arXiv:2607.00810 2026 Geometry 1 ideas extracted · analyzed Aug 30, 2026
The paper provides a variational construction for codimension-two objects by minimizing a fractional Sobolev seminorm over sphere-valued maps subject to an exact Jacobian-current constraint. The transferable asset is the combination of nonlocal smoothness and topological defect control: the representation can remain smooth away from prescribed codimension-two defects while preserving integer winding structure. A practical neural adaptation is a topology-aware implicit field or feature map whose fractional spectral energy is regularized while its discrete Jacobian or winding current is matched to a target. This is most promising for neural implicit geometry, segmentation, and generative fields where ordinary total-variation or gradient penalties erase holes, filaments, and vortex-like structures.
Train a neural field to represent a sphere-valued phase or feature map with a prescribed codimension-two defect set. Add a fractional Sobolev energy to suppress high-frequency oscillations, but enforce topology through a discrete Jacobian or winding-current loss so that smoothing cannot remove holes, filaments, or vortex defects.