Generalization of Sweeping Nets to Higher-Dimensional Singularities

The sweeping net method has been an effective tool for approximatingsingularities in two-dimensional manifolds. In this paper, we propose aconjecture that extends the sweeping net method to higher-dimensionalmanifolds with isolated singularities. We present a detailed formulation ofthe conjecture, discuss the key challenges in proving it, and explore poten-tial approaches using local coordinate analysis, geometric measure theory,and multi-dimensional generalizations of trigonometric functions. Thiswork aims to open new avenues in the study of singularities on manifoldsand stimulate further research in this area.