Non-Tonelli Finsler Geometry of Exotic Superconductivity: Metastable Vortex Phases and Geometric Phase Transitions
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arXiv
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| Format: | Preprint |
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2025
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| author | Fakhri, Y. Alipour |
| author_facet | Fakhri, Y. Alipour |
| contents | We develop a thermally coupled Ginzburg-Landau theory on \emph{Weakly Non-Tonelli (WNT) Finsler manifolds}, extending classical vortex analysis beyond the Tonelli convexity paradigm. The WNT framework weakens global $1$-homogeneity and strict convexity while preserving superlinearity and local ellipticity, enabling a geometric treatment of superconductors whose anisotropic energy landscapes are nonconvex and temperature-dependent. Within this setting, we construct the generalized Legendre correspondence, Hamiltonian metric, and WNT Laplacian, proving existence and sharp Coulomb asymptotics of the three-dimensional Green kernel. We then establish the $Γ$--convergence of the WNT-GL energy and identify metastable vortex filaments minimizing a renormalized geometric functional. Finally, a dynamic $Γ$-limit yields an effective filament flow governed by the WNT Finsler curvature and thermally induced geometric forces, predicting curvature focusing and phase bifurcation at a critical transition temperature $T_c$. This theory unifies convex-analytic, geometric, and physical perspectives, showing that Non-Tonelli Finsler structures form a natural analytic bridge between classical Finsler geometry, anisotropic variational models, and the nonlinear thermodynamics of exotic superconductivity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_12000 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Non-Tonelli Finsler Geometry of Exotic Superconductivity: Metastable Vortex Phases and Geometric Phase Transitions Fakhri, Y. Alipour Mathematical Physics (primary)35J60, 53B40, 82D55, (Secondary) 35K55, 35Q56, 49J45, 58J35, 35B40, 35A15 We develop a thermally coupled Ginzburg-Landau theory on \emph{Weakly Non-Tonelli (WNT) Finsler manifolds}, extending classical vortex analysis beyond the Tonelli convexity paradigm. The WNT framework weakens global $1$-homogeneity and strict convexity while preserving superlinearity and local ellipticity, enabling a geometric treatment of superconductors whose anisotropic energy landscapes are nonconvex and temperature-dependent. Within this setting, we construct the generalized Legendre correspondence, Hamiltonian metric, and WNT Laplacian, proving existence and sharp Coulomb asymptotics of the three-dimensional Green kernel. We then establish the $Γ$--convergence of the WNT-GL energy and identify metastable vortex filaments minimizing a renormalized geometric functional. Finally, a dynamic $Γ$-limit yields an effective filament flow governed by the WNT Finsler curvature and thermally induced geometric forces, predicting curvature focusing and phase bifurcation at a critical transition temperature $T_c$. This theory unifies convex-analytic, geometric, and physical perspectives, showing that Non-Tonelli Finsler structures form a natural analytic bridge between classical Finsler geometry, anisotropic variational models, and the nonlinear thermodynamics of exotic superconductivity. |
| title | Non-Tonelli Finsler Geometry of Exotic Superconductivity: Metastable Vortex Phases and Geometric Phase Transitions |
| topic | Mathematical Physics (primary)35J60, 53B40, 82D55, (Secondary) 35K55, 35Q56, 49J45, 58J35, 35B40, 35A15 |
| url | https://arxiv.org/abs/2512.12000 |