Geometric–Spectral Quantitative Rigidity of Two-Dimensional Conformal Metrics via Calabi Energy and Jacobi Operator Coercivity

A9 — Calabi Energy and Quantitative Rigidity in Two-Dimensional Conformal Geometry This work establishes a geometric–spectral framework for two-dimensional conformal metrics in which deviations from constant curvature are quantified by the Calabi energy restricted to a conformal class. The main result proves a bilateral equivalence between curvature variance and the H²-norm of the conformal factor, derived from the coercivity of the linearized Jacobi operator under the spectral condition λ₁(J) > 0. The analysis combines elliptic regularity, nonlinear estimates, and spectral characterization of the optimal constant, providing an intrinsic quantitative stability criterion relevant to conformal uniformization, graded-index optics, and hyperbolic geometric models in two dimensions. C1 — Quantitative Rigidity of Conformal Metrics on Bounded Planar Domains with Boundary This article extends quantitative rigidity estimates for two-dimensional conformal metrics to bounded planar domains with Dirichlet boundary conditions. For conformal perturbations g = e²ψg₀ with ψ ∈ H²∩H¹₀, it is shown that intrinsic curvature variance bilaterally controls the H²-norm of the deformation, with constants depending only on the domain and the reference geometry. The core mechanism is the spectral coercivity of the linearized operator L = −Δg₀ + 2κ₀ ensured by the first Dirichlet eigenvalue, enabling the transfer of rigidity results from compact surfaces to geometries with boundary and supporting applications in GRIN optics, truncated hyperbolic disks, and computational uniformization.