Stability of measures on Kähler manifolds

Let (M, ω) be a Kähler manifold and let K be a compact
group that acts on M in a Hamiltonian fashion. We study
the action of K C on probability measures on M . First of all
we identify an abstract setting for the momentum mapping
and give numerical criteria for stability, semi-stability and
polystability. Next we apply this setting to the action of K C
on measures. We get various stability criteria for measures on
Kähler manifolds. The same circle of ideas gives a very general
surjectivity result for a map originally studied by Hersch and
Bourguignon–Li–Yau.