1. The main result and some consequences. In 1956 E. Calabi [6] attacked the classification problem of compact euclidean space forms by means of a special construction, called the Calabi construction (see Wolf [14, p. 124]). Here we announce that the construction can be extended to compact riemannian manifolds whose Ricci curvature tensor is zero (Ricci flat). Of course, it is not known if ther...

We prove that any gradient shrinking Ricci soliton has at most Euclidean volume growth. This improves a recent result of H.-D. Cao and D. Zhou by removing a condition on the growth of scalar curvature. A complete Riemannian manifold M of dimension n is called gradient shrinking Ricci soliton if there exists f ∈ C (M) and a constant ρ > 0 such that Rij +∇i∇jf = ρgij , where Rij denotes the Ricci...

We study complete noncompact long time solutions (M, g(t)) to the Kähler-Ricci flow with uniformly bounded nonnegative holomorphic bisectional curvature. We will show that when the Ricci curvature is positive and uniformly pinched, i.e. Ri̄ ≥ cRgi̄ at (p, t) for all t for some c > 0, then there always exists a local gradient Kähler-Ricci soliton limit around p after possibly rescaling g(t) alon...

The Ricci flow is an evolution system on metrics. For a given metric as initial data, its local existence and uniqueness on compact manifolds was first established by Hamilton [8]. Later on, De Turck [4] gave a simplified proof. In the later of 80's, Shi [20] generalized the local existence result to complete noncompact manifolds. However, the uniqueness of the solutions to the Ricci flow on co...

In this note, we obtain a sharp volume estimate for complete gradient Ricci solitons with scalar curvature bounded below by a positive constant. Using Chen-Yokota’s argument we obtain a local lower bound estimate of the scalar curvature for the Ricci flow on complete manifolds. Consequently, one has a sharp estimate of the scalar curvature for expanding Ricci solitons; we also provide a direct ...

We present numerical visualizations of Ricci Flow of surfaces and 3-dimensional manifolds of revolution. Ricci rot is an educational tool which visualizes surfaces of revolution moving under Ricci flow. That these surfaces tend to remain embedded in R3 is what makes direct visualization possible. The numerical lessons gained in developing this tool may be applicable to numerical simulation of R...

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 a recent paper, we have pointed out a relation between the Killing spinor and Einstein equations. Using this relation, new brane solutions of D = 11 and D = 10 type IIB supergravity theories are constructed. It is shown that in a brane solution, the flat world-volume directions, the smeared transverse directions and the sphere located at a fixed radial distance can be replaced with any Loren...

This is the second paper in a series of works devoted to nonholonomic Ricci flows. By imposing non–integrable (nonholonomic) constraints on the Ricci flows of Riemannian metrics we can model mutual transforms of generalized Finsler–Lagrange and Riemann geometries. We verify some assertions made in the first partner paper and develop a formal scheme in which the geometric constructions with Ricc...

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...