Defining Functions for Unbounded $C^m$ Domains

For a domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$, we introduce the concept of a uniformly $C^m$ defining function. We characterize uniformly $C^m$ defining functions in terms of the signed distance function for the boundary and provide a large class of examples of unbounded domains with uniformly $C^m$ defining functions. Some of our results extend results from the bounded case.