  We consider a free energy functional on Cartan-Hadamard manifolds, and
investigate the existence of its global minimizers. The energy functional
consists of two components: an entropy (or internal energy) and an interaction
energy modelled by an attractive potential. The two components have competing
effects, as they favour spreading by linear diffusion and blow-up by nonlocal
attractive interactions, respectively. We find necessary and sufficient
conditions for existence of ground states for manifolds with sectional
curvatures bounded above and below, respectively. In particular, for general
Cartan-Hadamard manifolds, superlinear growth at infinity of the attractive
potential prevents the spreading. The behaviour can be relaxed for homogeneous
manifolds, for which only linear growth of the potential is sufficient for this
purpose.
