Bifurcation and local rigidity of homogeneous solutions to the Yamabe problem on spheres

We study existence and non-existence of constant scalar curvature metrics conformal and arbitrarily close to homogeneous metrics on spheres, using variational techniques. This describes all critical points of the Hilbert-Einstein functional on such conformal classes, near homogeneous metrics. Both bifurcation and local rigidity type phenomena are obtained for 1-parameter families of U(n+1), Sp(n+1) and Spin(9)-homogeneous metrics.