Asymptotic Cones of Embedded Singular Spaces

TL;DR

This paper extends the concept of asymptotic directions to singular spaces within Riemannian manifolds using geometric measure theory, providing formulas, convergence results, and applications to surface approximation.

Contribution

It introduces asymptotic cones for singular subspaces, generalizing classical notions, with explicit formulas for polyhedra and convergence theorems for approximations.

Findings

Explicit formulas for asymptotic cones of polyhedra in E^3

Convergence of asymptotic cones for sequences approaching smooth submanifolds

Application to approximate asymptotic lines on triangulated surfaces

Abstract

We use geometric measure theory to introduce the notion of asymptotic cones associated with a singular subspace of a Riemannian manifold. This extends the classical notion of asymptotic directions usually defined on smooth submanifolds. We get a simple expression of these cones for polyhedra in E^3, as well as convergence and approximation theorems. In particular, if a sequence of singular spaces tends to a smooth submanifold, the corresponding sequence of asymptotic cones tends to the asymptotic cone of the smooth one for a suitable distance function. Moreover, we apply these results to approximate the asymptotic lines of a smooth surface when the surface is approximated by a triangulation.