Related papers: On Normalized Ricci Flow and Smooth Structures on …
We study solutions to generalized Ricci flow on four-manifolds with a nilpotent, codimension $1$ symmetry. We show that all such flows are immortal, and satisfy type III curvature and diameter estimates. Using a new kind of monotone energy…
This project serves to analyze the behavior of Ricci Flow in five dimensional manifolds. Ricci Flow was introduced by Richard Hamilton in 1982 and was an essential tool in proving the Geometrization and Poincare Conjectures. In general,…
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 present in this paper a general approach to study the Ricci flow on homogeneous manifolds. Our main tool is a dynamical system defined on a subset H(q,n) of the variety of (q+n)-dimensional Lie algebras, parameterizing the space of all…
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…
Using a recently developed piecewise flat method, numerical evolutions of the Ricci flow are computed for a number of manifolds, using a number of different mesh types, and shown to converge to the expected smooth behaviour as the mesh…
In this article, we produce infinite families of 4-manifolds with positive first betti numbers and meeting certain conditions on their homotopy and smooth types so as to conclude the non-vanishing of the stable cohomotopy Seiberg-Witten…
We investigate the properties of the combinatorial Ricci flow for surfaces, both forward and backward -- existence, uniqueness and singularities formation. We show that the positive results that exist for the smooth Ricci flow also hold for…
In this paper we study the evolution of almost non-negatively curved (possibly singular) three dimensional metric spaces by Ricci flow. The non-negatively curved metric spaces which we consider arise as limits of smooth Riemannian manifolds…
We construct smooth solutions to Ricci flow starting from a class of singular metrics and give asymptotics for the forward evolution. The singular metrics heal with a set of points (of codimension at least three) coming out of the singular…
We verify a conjecture of Perelman, which states that there exists a canonical Ricci flow through singularities starting from an arbitrary compact Riemannian 3-manifold. Our main result is a uniqueness theorem for such flows, which,…
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…
This paper studies the Ricci flow on closed manifolds admitting harmonic spinors. It is shown that Perelman's Ricci flow entropy can be expressed in terms of the energy of harmonic spinors in all dimensions, and in four dimensions, in terms…
In this paper, we study the Ricci flow on a closed manifold and finite time interval $[0,T)~(T < \infty)$ on which certain integral curvature energies are finite. We prove that in dimension four, such flow converges to a smooth Riemannian…
We study the behavior of the normalized Ricci flow of invariant Riemannian homogeneous metrics at infinity for generalized Wallach spaces, generalized flag manifolds with four isotropy summands and second Betti number equal to one, and the…
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…
We exhibit the first examples of closed 4-manifolds with nonnegative sectional curvature that lose this property when evolved via Ricci flow.
We construct a class of monotonic quantities along the normalized Ricci flow on closed n-dimensional manifolds.
The stability of a recently developed piecewise flat Ricci flow is investigated, using a linear stability analysis and numerical simulations, and a class of piecewise flat approximations of smooth manifolds is adapted to avoid an inherent…
In the following work, an attempt to conciliate the Ricci flow conjecture and the Cobordism conjecture, stated as refinements of the Swampland distance conjecture and of the No global symmetries conjecture respectively, is presented. This…