Rigidity of conformal functionals on spheres

In this paper we investigate the nature of stationary points of functionals on the space of Riemannian metrics on a smooth compact manifold. Special cases are spectral invariants associated with Laplace or Dirac operators such as functional determinants, and the total Q-curvature. When the functional is invariant under conformal changes of the metric, and the manifold is the standard n-sphere, we apply methods from representation theory to give a universal form of the Hessian of the functional at a stationary point. This reveals a very strong rigidity in the local structure of any such functional. As a corollary this gives a new proof of the results of K. Okikiolu (Ann. Math., 2001) on local maxima and minima for the determinant of the conformal Laplacian, and we obtain results of the same type in general examples.