Smooth Approximation of Lipschitz Functions on Finsler Manifolds

M I Garrido,Y C Rangel,J A Jaramillo

doi:10.1155/2013/164571

Abstract

We study the smooth approximation of Lipschitz functions on Finsler manifolds, keeping control on the corresponding Lipschitz constants. We prove that, given a Lipschitz functionf:M→ℝdefined on a connected, second countable Finsler manifoldM, for each positive continuous functionε:M→(0,∞)and eachr>0, there exists aC1-smooth Lipschitz functiong:M→ℝsuch that|f(x)-g(x)|≤ε(x), for everyx∈M, andLip(g)≤Lip(f)+r. As a consequence, we derive a completeness criterium in the class of what we call quasi-reversible Finsler manifolds. Finally, considering the normed algebraCb1(M)of allC1functions with bounded derivative on a complete quasi-reversible Finsler manifoldM, we obtain a characterization of algebra isomorphismsT:Cb1(N)→Cb1(M)as composition operators. From this we obtain a variant of Myers-Nakai Theorem in the context of complete reversible Finsler manifolds.

Highlights

There are many geometrically significant functions on a Riemannian manifold which are typically Lipschitz but not smooth, as it is the case, for example, of distance functions
We study the smooth approximation of Lipschitz functions on Finsler manifolds, keeping control on the corresponding Lipschitz constants
It is interesting to study the regularization and smooth approximation of Lipschitz functions on Riemannian manifolds. This has been done in the classical work of Greene and Wu [1], where in particular it is proved that every Lipschitz real function on a Riemannian manifold can be approximated, in the C0-fine topology, by smooth Lipschitz functions whose Lipschitz constants can be made arbitrarily close to the Lipschitz constant of the original function

Summary

Introduction

There are many geometrically significant functions on a Riemannian manifold which are typically Lipschitz but not smooth, as it is the case, for example, of distance functions. This has been done in the classical work of Greene and Wu [1], where in particular it is proved that every Lipschitz real function on a (connected, second countable, and finite dimensional) Riemannian manifold can be approximated, in the C0-fine topology, by smooth Lipschitz functions whose Lipschitz constants can be made arbitrarily close to the Lipschitz constant of the original function This result has been extended in [2] to the case of infinite-dimensional Riemannian manifolds, where some interesting applications are given. We prove in Theorem 8 that every Lipschitz real function on a connected, second countable Finsler manifold can be approximated, in the C0-fine topology, by C1-smooth Lipschitz functions with Lipschitz constants arbitrarily close to the Lipschitz constant of the original function This approximation result has been used in [4] in order to obtain a version of the Myers-Nakai Theorem for reversible Finsler manifolds (that is, in the case that the Finsler structure is absolutely homogeneous). From this we obtain a variant of Myers-Nakai Theorem in the context of complete reversible Finsler manifolds

Preliminaries

Mean Value Inequality

Smooth Approximation of Lipschitz Functions

Quasi-Reversible Manifolds and a Completeness Criterium

Algebras of Differentiable Functions on Finsler Manifolds