Properties of gradient maps associated with action of real reductive group

Let (Z, omega) be a Kahler manifold and let U be a compact connected Lie group with Lie algebra u acting on Z and preserving omega. We assume that the U -action extends holomorphically to an action of the complexified group U (c) and the U -action on Z is Hamiltonian. Then there exists a U-equivariant momentum map mu : Z -> u. If G subset of U (c) is a closed subgroup such that the Cartan decomposition U (c) = Uexp(p) induces a Cartan decomposition G = Kexp(p), where K = U boolean AND G, p = g boolean AND iu and g = k circle plus p is the Lie algebra of G, there is a corresponding gradient map mu p : Z-iota p. If X is a G -invariant compact and connected real submanifold of Z, we may consider mu p as a mapping mu (p) : X -> p. Given an Ad(K)-invariant scalar product on p, we obtain a Morse like function f = 1/2 || mu (p) || on X. We point out that, without the assumption that X is a real analytic manifold, the Lojasiewicz gradient inequality holds for f. Therefore, the limit of the negative gradient flow of f exists and it is unique. Moreover, we prove that any G -orbit collapses to a single K -orbit and two critical points of f which are in the same G -orbit belong to the same K -orbit. We also investigate convexity properties of the gradient map mu p in the Abelian case. In particular, we study two-orbit variety X and we investigate topological and cohomological properties of X.