TY - JOUR
AU - Hayano, Kenta
PY - 2022/06/11
Y2 - 2022/11/29
TI - Stability of non-proper functions
JF - MATHEMATICA SCANDINAVICA
JA - Math. Scand.
VL - 128
IS - 2
SE - Articles
DO - 10.7146/math.scand.a-132211
UR - https://www.mscand.dk/article/view/132211
SP -
AB - <p>The purpose of this paper is to give a sufficient condition for (strong) stability of non-proper smooth functions (with respect to the Whitney 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 is strongly stable if (and only if) it is quasi-proper. This result yields existence of a strongly stable but not infinitesimally stable function. Applying our result on stability, we give a sufficient condition for stability of Nash functions, and show that any Nash function becomes stable after a generic linear perturbation.</p>
ER -