A Residual-Quadratic Rigidity Theorem on Compact Hyperbolic Surfaces

We establish local 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 by R (ψ) = K (g) + κ₀, which can be written 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 expressed 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 for every sequence (ψₙ) with η (ψₙ) → 0 and satisfying a uniform smallness condition, one has ψₙ → 0 strongly in H² (M). This establishes local rigidity of the constant-curvature metric within the conformal class. The argument is based on a structural decomposition of the functional into a weighted quadratic term and a nonlinear remainder, combined with quantitative exponential integrability via the sharp Trudinger–Moser–Fontana inequality and elliptic regularity. 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 linear operator L, showing that degeneracy of the weight cannot concentrate along low-energy directions in the small-energy regime. As a consequence, small residual energy enforces control of the linear component Lψ in the unweighted L² space, yielding convergence in H² (M). The result provides a quantitative local stability statement for the curvature residual functional in dimension two. Global coercivity on the full H² space does not hold, due to the presence of concentration phenomena, and lies outside the scope of this work.