We present new non-Ricci-flat Kähler metrics with U(N) and O(N) isometries as target manifolds of conformally invariant sigma models with an anomalous dimension. They are so-called Ricci solitons, special solutions to a Ricci-flow equation. These metrics explicitly contain the anomalous dimension and reduce to Ricci-flat Kähler metrics on the canonical line bundles over certain coset spaces in ...

In this paper, using the local Ricci flow, we prove the short-time existence of the Ricci flow on noncompact manifolds, whose Ricci curvature has global lower bound and sectional curvature has only local average integral bound. The short-time existence of the Ricci flow on noncompact manifolds was studied by Wan-Xiong Shi in 1990s, who required a point-wise bound of curvature tensors. As a coro...

A Ricci surface is a Riemannian 2-manifold (M, g) whose Gaussian curvature K satisfies K∆K+g(dK, dK)+4K = 0. Every minimal surface isometrically embedded in R is a Ricci surface of non-positive curvature. At the end of the 19 century Ricci-Curbastro has proved that conversely, every point x of a Ricci surface has a neighborhood which embeds isometrically in R as a minimal surface, provided K(x)...

Suppose {(M, g(t)), 0 ≤ t <∞} is a Kähler Ricci flow solution on a Fano surface. If |Rm| is not uniformly bounded along this flow, we can blowup at the maximal curvature points to obtain a limit complete Riemannian manifold X. We show that X must have certain topological and geometric properties. Using these properties, we are able to prove that |Rm| is uniformly bounded along every Kähler Ricc...

We study a class of asymptotically cylindrical Ricci-flat Kähler metrics arising on quasiprojective manifolds. Using the Calabi–Yau geometry and analysis and the Kodaira–Kuranishi–Spencer theory and building up on results of N.Koiso, we show that under rather general hypotheses any local asymptotically cylindrical Ricci-flat deformations of such metrics are again Kähler, possibly with respect t...

where λ is the soliton constant. In [2], Lauret proves the existence of many left invariant Ricci solitons on nilpotent Lie groups. The first explicit construction of Lauret solitons has been obtained by Baird and Danielo in [4]. In particular they show that the soliton structure on Nil is of nongradient type. Remarkably this is the first example of nongradient Ricci soliton. In [5], it is prov...

We first present the natural definitions of horizontal differential, divergence (as an adjoint operator), and a $p$-harmonic form on Finsler manifold. Next, we prove Hodge-type theorem for manifold in sense that $p$-form is harmonic if only Laplacian vanishes. This viewpoint provides new appropriate definition vector fields geometry. approach leads to Bochner-Yano type classification based Ricc...

where Rαβ(x, t) denotes the Ricci curvature tensor of the metric gαβ(x, t). One of the main problems in differential geometry is to find canonical structure on manifolds. The Ricci flow introduced by Hamilton [8] is an useful tool to approach such problems. For examples, Hamilton [10] and Chow [7] used the convergence of the Ricci flow to characterize the complex structures on compact Riemann s...

We study the consequences of timelike and spaccelike conformal Ricci and conformal matter collineations for anisotropic fluid in the context of General Relativity. Necessary and sufficient conditions are derived for a spacetime with anisotropic fluid to admit conformal Ricci and conformal matter collineations parallel to u and x. These conditions for timelike and spacelike conformal Ricci and c...

Ricci curvature was proposed by Ollivier in a general framework of metric measure spaces, and it has been studied extensively in the context of graphs in recent years. In this paper we obtain the exact formulas for Ollivier’s Ricci-curvature for bipartite graphs and for the graphs with girth at least 5. These are the first formulas for Ricci-curvature that hold for a wide class of graphs, and e...