Killing fields generated by multiple solutions to the Fischer–Marsden equation II

Abstract

In the process of finding Einstein metrics in dimension [Formula: see text], we can search metrics critical for the scalar curvature among fixed-volume metrics of constant scalar curvature on a closed oriented manifold. This leads to a system of PDEs (which we call the Fischer–Marsden Equation, after a conjecture concerning this system) for scalar functions, involving the linearization of the scalar curvature. The Fischer–Marsden conjecture said that, if the equation admits a solution, the underlying Riemannian manifold is Einstein. Counter-examples are known by Kobayashi and Lafontaine, and by our first paper. Multiple solutions to this system yield Killing vector fields. We showed in our first paper that the dimension of the solution space [Formula: see text] can be at most [Formula: see text], with equality implying that [Formula: see text] is a sphere with constant sectional curvatures. Moreover, we also showed there that the identity component of the isometry group has a factor [Formula: see text]. In this second paper, we apply our results in the first paper to show that either [Formula: see text] is a standard sphere or the dimension of the space of Fischer–Marsden solutions can be at most [Formula: see text].