相关论文: Dynamical stability of Pluriclosed and Generalized…
In this paper, I computed the second variation formula of the generalized Einstein-Hilbert functional and prove that a Bismut-flat, Einstein manifold is linearly stable under some curvature assumption. In the last part of the paper, I prove…
Motivated by M\"uller-Haslhofer results on the dynamical stability and instability of Ricci-flat metrics under the Ricci flow, we obtain dynamical stability and instability results for pairs of Ricci-flat metrics and vanishing 3-forms under…
We consider dynamical stability for a modified Ricci flow equation whose stationary solutions include Einstein and Ricci soliton metrics. Our focus is on homogeneous metrics on non-compact manifolds. Following the program of Guenther,…
We consider a modified Ricci flow equation whose stationary solutions include Einstein and Ricci soliton metrics, and we study the linear stability of those solutions relative to the flow. After deriving various criteria that imply linear…
In this paper, we continue to study the generalized Ricci flow. We give a criterion on steady gradient Ricci soliton on complete and noncompact Riemannian manifolds that is Ricci-flat, and then introduce a natural flow whose stable points…
In this announcement, we exhibit the second variation of Perelman's $\lambda$ and $\nu$ functionals for the Ricci flow, and investigate the linear stability of examples. We also define the "central density" of a shrinking Ricci soliton and…
We consider the volume-normalized Ricci flow close to compact shrinking Ricci solitons. We show that if a compact Ricci soliton $(M,g)$ is a local maximum of Perelman's shrinker entropy, any normalized Ricci flow starting close to it exists…
In previous work, the authors studied the linear stability of algebraic Ricci solitons on simply connected solvable Lie groups (solvsolitons), which are stationary solutions of a certain normalization of Ricci flow. Many examples were shown…
We prove dynamical stability and instability theorems for compact Einstein metrics under the Ricci flow. We give a nearly complete charactarization of dynamical stability and instability in terms of the conformal Yamabe invariant and the…
In this paper, we continue investigating the second variation of Perelman's $\nu$-entropy for compact shrinking Ricci solitons. In particular, we improve some of our previous work in "H.-D. Cao and M. Zhu, Math. Ann. 353 (2012), No. 3,…
We show that every gradient shrinking soliton of the generalized Ricci flow on compact manifold is a Ricci soliton. And we prove that the pluriclosed soliton is gradient Kahler-Ricci soliton under a broad cohomological condition. Moreover,…
We review recent results relating linear stability to dynamical stability and the scalar curvature rigidity of Einstein manifolds. We discuss closed and open Einstein manifolds as well as complete noncompact Einstein manifolds which are…
We introduce certain spherically symmetric singular Ricci solitons and study their stability under the Ricci flow from a dynamical PDE point of view. The solitons in question exist for all dimensions $n+1\ge 3$, and all have a point…
We study stability of non-compact gradient Kaehler-Ricci flow solitons with positive holomorphic bisectional curvature. Our main result is that any compactly supported perturbation and appropriately decaying perturbations of the Kaehler…
The linear stability of warped product Einstein metrics as fixed points of the Ricci flow is investigated. We generalise the results of Gibbons, Hartnoll and Pope and show that in sufficiently low dimensions, all warped product Einstein…
We consider four extended Ricci flow systems---that is, Ricci flow coupled with other geometric flows---and prove dynamical stability of certain classes of stationary solutions of these flows. The systems include Ricci flow coupled with…
We study the stability of non compact steady and expanding gradient Ricci solitons. We first show that strict linear stability implies dynamical stability. Then we give various sufficient geometric conditions ensuring the strict linear…
The Ricci flow on the 2-sphere with marked points is shown to converge in all three stable, semi-stable, and unstable cases. In the stable case, the flow was known to converge without any reparametrization, and a new proof of this fact is…
We prove that if an ALE Ricci-flat manifold $(M,g)$ is linearly stable and integrable, it is dynamically stable under Ricci flow, i.e. any Ricci flow starting close to g exists for all time and converges modulo diffeomorphism to an ALE…
We provide a comparatively simple proof of the dynamical stability of Ricci flow near a linearly stable Ricci-flat ALE metric with integrable deformations. Our proof relies on the equivalence between integrability and an…