Convergence of a K\"ahler-Ricci flow

In this paper we prove that for a given K\"ahler-Ricci flow with uniformly bounded Ricci curvatures in an arbitrary dimension, for every sequence of times $t_i$ converging to infinity, there exists a subsequence such that $(M,g(t_i + t))\to (Y,\bar{g}(t))$ and the convergence is smooth outside a singular set (which is a set of codimension at least 4) to a solution of a flow. We also prove that in the case of complex dimension 2, without any curvature assumptions we can find a subsequence of times such that we have a convergence to a K\"ahler-Ricci soliton, away from finitely many isolated singularities

Convergence of Kahler-Einstein orbifolds

We proved the convergence of a sequence of 2 dimensional comapct Kahler-Einstein orbifolds with rational quotient singularities and with some uniform bounds on the volumes and on the Euler characteristics of our orbifods to a Kahler-Einstein 2-dimensional orbifold. Our limit orbifold can have worse singularities than the orbifolds in our sequence. We will also derive some estimates on the norms of the sections of plurianticanonical bundles of our orbifolds in the sequence that we are considering and our limit orbifold

Curvature tensor under the Ricci flow

Consider the unnormalized Ricci flow $(g_{ij})_t = -2R_{ij}$ for $t\in [0,T)$, where $T < \infty$. Richard Hamilton showed that if the curvature operator is uniformly bounded under the flow for all times $t\in [0,T)$ then the solution can be extended beyond $T$. We prove that if the Ricci curvature is uniformly bounded under the flow for all times $t\in [0,T)$, then the curvature tensor has to be uniformly bounded as well

Convergence of the Ricci flow toward a unique soliton

We will consider a {\it $\tau$-flow}, given by the equation $\frac{d}{dt}g_{ij} = -2R_{ij} + \frac{1}{\tau}g_{ij}$ on a closed manifold $M$, for all times $t\in [0,\infty)$. We will prove that if the curvature operator and the diameter of $(M,g(t))$ are uniformly bounded along the flow and if one of the limit solitons is integrable, then we have a convergence of the flow toward a unique soliton, up to a diffeomorphism

Limiting behaviour of the Ricci flow

We will consider a {\it $\tau$-flow}, given by the equation $\frac{d}{dt}g_{ij} = -2R_{ij} + \frac{1}{\tau}g_{ij}$ on a closed manifold $M$, for all times $t\in [0,\infty)$. We will prove that if the curvature operator and the diameter of $(M,g(t))$ are uniformly bounded along the flow, then we have a sequential convergence of the flow toward the solitons

Compactness results for the K\"ahler-Ricci flow

We consider the K\"ahler-Ricci flow $\frac{\partial}{\partial t}g_{i\bar{j}} = g_{i\bar{j}} - R_{i\bar{j}}$ on a compact K\"ahler manifold $M$ with $c_1(M) > 0$, of complex dimension $k$. We prove the $\epsilon$-regularity lemma for the K\"ahler-Ricci flow, based on Moser's iteration. Assume that the Ricci curvature and \int_M |\rem|^k dV_t are uniformly bounded along the flow. Using the $\epsilon$-regularity lemma we derive the compactness result for the K\"ahler-Ricci flow. Under our assumptions, if $k \ge 3$ in addition, using the compactness result we show that |\rem| \le C holds uniformly along the flow. This means the flow does not develop any singularities at infinity. We use some ideas of Tian from \cite{Ti} to prove the smoothing property in that case

Linear and dynamical stability of Ricci flat metrics

We can talk about two kinds of stability of the Ricci flow at Ricci flat metrics. One of them is a linear stability, defined with respect to Perelman's functional $\mathcal{F}$. The other one is a dynamical stability and it refers to a convergence of a Ricci flow starting at any metric in a neighbourhood of a considered Ricci flat metric. We show that dynamical stability implies linear stability. We also show that a linear stability together with the integrability assumption imply dynamical stability. As a corollary we get a stability result for $K3$ surfaces part of which has been done in \cite{dan2002}

Ricci flow on three-dimensional manifolds with symmetry

We describe the Ricci flow on two classes of compact three-dimensional manifolds: 1. Warped products with a circle fiber over a two-dimensional base. 2. Manifolds with a free local isometric U(1) x U(1) action.Comment: final versio

Asymptotic behavior of Type III mean curvature flow on noncompact hypersurfaces

In this paper, we introduce a monotonicity formula for the mean curvature flow which is related to self-expanders. Then we use the monotonicity to study the asymptotic behavior of Type III mean curvature flow on noncompact hypersurfaces.Comment: 10 pages, a global version of functional is introduced, more applications are foun

On gradient Ricci solitons

In the first part of the paper we derive integral curvature estimates for complete gradient shrinking Ricci solitons. Our results and the recent work of Lopez-Rio imply rigidity of gradient shrinking Ricci solitons with harmonic Weyl tensor. In the second part of the paper we address the issue of existence of harmonic functions on gradient shrinking K\"{a}hler and gradient steady Ricci solitons and show that if the total energy of a harmonic function on such a manifold is finite then the function is constant. Consequences to the structure of the manifold at infinity are also discussed.Comment: to appear in J. Geom. Ana