Homotopical Stability of Isolated Critical Points of Continuous Functionals

We give a simple proof of the so-called Potential well theorem of Ioffe and Schwartzman, estimating the size of the potential well associated with a local minimum of a continuous functional defined on a complete metric space. Applications to the homotopical stability of an isolated local minimum and to an abstract bifurcation result, as in Ioffe and Schwartzman, are described. We also establish a result on homotopical stability of arbitrary isolated critical points of continuous functionals, thus extending a result of Chang.

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