The mixed Hessian develops exactly one logarithmically diverging eigenvalue per symmetry block at the analytic threshold zeta_c, with the remaining spectrum bounded, and scalar Gram functions continue regularly past the geometric threshold.
Fredholm eigenvalue for a quasi-circle and Grunsky functionals.Annales Fennici Mathematici, 35(2):581–593
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Spectral Structure of the Mixed Hessian of the Dispersionless Toda $\tau$-Function
The mixed Hessian develops exactly one logarithmically diverging eigenvalue per symmetry block at the analytic threshold zeta_c, with the remaining spectrum bounded, and scalar Gram functions continue regularly past the geometric threshold.