Smooth Approximation of Lipschitz functions on Riemannian manifolds

Details Smooth Approximation of Lipschitz functions on Riemannian manifolds Smooth Approximation of Lipschitz functions on Riemannian manifolds

2006-02-02

arxiv.org/abs/math/0602051v1

Mathematics

Differential Geometry

10 pages

Scientific paper

We show that for every Lipschitz function $f$ defined on a separable Riemannian manifold $M$ (possibly of infinite dimension), for every continuous $\epsilon:M\to (0,+\infty)$, and for every positive number $r>0$, there exists a $C^\infty$ smooth Lipschitz function $g:M\to\mathbb{R}$ such that $|f(p)-g(p)|\leq\epsilon(p)$ for every $p\in M$ and $\textrm{Lip}(g)\leq\textrm{Lip}(f)+r$. 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.

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