Global H²-Rigidity for a Curvature Residual Functional on Compact Hyperbolic Surfaces

We establish global H²-rigidity for a curvature residual functional on compact hyperbolic surfaces. Let (M, g₀) be a smooth compact two-dimensional manifold with constant negative curvature K (g₀) = −κ₀ < 0, and consider conformal metrics of the form g = exp (2ψ) g₀ with ψ in H² (M) and zero mean. The curvature residual is defined as R (ψ) = K (g) + κ₀, which can be expressed in terms of ψ as R (ψ) = exp (−2ψ) −Δ (g₀) ψ + κ₀ (exp (2ψ) − 1). We study the associated residual quadratic functional η (ψ) = ∫M (K (g) + κ₀) ² dμg, equivalently written as η (ψ) = ∫M exp (−2ψ) |Lψ + N (ψ) |² dμ (g₀), where L = −Δ (g₀) + 2κ₀ and N (ψ) = κ₀ (exp (2ψ) − 1 − 2ψ). The main result proves that η (ψₙ) → 0 implies ψₙ → 0 strongly in H² (M), establishing global rigidity without any smallness assumption. The argument is based on a structural decomposition of the functional into a weighted quadratic term and a nonlinear remainder, combined with uniform control of the mass and exponential integrability of ψ. A key difficulty arises from the endogenous weight exp (−2ψ), which prevents direct comparison between weighted and unweighted norms. This obstruction is resolved through a precise analysis of the interaction between the weight and the operator L, showing that degeneracy of the weight cannot align with low-energy modes. As a consequence, small residual energy enforces strong control of the linear component Lψ in the unweighted L² space, leading to convergence in H² via elliptic regularity. The result provides a complete global stability theory for the curvature residual functional in dimension two and can be interpreted as a quantitative rigidity statement for the Liouville-type curvature equation. More broadly, it illustrates how nonlinear geometric problems can be reduced to spectral control of an associated second-order operator through a residual quadratic framework. This work contributes to the analysis of nonlinear elliptic equations, geometric variational problems, and conformal geometry, and identifies a structural mechanism that may extend to other residual-based functionals in geometric analysis.