The Conformal Signature of Metric Spaces

This paper introduces the concept of a conformal signature for metric spaces, a novel invariant designed to capture the local and global scaling properties of a space beyond the scope of smooth manifolds. Traditional conformal geometry relies on differentiability, limiting its application to settings like fractal spaces, Carnot groups, or data clouds. The conformal signature is defined as a distribution or measure that characterizes the infinitesimal behavior of the metric under rescaling at almost every point. We develop a methodology based on the theory of quasi-conformal mappings and analysis on metric spaces to construct this signature. The paper demonstrates that for conformally flat spaces like Euclidean space, the signature is trivial, represented by a Dirac delta measure. In contrast, for spaces with complex local geometry, such as the Sierpinski carpet or the Heisenberg group, the signature is a non-trivial measure that reveals their intrinsic, anisotropic scaling structure. We prove that the conformal signature is invariant under quasi-conformal maps, establishing it as a robust geometric invariant. This new tool offers a finer distinction between metric spaces than classical invariants like Hausdorff or conformal dimension, with potential applications in geometric group theory, data analysis, and theoretical physics.