A deformation lemma on aC 1 manifold

Abstract

The classical construction of deformations by mean of pseudo-gradient vector fields requires theC1,1 regularity. Here, we are concerned with a deformation lemma for aC1 function on a manifold defined by aC1 functional. We will assume some coupled Palais-Smale conditions between the two functions. The deformation is constructed with the help of integral lines of pseudo-gradient vector fields on a foliation of the manifold. Three different constructions are used for a sub-manifold of codimension 1 in finite dimension, then in infinite dimension and lastly a sub-manifold of any finite codimension in an infinite dimensional Banach space.