Abstract. We introduce singular Ricci flows, which are Ricci flow spacetimes subject to certain asymptotic conditions. We consider the behavior of Ricci flow with surgery starting from a fixed initial compact Riemannian 3-manifold, as the surgery parameter varies. We prove that the flow with surgery subconverges to a singular Ricci flow as the surgery parameter tends to zero. We establish a num...

We show that Perelman’s W functional on Kahler manifolds has a natural counterpart on Sasaki manifolds. We prove, using this functional, that Perelman’s results on Kahler-Ricci flow (the first Chern class is positive) can be generalized to Sasaki-Ricci flow, including the uniform bound on the diameter and the scalar curvature along the flow. We also show that positivity of transverse bisectiona...

The purpose of this paper is to introduce the Ricci Yang-Mills soliton equations on nilpotent Lie groups. In the 2-step nilpotent setting, we show that these equations are strictly weaker than the Ricci soliton equations. Using techniques from Geometric Invariant Theory, we develop a procedure to build many different kinds of Ricci Yang-Mills solitons. We finish this note by producing examples ...

We show the properties of the blowup limits of Kähler Ricci flow solutions on Fano surfaces if Riemannian curvature is unbounded. As an application, on every toric Fano surface, we prove that Kähler Ricci flow converges to a Kähler Ricci soliton metric if the initial metric has toric symmetry. Therefore we give a new Ricci flow proof of existence of Kähler Ricci soliton metrics on toric surfaces.

In this article, we thoroughly investigate the stability inequality for Ricci-flat cones. Perhaps most importantly, we prove that the Ricci-flat cone over CP 2 is stable, showing that the first stable non-flat Ricci-flat cone occurs in the smallest possible dimension. On the other hand, we prove that many other examples of Ricci-flat cones over 4-manifolds are unstable, and that Ricci-flat cone...

We call a Gromov-Hausdorff limit of complete Riemannian manifolds with a lower bound of Ricci curvature a Ricci limit space. In this paper, we prove that any Ricci limit space has integral Hausdorff dimension provided that its Hausdorff dimension is not greater than two. We also classify one-dimensional Ricci limit spaces.

Let P be a principal U(1)-bundle over a closed manifold M . On P , one can define a modified version of the Ricci flow called the Ricci Yang-Mills flow, due to these equations being a coupling of Ricci flow and the Yang-Mills heat flow. We use maximal regularity theory and ideas of Simonett concerning the asymptotic behavior of abstract quasilinear parabolic partial differential equations to st...