Related papers: Ricci flow with surgery in higher dimensions

In this second part of a series of papers on the long-time behavior of Ricci flows with surgery, we establish a bound on the evolution of the infimal area of simplicial complexes inside a 3-manifold under the Ricci flow. This estimate…

We exhibit a concentration-collapse decomposition of singularities of fourth order curvature flows, including the $L^2$ curvature flow and Calabi flow, in dimensions $n \leq 4$. The proof requires the development of several new a priori…

In this paper we construct solutions to Ricci DeTurck flow in four dimensions on closed manifolds which are instantaneously smooth but whose initial values $g$ are (possibly) non-smooth Riemannian metrics whose components in smooth…

We decribe and announce some results (joint with G. Besson, L. Bessieres, M. Boileau and J.Porti) about the geometry and topology of 3-manifolds. Most of the article is primarily intended as an introduction for nonexperts to geometrization…

Assume $M$ is a closed 3-manifold whose universal covering is not $S^3$. We show that the obstruction to extend the Ricci flow is the boundedness $L^{3/2}$-norm of the scalar curvature $R(t)$, i.e, the Ricci flow can be extended over time…

In this paper, we study the evolution of L2 p-forms under Ricci flow with bounded curvature on a complete non-compact or a compact Riemannian manifold. We show that under curvature pinching conditions on such a manifold, the L2 norm of a…

We study the long time behaviour of Ricci flow with bubbling-off on a possibly noncompact $3$-manifold of finite volume whose universal cover has bounded geometry. As an application, we give a Ricci flow proof of Thurston's hyperbolisation…

This is a survey of some of the recent developments on the geometric and analytic aspects of the Anomaly flow. It is a flow of $(2,2)$-forms on a $3$-fold which was originally motivated by string theory and the need to preserve the…

In this note, we study the problem of uniqueness of Ricci flow on complete noncompact manifolds. We consider the class of solutions with curvature bounded above by C/t when t > 0. In paricular, we proved uniqueness if in addition the…

In this paper we prove localised weighted curvature integral estimates for solutions to the Ricci flow in the setting of a smooth four dimensional Ricci flow or a closed $n$-dimensional K\"ahler Ricci flow. These integral estimates improve…

The geometric description of incompressible hydrodynamics, as geodesic motion on the infinite-dimensional group of volume-preserving diffeomorphisms, enables notions of curvature in the study of fluids in order to study stability. Formulas…

In this paper, we study curvature behavior at the first singular time of solution to the Ricci flow on a smooth, compact n-dimensional Riemannian manifold $M$, $\frac{\partial}{\partial t}g_{ij} = -2R_{ij}$ for $t\in [0,T)$. If the flow has…

We consider Ricci flow invariant cones C in the space of curvature operators lying between nonnegative Ricci curvature and nonnegative curvature operator. Assuming some mild control on the scalar curvature of the Ricci flow, we show that if…

I survey some of the developments in the theory of Ricci flow and its applications from the past decade. I focus mainly on the understanding of Ricci flows that are permitted to have unbounded curvature in the sense that the curvature can…

We study the Ricci flow on $\mathbb{R}^{4}$ starting at an SU(2)-cohomogeneity 1 metric $g_{0}$ whose restriction to any hypersphere is a Berger metric. We prove that if $g_{0}$ has no necks and is bounded by a cylinder, then the solution…

We study deformations of Riemannian metrics on a given manifold equipped with a codimension-one foliation subject to quantities expressed in terms of its second fundamental form. We prove the local existence and uniqueness theorem and…

We consider the K\"ahler Ricci flow on a smooth minimal model of general type, we show that if the Ricci curvature is uniformly bounded below along the K\"ahler-Ricci flow, then the diameter is uniformly bounded. As a corollary we show that…

We consider solutions (M,g(t)), 0 <= t <T, to Ricci flow on compact, four dimensional manifolds without boundary. We prove integral curvature estimates which are valid for any such solution. In the case that the scalar curvature is bounded…

As part of the general investigation of Ricci flow on complete surfaces with finite total curvature, we study this flow for surfaces with asymptotically conical (which includes as a special case asymptotically Euclidean) geometries. After…

We investigate the convergence of the mean curvature flow of arbitrary codimension in Riemannian manifolds with bounded geometry. We prove that if the initial submanifold satisfies a pinching condition, then along the mean curvature flow…