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The aim is to quantify deviations from constant curvature in two-dimensional conformal metrics using Calabi energy and spectral analysis.

February 21, 2026Open Access

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

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The aim is to quantify deviations from constant curvature in two-dimensional conformal metrics using Calabi energy and spectral analysis.
Developed a geometric-spectral framework for analyzing conformal metrics
Examined the linearized Jacobi operator under specific spectral conditions
Applied elliptic regularity and nonlinear estimates to derive results
Extended analysis to bounded planar domains with Dirichlet boundary conditions
Proved a bilateral equivalence between curvature variance and the H²-norm of the conformal factor
Demonstrated that intrinsic curvature variance controls the H²-norm of deformations with boundary constraints
Established spectral coercivity as a key factor in transferring rigidity results to geometries with boundaries

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This work establishes a geometric-spectral framework for two-dimensional conformal metrics, indicating a method for quantifying curvature variance.

Mário César Garms Thimoteo (2026) studied this question.

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This work establishes a geometric-spectral framework for two-dimensional conformal metrics, indicating a method for quantifying curvature variance.