curvature in the 3-sphere. We carry out the corresponding construction for timelike surfaces in anti-de Sitter 3-space ℍ31≅ SL(2,ℝ): those of constant mean curvature, whose Gauss map is harmonic for the conformal structure of the first fundamental form, and those of constant Gaussian curvature K>−1, K≠0, whose Gauss map is then an immersion and is harmonic for the second fundamental form. For both classes we prove the harmonicity characterization, recover the surfaces from extended frames by Sym-Bobenko-type formulas, and solve the geometric Cauchy problem by the generalized DPW method, with potentials written explicitly in terms of the data and no normalization of the Gauss map. We then show that the two constructions differ only by a normalization: after the substitution λ=ζ2 and a constant gauge, a constant mean curvature extended frame F̂ is an extended frame of its Gauss map, and the maps F̂|λ=e4θexp(θe3)F̂|λ=1−1 and e3F̂|λ=e4θe3exp(θe3)F̂|λ=1−1, with e3=diag(−1,1), are timelike surfaces of constant Gaussian curvature 1/sinh2θ and −1/cosh2θ wherever the constant mean curvature surface is not flat. The first is the classical parallel surface, the second its polar surface. Explicit families, with extended frames and surfaces in closed form, illustrate the constructions.
Related
- Hamiltonian stationary Lagrangian surfaces of constant Gaussian curvature in complex space forms
- Biharmonic Lagrangian surfaces with constant Gaussian curvature in $2$-dimensional complex space forms
- Hamiltonian stationary Lagrangian surfaces in complex space forms with constant-length gradient of the mean curvature
- Hopf type theorem for surfaces of constant weighted mean curvature in Riemannian manifolds
- Determinants of Dirichlet and Neumann Laplacians on Constant Curvature Surfaces with Geodesic Polygonal Boundary
BibTeX
@misc{bravogadea2026timelike,
title = {{Timelike surfaces of constant mean and constant Gaussian curvature in anti-de Sitter 3-space via loop groups}},
author = {Jorge Bravo-Gadea},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.33504},
eprint = {2609.33504},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33504},
}
RIS
TY - UNPB
TI - Timelike surfaces of constant mean and constant Gaussian curvature in anti-de Sitter 3-space via loop groups
AU - Jorge Bravo-Gadea
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2609.33504
UR - https://arxiv.org/abs/2609.33504
AB - The loop group description of surfaces whose Gauss map is Lorentz harmonic is known for timelike surfaces of constant mean curvature in the flat Lorentzian 3-space, and for surfaces of constant Gaussian curvature in the 3-sphere. We carry out the corresponding construction for timelike surfaces in anti-de Sitter 3-space ℍ^3_1≅ SL(2,ℝ): those of constant mean curvature, whose Gauss map is harmonic for the conformal structure of the first fundamental form, and those of constant Gaussian curvature K>−1, K≠0, whose Gauss map is then an immersion and is harmonic for the second fundamental form. For both classes we prove the harmonicity characterization, recover the surfaces from extended frames by Sym-Bobenko-type formulas, and solve the geometric Cauchy problem by the generalized DPW method, with potentials written explicitly in terms of the data and no normalization of the Gauss map. We then show that the two constructions differ only by a normalization: after the substitution λ=ζ^2 and a constant gauge, a constant mean curvature extended frame F̂ is an extended frame of its Gauss map, and the maps F̂|_λ=e^4θexp(θe_3)F̂|_λ=1^−1 and e_3F̂|_λ=e^4θe_3exp(θe_3)F̂|_λ=1^−1, with e_3=diag(−1,1), are timelike surfaces of constant Gaussian curvature 1/sinh^2θ and −1/cosh^2θ wherever the constant mean curvature surface is not flat. The first is the classical parallel surface, the second its polar surface. Explicit families, with extended frames and surfaces in closed form, illustrate the constructions.
ER -