The Canonical Operator in Two-Dimensional Conformal Geometry: Uniqueness, Spectral Shift, and Relative Determinant

This work establishes a precise mathematical characterization of the canonical stability operator associated with curvature energy in two-dimensional conformal geometry. Let (M, g₀) be a compact surface with constant curvature K₆䃐 = −κ₀, κ₀ > 0. Consider conformal metrics: g = e^ (2ψ) g₀. The curvature residual is defined by: R (ψ) = Kg + κ₀, and induces the quadratic energy: η (g) = ||R (ψ) ||². The analysis is restricted to the local quadratic regime near ψ = 0. The first result is an algebraic identification: the curvature residual belongs to the quadratic variational class F (u) = ||R (u) ||², with linearization: L = Jcan = −Δ₆䃐 + 2κ₀. The second variation satisfies: Hess η = 2Jcan², which is a direct instance of the universal structure established for quadratic variational functionals. This step is a verification of consistency and not a new theorem. The second result establishes canonicity. Consider the class of second-order elliptic operators on (M, g₀) of the form: J = −Δ₆䃐 + V, acting on scalar functions, where V is a smooth potential. Within the class of admissible stability operators in dimension two, defined by covariance under conformal perturbations and dependence only on intrinsic geometric data, the operator: Jcan = −Δ₆䃐 + 2κ₀ is uniquely determined. The potential V is entirely fixed by the intrinsic curvature, with no free parameters. The third result develops the relative spectral zeta function associated with a pair of operators: Jcan and J̃ = −Δ₆䃐 + κ₀. The relative zeta function is defined for Re (s) > 0 by: ζ₉₂₀₍ / ₉̃ (s) = Tr (Jcan^ (−s) − J̃^ (−s) ), and admits analytic continuation to s = 0. The functional determinant satisfies the identity: log det_ζ Jcan − log det_ζ J̃ = −ζ'₉₂₀₍ / ₉̃ (0). This provides a complete and intrinsic characterization of the spectral shift between the canonical operator and its baseline counterpart. The analysis relies on elliptic operator theory, spectral theory of self-adjoint operators, and zeta regularization techniques. All results are self-contained and formulated within the local quadratic regime, without reference to external frameworks or physical models. Author: Mario César Garms ThimoteoEmail: mariothimoteo@hotmail. comDOI (Zenodo): 10. 5281/zenodo. 19360299