Abstract : In the first part of the manuscript we study some compactifications. Given a nonpositively curved symmetric space we search for its differentiable compactifications, that is its embedding into a manifold with boundary where the action of the isometry group extends differentiably up to the boundary. The main results are : the classification of such compactifications of the real hyperbolic space, and the nonexistence of such compactifications for higher rank spaces.
In the second part we are concerned with holomorphic fillings. Given a compact R manifold M and a subgroup F of automorphisms, we ask which compact complex manifolds with boundary X have boundary isomorphic to M and admit a prolongation of the action of F. Under assumption on the convexity and dimension of M and on the size of F, we prove a unicity result (up to blow-up).