This work establishes a bilateral coercivity estimate for the Jacobi operator in the conformal class of compact hyperbolic surfaces, providing a quantitative rigidity result in geometric analysis. Let (M, g₀) be a compact Riemann surface of genus γ ≥ 2 with hyperbolic metric satisfying K₆䃐 = −κ₀, κ₀ > 0. Consider conformal perturbations g = e^ (2ψ) g₀ with ψ ∈ H² (M) and ∫M ψ dμ₆䃐 = 0. The curvature deviation functional is defined by η (g) = ∫M (Kg + κ₀) ² dμg. At the linearized level, the structure is governed by the Jacobi operator J = −Δ₆䃐 + 2κ₀, with associated quadratic functional F (ψ) = ||Jψ||²₋ℂ. In the perturbative regime, the main result establishes the bilateral estimate: c · ||ψ||²₇ℂ ≤ η (g) ≤ C · ||ψ||²₇ℂ, with constants depending only on the background geometry. The rigidity mechanism is entirely determined by the spectral gap λ₁ (J) = μ₁ + 2κ₀ > 0, where μ₁ is the first positive eigenvalue of the Laplace–Beltrami operator. This condition is both necessary and sufficient for the validity of the estimate at the local level. The work distinguishes explicitly between the operator J (order 2), arising from linearization, and the operator A = J² (order 4), governing the Hessian D²η = 2J². The functional η (g) is shown to be equivalent to the squared graph norm of J, providing a complete characterization of the quadratic structure of conformal deformations. The paper also introduces an abstract variational framework for functionals of the form F (u) = ||R (u) ||², showing that the associated quadratic behavior is fully determined by the operator A = L*L, where L = DR (u₀). The analysis relies on elliptic regularity, Sobolev embeddings, and spectral theory. The result is strictly local and does not address global nonlinear behavior. Author: Mario César Garms Thimoteo Email: mariothimoteo@hotmail. com DOI (Zenodo): 10. 5281/zenodo. 19359283
Mário César Garms Thimoteo (Fri,) studied this question.