Bilateral Coercivity and Dirichlet Rigidity via the Spectral Gap of the Linearized Curvature Operator

This work establishes a quantitative rigidity result for conformal metrics under Dirichlet boundary conditions, based on the spectral properties of the associated linearized curvature operator. Let (M, g₀) be a compact Riemannian surface with boundary, equipped with a reference metric of constant curvature K₆䃐 = −κ₀, κ₀ > 0. Consider conformal perturbations of the form: g = e^ (2ψ) g₀, with ψ ∈ H² (M) satisfying Dirichlet boundary condition: ψ|∂₌ = 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 operator: L₆䃐 = −Δ₆䃐 + 2κ₀, with associated quadratic functional: F (ψ) = ||L₆䃐 ψ||²₋ℂ. In the perturbative regime, the main result establishes the bilateral estimate: c · ||ψ||²₇ℂ ≤ η (g) ≤ C · ||ψ||²₇ℂ, where c, C > 0 depend only on the background geometry and boundary conditions. The rigidity mechanism is fully determined by the spectral gap: λ₁ (L₆䃐) > 0, ensured by the Dirichlet condition, which removes the kernel and guarantees strict coercivity. The second variation satisfies: D²η = 2L₆䃐², showing that the quadratic structure is governed by a fourth-order positive operator. The functional η (g) is equivalent to the squared graph norm of L₆䃐, providing a complete characterization of the linearized geometry under boundary constraints. The analysis relies on elliptic regularity, Sobolev estimates, and spectral theory of self-adjoint operators. The result is strictly local and does not address global nonlinear rigidity. Author: Mario César Garms Thimoteo Email: mariothimoteo@hotmail. com DOI (Zenodo): 10. 5281/zenodo. 19359675