Gradient Kähler-Ricci Solitons and Periodic Orbits

Details Gradient Kähler-Ricci Solitons and Periodic Orbits Gradient Kähler-Ricci Solitons and Periodic Orbits

1998-07-02

arxiv.org/abs/math/9807009v1

Mathematics

Differential Geometry

12 pages, LaTeX, to appear in Comm. Anal. Geom

Scientific paper

We study Hamiltonian dynamics of gradient Kaehler-Ricci solitons that arise as limits of dilations of singularities of the Ricci flow on compact Kaehler 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 V_f of f, with respect to the Kaehler form of the gradient soliton metric, admits a periodic orbit. The latter should be impotant in the study of singularities of the Ricci flow on compact Kaehler manifolds in light of the ``little loop lemma'' principle due to the second author.

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