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.
The Bergman kernel for circular multiply connected domains.Pacific Journal of Mathematics, 233(1):145–157, 2007
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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.