In this article we give a brief survey of breather and soliton solutions to the Ricci flow and prove a no breather and soliton theorem for homogeneous solutions.

In this paper, we prove a theorem on convergence of Kähler-Ricci flow on a compact Kähler manifold M which admits a Kähler-Ricci soliton. A Kähler metric h is called a Kähler-Ricci soliton if its Kähler form ωh satisfies equation Ric(ωh)− ωh = LXωh, where Ric(ωh) is the Ricci form of h and LXωh denotes the Lie derivative of ωh along a holomorphic vector field X on M . As usual, we denote a Kähl...

In this paper, we prove that Kähler-Ricci flow converges to a Kähler-Einstein metric (or a Kähler-Ricci soliton) in the sense of Cheeger-Gromov as long as an initial Kähler metric is very closed to gKE (or gKS) if a compact Kähler manifold with c1(M) > 0 admits a Kähler Einstein metric gKE (or a Kähler-Ricci soliton gKS). The result improves Main Theorem in [TZ3] in the sense of stability of Kä...

We show that if two gradient Ricci solitons are asymptotic along some end of each to the same regular cone ((0,∞)× Σ, dr + r2gΣ), then the soliton metrics must be isometric on some neighborhoods of infinity of these ends. Our theorem imposes no restrictions on the behavior of the metrics off of the ends in question and in particular does not require their geodesic completeness. As an applicatio...

The Ricci flow ∂g/∂t = −2Ric(g) is an evolution equation for Riemannian metrics. It was introduced by Richard Hamilton, who has shown in several cases ([7], [8], [9]) that the flow converges, up to re-scaling, to a metric of constant curvature. However, “soliton” solutions to the flow give examples where the Ricci flow does not uniformize the metric, but only changes it by diffeomorphisms. Soli...

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 respe...

We investigate Riemannian (non-Kähler) Ricci flow solutions that develop finite-time Type-I singularities with the property that parabolic rescalings at the singularities converge to singularity models taking the form of shrinking Kähler–Ricci solitons. More specifically, the singularity models for these solutions are given by the “blowdown soliton” discovered in [FIK03]. Our results support th...

In this note, using Calabi’s method, we construct rotationally symmetric KählerRicci solitons on the total space of direct sum of fixed hermitian line bundle and its projective compactification, where the curvature of hermitian line bundle is Kähler-Einstein. These examples generalize the construction of Koiso, Cao and Feldman-Ilmanen-Knopf. 1 A little motivation In [1], the authors constructed...

For n+1 ≥ 3, we construct complete solutions to Ricci flow on R which encounter global singularities at a finite time T . The singularities are forming arbitrarily slowly with the curvature blowing up arbitrarily fast at the rate (T − t)−2λ for λ ≥ 1. Near the origin, blow-ups of such a solution converge uniformly to the Bryant soliton. Near spatial infinity, blow-ups of such a solution converg...