×

The Morse-Sard-Brown theorem for functionals on bounded Fréchet-Finsler manifolds. (English) Zbl 1338.58027

Summary: In this paper we study Lipschitz-Fredholm vector fields on bounded Fréchet-Finsler manifolds. In this context we generalize the Morse-Sard-Brown theorem, asserting that if \(M\) is a connected smooth bounded Fréchet-Finsler manifold endowed with a connection \(\mathcal{K}\) and if \(\xi\) is a smooth Lipschitz-Fredholm vector field on \(M\) with respect to \(\mathcal{K}\) which satisfies condition (WCV), then, for any smooth functional \(l\) on \(M\) which is associated to \(\xi\), the set of the critical values of \(l\) is of first category in \(\mathbb{R}\). Therefore, the set of the regular values of \(l\) is a residual Baire subset of \(\mathbb{R}\).

MSC:

58K05 Critical points of functions and mappings on manifolds
58B20 Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds
58B15 Fredholm structures on infinite-dimensional manifolds
PDFBibTeX XMLCite
Full Text: arXiv Link