Gradient Kahler-Ricci solitons and periodic orbits

We study Hamiltonian dynamics of gradient Kahler-Ricci solitons that arise as limits of dilations of singularities of the Ricci how on compact Kahler manifolds. Our main result is that the underlying spaces of such gradient s olitons must be Stein manifolds. Moreover, on all most all energy surfaces of the potential function f of such a soliton, the Hamiltonian vector field V-f of f, with respect to the Kahler form of the gradient soliton metric, admits a periodic orbit. The latter should be of significance in the study of singularities of the Ricci flow on compact Kahler manifolds in light of the "little loop lemma" principle in [10].