Stability of non-proper functions

TL;DR

This paper establishes conditions under which non-proper smooth functions, especially Morse and Nash functions, are stable or strongly stable, highlighting the roles of end-triviality and quasi-properness in their stability properties.

Contribution

It provides new criteria for the stability and strong stability of non-proper functions, including Morse and Nash functions, based on end-triviality and quasi-properness.

Findings

Morse functions are stable if end-trivial at discriminant points.

Strong stability is characterized by quasi-properness of Morse functions.

Any Nash function becomes stable after a generic linear perturbation.

Abstract

The purpose of this paper is to give a sufficient condition for (strong) stability of non-proper smooth functions (with respect to the Whitney $C^\infty$-topology). We show that a Morse function is stable if it is end-trivial at any point in its discriminant, where end-triviality (which is also called local triviality at infinity) is a property concerning behavior of functions around the ends of the source manifolds. We further show that a Morse function $f:N\to \mathbb{R}$ is strongly stable (i.e. there exists a continuous mapping $g\mapsto (\Phi_g,\phi_g)\in\operatorname{Diff}(N)\times \operatorname{Diff}(\mathbb{R})$ such that $\phi_g\circ g\circ \Phi_g =f$ for any $g$ close to $f$) if (and only if) $f$ is quasi-proper. This result yields existence of a strongly stable but not infinitesimally stable function. Applying our result on stability, we give a reasonable sufficient condition for stability of Nash functions, and show that any Nash function becomes stable after a generic linear perturbation.