Non-Tonelli Finsler Geometry of Exotic Superconductivity: Metastable Vortex Phases and Geometric Phase Transitions

TL;DR

This paper extends Finsler geometry to Weakly Non-Tonelli manifolds to analyze exotic superconductors with nonconvex, temperature-dependent energy landscapes, revealing vortex phase transitions and geometric forces.

Contribution

It introduces a novel WNT Finsler framework for superconductivity, generalizing classical models to nonconvex, thermally influenced energy landscapes and deriving new geometric and variational results.

Findings

Existence of Green kernel with Coulomb asymptotics

Identification of metastable vortex filaments

Prediction of phase bifurcation at critical temperature T_c

Abstract

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 $\Gamma$--convergence of the WNT-GL energy and identify metastable vortex filaments minimizing a renormalized geometric functional. Finally, a dynamic $\Gamma$-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.