Artigo A

This paper studies the local second-order structure of curvature residuals in two-dimensional conformal geometry. For conformal metrics of the form g = e^ (2ψ) g₀ on a compact hyperbolic surface, it proves that the linearization of the curvature residual R (ψ) = Kg + κ₀ is the canonical operator Jcan = −Δg₀ + 2κ₀, and that the Hessian of the squared-residual energy is governed by the closed quadratic form D²η (0) (h, h) = 2 ‖Jcan h‖²L² (M, g₀), with associated self-adjoint operator A = Jcan². The paper also establishes the spectral shift Jcan = J̃ + κ₀ I, where J̃ = −Δg₀ + κ₀ is the Liouville stability operator, and derives the corresponding relative zeta-determinant identity. The results are local, perturbative, and restricted to second order. Author: Mario César Garms Thimoteo E-mail: mariothimoteo@hotmail. com DOI: 10. 5281/zenodo. 19464156