相关论文: A remark on odd dimensional normalized Ricci flow

On a smooth closed oriented 4-manifold $M$ with a smooth action by a finite group $G$, we show that a $G$-monopole class gives the $L^2$-estimate of the Ricci curvature of a $G$-invariant Riemannian metric, and derive a topological…

This is a revised version of our short note [arxiv.math.DG/0403065] where we discuss the monotonicity of the eigen-values of the Laplacian operator to the Ricci-Hamilton flow on a compact or a complete non-compact Riemannian manifold. We…

We show that an orientable 3-dimensional manifold M admits a complete riemannian metric of bounded geometry and uniformly pos- itive scalar curvature if and only if there exists a finite collection F of spherical space-forms such that M is…

Let $g(t)$, $t\in [0, +\infty)$, be a solution of the normalized K\"ahler-Ricci flow on a compact K\"ahler $n$-manifold $M$ with $c_{1}(M)>0$ and initial metric $g (0)\in 2\pi c_{1}(M)$. If there is a constant $C$ independent of $t$ such…

In this survey article, we discuss some topics on self-similar solutions to the Ricci flow and the mean curvature flow. Self-similar solutions to the Ricci flow are known as Ricci solitons. In the first part of this paper we discuss a lower…

Let (M,g) be a three-dimensional steady gradient Ricci soliton which is non-flat and \kappa-noncollapsed. We prove that (M,g) is isometric to the Bryant soliton up to scaling. This solves a problem mentioned in Perelman's first paper.

We study $n$-dimensional Ricci flows with non-negative Ricci curvature where the curvature is pointwise controlled by the scalar curvature and bounded by $C/t$, starting at metric cones which are Reifenberg outside the tip. We show that any…

We consider the normalized Ricci flow $\del_t g = (\rho - R)g$ with initial condition a complete metric $g_0$ on an open surface $M$ where $M$ is conformal to a punctured compact Riemann surface and $g_0$ has ends which are asymptotic to…

In the paper, we study evolution equations of the scalar and Ricci curvatures under the Hamilton's Ricci flow on a closed manifold and on a complete noncompact manifold. In particular, we study conditions when the Ricci flow is trivial and…

In this paper, we study monotonicity of eigenvalues of Laplacian-type operator $-\Delta+cR$, where $c$ is a constant, along the Ricci-Bourguignon flow. For $c\neq0$, We derive monotonicity of the lowest eigenvalue of Laplacian-type operator…

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…

We estimate from above the rate at which a solution to the normalized Ricci flow on a closed manifold may converge to a limit soliton. Our main result implies that any solution which converges modulo diffeomorphisms to a soliton faster than…

Suppose $G$ is a compact Lie group, $H$ is a closed subgroup of $G$, and the homogeneous space $G/H$ is connected. The paper investigates the Ricci flow on a manifold $M$ diffeomorphic to $[0,1]\times G/H$. First, we prove a short-time…

In this paper we investigate a kind of generalized Ricci flow which possesses a gradient form. We study the monotonicity of the given function under the generalized Ricci flow and prove that the related system of partial differential…

In this article, we shall investigate the relationship between the existence or non-existence of non-singular solutions to the normalized Ricci flow and smooth structures on closed 4-manifolds, where non-singular solutions to the normalized…

B Wilking has recently shown that one can associate a Ricci flow invariant cone of curvature operators $C(S)$, which are nonnegative in a suitable sense, to every $Ad_{SO(n,\C)}$ invariant subset $S \subset {\bf so}(n,\C)$. For curvature…

We show the existence of a solution to the Ricci flow with a compact length space of bounded curvature, i.e., a space that has curvature bounded above and below in the sense of Alexandrov, as its initial condition. We show that this flow…

For an immortal Ricci flow on an $m$-dimensional $(m\ge 3)$ closed manifold, we show the following convergence results: (1) if the curvature and diameter are uniformly bounded, then any unbounded sequence of time slices sub-converges to a…

We demonstrate that any four-dimensional shrinking Ricci soliton $(\mathcal B \times {\mathbb S^2}, g)$, where $\mathcal B$ is any two-dimensional complete noncompact surface and $g$ is a warped product metric over the base $\mathcal B$,…

We study the Ricci flow out of spaces with edge type conical singularities along a closed, embedded curve. Under the additional assumption that for each point of the curve, our space is locally modelled on the product of a fixed positively…