  We investigate CAT(0) metric spaces whose associated Tits boundary is
compact. Prominent examples of such spaces are of course the euclidean ones.
However there exist non trivial geodesically complete CAT(0) spaces with
compact Tits boundary that are far from being flat. We show that if the
isometry group of such a space acts cocompactly, then the space is "as flat as
possible", i.e. the boundary minimal subspace must be flat. Our result relies
on the existence of hyperbolic isometries, which is a problem itself. We hence
add the proof of the following existence proposition: a cocompact CAT(0) space
either admits a hyperbolic isometry or a convex embedded flat plane.

Metric Geometry