A geometric framework for integrable arc-length-preserving flows of spacelike curves in the Anti-de Sitter space ($ \mathrm{AdS}_3 $) was constructed. In the Frenet–Serret formalism, the constant negative background curvature added the position-vector term $ \boldsymbol{\gamma} $ to the structural equation, $ \partial_s T = \kappa N_1 + \boldsymbol{\gamma} $. Via a zero-curvature representation, this yielded local evolution equations for the curvature $ \kappa $ and torsion $ \tau $: the Betchov–Da Rios system for non-planar curves and a modified Korteweg-de Vries (mKdV) equation for planar ones. Globally, the monodromy matrix provided a topological invariant for closed loops. For helices of constant curvature with characteristic frequencies $ \omega_1, \omega_2 $, closure imposed the quantization conditions
$ \omega_1 L = 2\pi m, \qquad \omega_2 L = 2\pi n \qquad (m, n\in\mathbb Z), $
which were preserved under the integrable flows and labeled the disconnected components of the moduli space. This monodromy invariant directly connected the geometry of spacelike curves in $ \mathrm{AdS}_3 $ to integrable systems and holographic observables.
Citation: Yanyan Li. Integrable flows and global closure properties of spacelike curves in the Anti-de Sitter space $ \mathrm{AdS}_3 $[J]. AIMS Mathematics, 2026, 11(8): 25795-25813. doi: 10.3934/math.20261033
A geometric framework for integrable arc-length-preserving flows of spacelike curves in the Anti-de Sitter space ($ \mathrm{AdS}_3 $) was constructed. In the Frenet–Serret formalism, the constant negative background curvature added the position-vector term $ \boldsymbol{\gamma} $ to the structural equation, $ \partial_s T = \kappa N_1 + \boldsymbol{\gamma} $. Via a zero-curvature representation, this yielded local evolution equations for the curvature $ \kappa $ and torsion $ \tau $: the Betchov–Da Rios system for non-planar curves and a modified Korteweg-de Vries (mKdV) equation for planar ones. Globally, the monodromy matrix provided a topological invariant for closed loops. For helices of constant curvature with characteristic frequencies $ \omega_1, \omega_2 $, closure imposed the quantization conditions
$ \omega_1 L = 2\pi m, \qquad \omega_2 L = 2\pi n \qquad (m, n\in\mathbb Z), $
which were preserved under the integrable flows and labeled the disconnected components of the moduli space. This monodromy invariant directly connected the geometry of spacelike curves in $ \mathrm{AdS}_3 $ to integrable systems and holographic observables.
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Yanyan Li. Integrable flows and global closure properties of spacelike curves in the Anti-de Sitter space $ \mathrm{AdS}_3 $[J]. AIMS Mathematics, 2026, 11(8): 25795-25813. doi: 10.3934/math.20261033
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