Tony Liimatainen

In this paper we give a natural condition for when a volumorphism on a Riemannian manifold $(M,g)$ is actually an isometry with respect to some other, optimal, Riemannian metric $h$. We consider the natural action of volumorphisms on the space $\M_\mu^s$ of all Riemannian metrics of Sobolev class $H^s$, $s>n/2$, with a fixed volume form $\mu$. An optimal Riemannian metric, for a given volumorphism, is a fixed point of this action in a certain complete metric space containing $\M_\mu^s$ as an isometrically embedded subset. We show that a fixed point exists if the orbit of the action is bounded. We also generalize a mean ergodic theorem and a fixed point theorem to the nonlinear setting of complete global Alexandrov nonpositive curvature spaces.

In this note we prove that a generic Riemannian manifold of dimension $\geq 3$ does not admit any nontrivial local conformal diffeomorphisms. This is a conformal analogue of a result of Sunada concerning local isometries, and makes precise the principle that generic manifolds in high dimensions do not have conformal symmetries. Consequently, generic manifolds of dimension $\geq 3$ do not admit nontrivial conformal Killing vector fields near any point. As an application to the inverse problem of Calderón on manifolds, this implies that generic manifolds of dimension $\geq 3$ do not admit limiting Carleman weights near any point.