Lipschitz Smoothings of Polyhedral Manifolds

TL;DR

This paper establishes that polyhedral manifolds in dimensions up to 4 with bounded geometry can be bi-Lipschitz equivalent to smooth Riemannian manifolds, with bounds on geometric properties explicitly related.

Contribution

It provides a converse to Bowditch's theorem in low dimensions, linking polyhedral and smooth geometries with explicit bounds.

Findings

Polyhedral manifolds in dimensions ≤4 are bi-Lipschitz homeomorphic to Riemannian manifolds.

Bounds on curvature, injectivity radius, and bi-Lipschitz constants are explicitly related.

The results extend the understanding of geometric smoothing of polyhedral manifolds in low dimensions.

Abstract

We use a recent result of C. Lange to obtain a converse to a theorem of B. Bowditch in dimension at most $4$. In particular, we show that, for $n \leq 4$, a polyhedral $n$-manifold $X$ with bounded geometry is $K$-bi-Lipschitz homeomorphic to a Riemannian manifold $M$. We bound the constant $K$, the curvature, and the injectivity radius of $M$ by the bounds on the geometry of $X$.