Smooth approximation of Lipschitz functions on Riemannian manifolds

Abstract

We show that for every Lipschitz function f defined on a separable Riemannian manifold M (possibly of infinite dimension), for every continuous ε : M → ( 0 , + ∞ ) , and for every positive number r > 0 , there exists a C ∞ smooth Lipschitz function g : M → R such that | f ( p ) − g ( p ) | ⩽ ε ( p ) for every p ∈ M and Lip ( g ) ⩽ Lip ( f ) + r . Consequently, every separable Riemannian manifold is uniformly bumpable. We also present some applications of this result, such as a general version for separable Riemannian manifolds of Deville–Godefroy–Zizler's smooth variational principle.