The problem of finding Kähler-Einstein metrics on a compact Kähler manifold has been the subject of intense study over the last few decades. In his solution to Calabi’s conjecture, Yau [Ya1] proved the existence of a Kähler-Einstein metric on compact Kähler manifolds with vanishing or negative first Chern class. An alternative proof of Yau’s theorem is given by Cao [Ca] using the Kähler-Ricci f...

Suppose M is a compact n-dimensional manifold, n ≥ 2, with a metric gij(x, t) that evolves by the Ricci flow ∂tgij = −2Rij in M × (0, T ). We will give a simple proof of a recent result of Perelman on the non-existence of shrinking breather without using the logarithmic Sobolev inequality. It is known that Ricci flow is a very powerful tool in understanding the geometry and structure of manifol...

In this paper, we prove a general maximum principle for the time dependent Lichnerowicz heat equation on symmetric tensors coupled with the Ricci flow on complete Riemannian manifolds. As an application we construct complete manifolds with bounded nonnegative sectional curvature of dimension greater than or equal to four such that the Ricci flow does not preserve the nonnegativity of the sectio...

Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multi...

Our goal in this paper is to obtain further information about the curvature of gradient shrinking Ricci solitons. This is important for a better understanding and ultimately for the classification of these manifolds. The classification of gradient shrinkers is known in dimensions 2 and 3, and assuming locally conformally flatness, in all dimensions n ≥ 4 (see [14, 13, 6, 15, 20, 12, 2]). Many o...

A regular edge but not vertex transitive graph is said to be semisym metric The study of semisymmetric graphs was initiated by Folkman who among others gave constructions of several in nite families such graphs In this paper a generalization of his construction for or ders a multiple of is proposed giving rise to some new families of semisymmetric graphs In particular one associated with the cy...

In this work we construct and analyze exact solutions describing Ricci flows and nonholonomic deformations of four dimensional (4D) Taub-NUT spacetimes. It is outlined a new geometric techniques of constructing Ricci flow solutions. Some conceptual issues on spacetimes provided with generic off–diagonal metrics and associated nonlinear connection structures are analyzed. The limit from gravity/...

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...

We derive the entropy formula for the linear heat equation on general Riemannian manifolds and prove that it is monotone non-increasing on manifolds with nonnegative Ricci curvature. As applications, we study the relation between the value of entropy and the volume of balls of various scales. The results are simpler version, without Ricci flow, of Perelman’s recent results on volume non-collaps...

We discuss the geometry of homogeneous Ricci solitons. After showing the nonexistence of compact homogeneous and noncompact steady homogeneous solitons, we concentrate on the study of left invariant Ricci solitons. We show that, in the unimodular case, the Ricci soliton equation does not admit solutions in the set of left invariant vector fields. We prove that a left invariant soliton of gradie...