We define a gradient Ricci soliton to be rigid if it is a flat bundle N×ΓR k where N is Einstein. It is known that not all gradient solitons are rigid. Here we offer several natural conditions on the curvature that characterize rigid gradient solitons. Other related results on rigidity of Ricci solitons are also explained in the last section.

In this paper, we will study the asymptotic geometry of 4-dimensional steady gradient Ricci solitons under condition that they dimension reduce to 3-manifolds. We show such either strongly a spherical space form S3/Γ or weakly 3-dimensional Bryant soliton. also soliton singularity models with nonnegative curvature outside compact set are Ricci-flat ALE 4-manifolds manifolds. As further applicat...

We show that a four-dimensional complete gradient shrinking Ricci soliton with positive isotropic curvature is either a quotient of S4 or a quotient of S3 × R. This gives a clean classification result removing the earlier additional assumptions in [14] by Wallach and the second author. The proof also gives a classification result on gradient shrinking Ricci solitons with nonnegative isotropic c...

Let (Y, d) be a Gromov-Hausdorff limit of closed shrinking Ricci solitons with uniformly upper bounded diameter and lower bounded volume. We prove that off a closed subset of codimension at least 2, Y is a smooth manifold satisfying a shrinking Ricci soliton equation.

In this paper, we define a reduced distance function based at a point at the singular time T < ∞ of a Ricci flow. We also show the monotonicity of the corresponding reduced volume based at time T, with equality iff the Ricci flow is a gradient shrinking soliton. Our curvature bound assumption is more general than the type I condition.

In each dimension N ≥ 3 and for each real number λ ≥ 1, we construct a family of complete rotationally symmetric solutions to Ricci flow on R which encounter a global singularity at a finite time T . The singularity forms arbitrarily slowly with the curvature blowing up arbitrarily fast at the rate (T − t)−(λ+1). Near the origin, blow-ups of such a solution converge uniformly to the Bryant soli...

Some aspects of curved BPS domain walls and their supersymmetric Lorentz invariant vacuums of four dimensional N = 1 supergravity coupled to a chiral multiplet are considered. In particular, the scalar manifold can be viewed as two dimensional Kähler-Ricci soliton which further implies that all quantities describing the walls and their vacuums have evolution with respect to the soliton. Then, t...

We study four dimensional chiral N = 1 supersymmetric theories on which the scalar manifold is described by Kähler geometry and further, it can be viewed as Kähler-Ricci soliton. All couplings and solutions, namely BPS domain walls and their supersymmetric Lorentz invariant vacuums turn out to be evolved with respect to the soliton. Two models are discussed, namely N = 1 theory on Kähler-Einste...

In this paper, we prove the compactness theorem for gradient Ricci solitons. Let (Mα, gα) be a sequence of compact gradient Ricci solitons of dimension n ≥ 4, whose curvatures have uniformly bounded L n 2 norms, whose Ricci curvatures are uniformly bounded from below with uniformly lower bounded volume and with uniformly upper bounded diameter, then there must exists a subsequence (Mα, gα) conv...

To every Ricci flow on a manifold M over a time interval I ⊂ R, we associate a shrinking Ricci soliton on the space-time M×I . We relate properties of the original Ricci flow to properties of the new higher-dimensional Ricci flow equipped with its own time-parameter. This geometric construction was discovered by consideration of the theory of optimal transportation, and in particular the result...