The Morse-Sard-Brown Theorem for Functionals on Bounded-Fr\'{e}chet-Finsler Manifolds

TL;DR

This paper extends the Morse-Sard-Brown theorem to Lipschitz-Fredholm vector fields on bounded Fréchet-Finsler manifolds, showing that critical values are rare and regular values form a large subset.

Contribution

It generalizes the Morse-Sard-Brown theorem to a broader class of infinite-dimensional manifolds with Lipschitz-Fredholm vector fields.

Findings

Critical values form a first category set in 

Regular values constitute a residual Baire subset of 

The theorem applies to smooth functionals associated with Lipschitz-Fredholm vector fields

Abstract

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