Let $M$ be a smooth compact $n$-manifold without boundary, $g$ some choice of Riemannian metric on $M$, and $\omega_g$ the volume form gotten from $g$. Say you're interested in finding extrema for quantities determined by a choice of Riemannian metric on $M$ . . . perhaps $\lambda_1(\int_M \omega_g)^{2/n}$, where $\lambda_1$ is the first nonzero eigenvalue of a Laplace-type operator, or maybe the zeta-regularized determinant of the conformal Laplacian.

The setting for this variational problem is the space of Riemannian metrics $\mathcal{M}$ on $M$. This is a tame Frechet manifold and is an open convex cone inside the tame Frechet space $\Gamma^\infty\!(S^2T^\ast\!M)$ of symmetric covariant 2-tensors on $M$ (Hamilton, 1982). These objects are standard in conformal geometry. In particular, if $M=S^{n}$, then $\Gamma^\infty\!(S^2T^\ast\!M)$ is the space of sections of a vector bundle associated to a representation of the conformal group of $M$, and variational problems can be addressed via the action of the conformal group (Møller & Ørsted, 2009).

My concern is whether there is an analogue of this picture for strongly pseudoconvex CR manifolds, in particular for odd-dimensional spheres. What is the tangent bundle of the space of strongly pseudoconvex CR structures? What is the tangent bundle of the space of *all* CR structures? Is the space of strongly pseudoconvex CR structures on the sphere a tame Frechet manifold sitting as an open convex cone inside some tame Frechet space of sections of some vector bundle over $S^{2n+1}$? Is this vector bundle associated to a representation of a Lie group?