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A deformation lemma on a C\(^ 1\) manifold. (Un lemme de déformation sur une sous-variété de classes C\(^ 1\).) (French. Abridged English version) Zbl 0801.57022
Let \(S = \{x \in E \mid g_ 1(x) = 1,\dots,g_ k(x) = 1\}\) where \(g_ i \in C^ 1(E,R)\), 1 is a regular value of \(g_ i\) and where the family of \(k\) submanifolds \(H_ i = \{g_ i = 1\}\) \(i = 1,\dots,k\) is transverse. Using also the numbers \(\| f'/s_ w \| = \inf_{\lambda_ 1,\dots,\lambda_ k\in R}\| f'(x) - \sum^ k_{i = 1} \lambda_ i g_ i' (x)\|\) and \(t_ g (R) = \inf_{\lambda_ 1,\dots,\lambda_ R \in S^{k - 1}} \| \sum^ k_{i = 1} \lambda_ i g_ i'(x)\|\) the author defines a condition similar to the Palais-Smale condition.
Then he proves a theorem which is some kind of deformation lemma, namely:
Let \(S = \{x \in E \mid g_ 1(x) = 1,\dots,g_ k(x) = 1\}\) be a \(C^ 1\) submanifold of a Banach space \(E\), let \(f\in C^ 1(U,R)\), where \(U\) is a neighborhood of \(S\). If \(f|_ S\) and \(g_ i' s\) satisfy the condition above at \(a\) which is a regular value of \(|_ S\), then there exists \(\widehat{\varepsilon} > 0\) such that for all \(0 < \varepsilon < \widehat{\varepsilon}\) there exists a homeomorphism \(h\) of \(S\) onto \(S\) such that
a) \(h(x) = x\) if \(f(x) \not\in [a - \widehat{\varepsilon}, a + \widehat{\varepsilon}]\),
b) \(f(h(x))\leq f(x)\) for all \(x \in S\),
c) \(f(h(x))\leq a - \varepsilon\) for all \(x\) such that \(f(x) \leq a + \varepsilon\),
d) If \(S\) is symmetrical \((S = -S)\) and if \(f\) is even then \(h\) is odd.
The case where \(\dim E < \infty\) is treated in a different way, and it is stronger. The author gives one outline of the proof for the case \(k = 1\) and \(\dim E = \infty\). He uses the integral lines of two pseudo-gradient vector fields for \(f\) and \(g\) in order to build the deformation \(h\). The complete details of the results of the paper appear in Manuscr. Math. 81, No. 3-4, 339-359 (1993)].

57R70 Critical points and critical submanifolds in differential topology
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