Morse theory for symmetric functionals on the sphere and an application to a bifurcation problem

Abstract

Abstract : This paper uses Conley's index to study the cortical points of a function f on a finite dimensional sphere in presence of a symmetry group. The authors prove a theorem which leads to a lower bound on the number of critical points of f when the group is finite, even if the action is not free. This investigation has been motivated by the following bifurcation problem Au = lambda, where A is a variational G-equivariant operator. An estimated on the number of 'branches' bifurcating from an eigenvalue of A'(0) is given. (Author)