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 Kg₀ = −κ₀, κ₀ > 0. Consider conformal perturbations of the form: g = e^(2ψ) g₀, with ψ ∈ H²(M) satisfying Dirichlet boundary condition: ψ|∂M = 0. The curvature deviation functional is defined by: η(g) = ∫_M (K_g + κ₀)² dμ_g. At the linearized level, the structure is governed by the operator: Lg₀ = −Δg₀ + 2κ₀, with associated quadratic functional: F(ψ) = ||Lg₀ ψ||²L². In the perturbative regime, the main result establishes the bilateral estimate: c · ||ψ||²H² ≤ η(g) ≤ C · ||ψ||²H², where c, C > 0 depend only on the background geometry and boundary conditions. The rigidity mechanism is fully determined by the spectral gap: λ₁(Lg₀) > 0, ensured by the Dirichlet condition, which removes the kernel and guarantees strict coercivity. The second variation satisfies: D²η = 2Lg₀², 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 Lg₀, 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