On some problems on smooth approximation and smooth extension of Lipschitz functions on Banach-Finsler Manifolds

TL;DR

This paper proves that Lipschitz functions on Riemannian and Finsler manifolds can be uniformly approximated by smooth functions, establishing conditions for such approximations and extensions, with implications for the structure of these manifolds.

Contribution

It introduces new approximation techniques for Lipschitz functions on Banach-Finsler manifolds and characterizes manifolds where these approximations are possible.

Findings

Lipschitz functions can be uniformly approximated by $C^1$-smooth functions with controlled Lipschitz constants.

Every Riemannian manifold is uniformly bumpable.

Conditions are provided for the existence of Lipschitz and $C^1$-smooth extensions of functions on submanifolds.

Abstract

Let us consider a Riemannian manifold $M$ (either separable or non-separable). We prove that, for every $\epsilon>0$, every Lipschitz function $f:M\rightarrow\mathbb R$ can be uniformly approximated by a Lipschitz, $C^1$-smooth function $g$ with $\Lip(g)\le \Lip(f)+\epsilon$. As a consequence, every Riemannian manifold is uniformly bumpable. The results are presented in the context of $C^\ell$ Finsler manifolds modeled on Banach spaces. Sufficient conditions are given on the Finsler manifold $M$ (and the Banach space $X$ where $M$ is modeled), so that every Lipschitz function $f:M\rightarrow \mathbb R$ can be uniformly approximated by a Lipschitz, $C^k$-smooth function $g$ with $\Lip(g)\le C \Lip(f)$ (for some $C$ depending only on $X$). Some applications of these results are also given as well as a characterization, on the separable case, of the class of $C^\ell$ Finsler manifolds satisfying the above property of approximation. Finally, we give sufficient conditions on the $C^1$ Finsler manifold $M$ and $X$, to ensure the existence of Lipschitz and $C^1$-smooth extensions of every real-valued function $f$ defined on a submanifold $N$ of $M$ provided $f$ is $C^1$-smooth on $N$ and Lipschitz with the metric induced by $M$.