Gradient Kähler-Ricci Solitons and Periodic Orbits

Abstract

We study Hamiltonian dynamics of gradient Kähler-Ricci solitons that arise as limits of dilations of singularities of the Ricci flow on compact Kähler manifolds. Our main result is that the underlying spaces of such gradient solitons must be Stein manifolds. Moreover, on all most all energy surfaces of the potential function f of such a soliton, the Hamiltonian vector field Vf of f, with respect to the Kähler 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 Kähler manifolds in light of the "little loop lemma" principle in [10].