Relation Between Ginzburg–Landau Stability and Renormalized-Energy Hessians

Determine how the stability and Morse index of Ginzburg–Landau solutions relate to the Hessian of the renormalized energy, including the effects of degenerate configurations and higher-degree vortex cores.

Background

The paper derives the critical-point condition for the limiting interior vortex configuration but does not analyze the second variation or dynamical/stability properties of the corresponding Ginzburg–Landau solutions.

The open question concerns whether the Hessian of the renormalized energy governs the stability and Morse index of the finite- parameter solutions, and whether degenerate configurations or higher-degree vortices introduce additional solvability conditions.

References

This would also raise the question of how the stability and Morse index of the GL solutions relate to the Hessian of the renormalized energy, while degenerate configurations and higher-degree vortex cores may require additional solvability conditions.

— The Asymptotic Theory of Ginzburg--Landau Critical Points on Domains with Corners  (2610.01410 - Han et al., 1 Oct 2026) in Section Conclusion and outlook