Can every Lie group be realized as conformal group of smooth manifolds

Question

I was reading a paper

Saerens, Rita; Zame, William R., The isometry groups of manifolds and the automorphism groups of domains, Trans. Am. Math. Soc. 301, 413-429 (1987). ZBL0621.32025.

Here they showed,

We prove that every compact Lie group can be realized as the (full) automorphism group of a strictly pseudoconvex domain and as the (full) isometry group of a compact, connected, smooth Riemannian manifold.

Now my question is:

Are their any papers or theorems which shows that every lie group can also be realized as the full Conformal Group of some smooth manifold.

Answer 1

As I mentioned in comments, I am not very knowledgable in this field. As far as I know the conformal group of a smooth manifold has been studied extensively in various contexts, and some specific cases are known.

1. Conformal group of n-dimensional Euclidean space:
   Lee, J. M. (2006). Introduction to Smooth Manifolds
2. Conformal group of Minkowski space and Anti-de Sitter space:
   Fulton, T., Rohrlich, F., & Witten, L. (1962). Conformal Invariance in Physics
   Bekaert, X., & Boulanger, N. (2006). The unitary representations of the Poincaré group in any spacetime dimension
   Barut, A. O., & Raczka, R. (1980). Theory of Group Representations and Applications
   Avis, S. J., Isham, C. J., & Storey, D. (1978). Quantum field theory in anti-de Sitter space-time
3. Conformal groups of compact, simply connected, and positively curved Riemannian manifolds:
   Obata, M. (1971). The conjectures on conformal transformations of Riemannian manifolds
   Ejiri, N. (1981). A negative answer to a conjecture of conformal transformations of Riemannian manifolds
   Gursky, M. J., & Viaclovsky, J. A. (2001). A new variational characterization of three-dimensional space forms

Answer 2

First of all, one cannot talk about conformal transformations of a smooth manifold: For the notion of conformality to be defined you need an extra structure besides a smooth atlas. I will work with the notion of conformality in Riemannian geometry.

Definition. Suppose that $(,ℎ)$ is a connected Riemannian manifold and $=\operatorname{Conf}(,ℎ)$, the full group of conformal automorphisms of $(,ℎ)$. The group $G$ is called inessential if there is a positive function $\rho$ on $M$ such that the group $G$ acts isometrically on the Riemannian manifold $(M, \rho h)$.

Lichnerowicz's conjectured and Ferrand proved in

Ferrand, Jacqueline, The action of conformal transformations on a Riemannian manifold, Math. Ann. 304, No. 2, 277-291 (1996). ZBL0866.53027.

that apart from the Riemannian manifolds conformal to the Euclidean space (with the flat metric) and the round sphere, all Riemannian manifolds have inessential groups of conformal transformations. Thus, apart from the two exceptions, the story reduces to the realization of the given noncompact Lie group as the full isometry group of a Riemannian manifold. However, it appears that this question is still unanswered, see an incomplete Mathoverflow discussion here.