Mollifier smoothing of $C^0$-Finsler structures

TL;DR

This paper introduces a mollifier smoothing technique for $C^0$-Finsler structures, enabling the approximation of these structures by smooth Finsler metrics while preserving key geometric properties.

Contribution

It develops a convolution-based smoothing method for $C^0$-Finsler structures and proves the convergence of important geometric connections and curvature tensors.

Findings

The smoothing $F_ ext{ε}$ converges uniformly to $F$ on compact sets.

Connections and curvature tensors of $F_ ext{ε}$ converge to those of $F$.

Application to piecewise smooth Riemannian and Finsler manifolds with nonzero curvature.

Abstract

A $C^0$-Finsler structure is a continuous function $F:TM \rightarrow [0,\infty)$ defined on the tangent bundle of a differentiable manifold $M$ such that its restriction to each tangent space is an asymmetric norm. We use the convolution of $F$ with the standard mollifier in order to construct a mollifier smoothing of $F$, which is a one parameter family of Finsler structures $F_\varepsilon$ (of class $\mathit{C}^\infty$ on $TM\backslash 0$) that converges uniformly to $F$ on compact subsets of $TM$. We prove that when $F$ is a Finsler structure, then the Chern connection, the Cartan connection, the Hashiguchi connection, the Berwald connection and the flag curvature of $F_\varepsilon$ converges uniformly on compact subsets to the corresponding objects of $F$. As an application of this mollifier smoothing, we study examples of two-dimensional piecewise smooth Riemannian manifolds with nonzero total curvature on a line segment. We also indicate how to extend this study to the correspondent piecewise smooth Finsler manifolds.