Persistent Conformal Geometry and Loewner Evolution

We introduce persistent conformal geometry , a framework connecting geometric function theory with topological data analysis. The method assigns to each normalized univalent mapping a scale‐dependent distortion field and studies the persistent homology of its sublevel‐set filtration. The resulting persistence diagrams record the appearance, interaction, and disappearance of coherent regions of conformal nonlinearity across all distortion thresholds. We establish stability estimates that control changes in these diagrams through analytic differences between mappings, and we relate the persistent features to Taylor coefficients, pre‐Schwarzian bounds, and Dirichlet energy. We also control simultaneous changes in the conformal map and observation radius. For Loewner chains whose normalized derivatives are absolutely continuous in , the construction produces a stable time‐dependent continuous family of persistence diagrams; no canonical tracking of individual points is asserted. We also provide a discretization bound and reproducible computations of diagrams, barcodes, bottleneck distances, and mesh refinement. The signatures depend on a chosen conformal base point and observation radius. Synthetic comparisons illustrate information beyond a scalar distortion maximum; applications to inverse boundary problems, morphology, porous media, and shape classification remain prospective rather than validated outcomes of this study.