I will discuss the metric behavior of the Kahler-Ricci flow on Hirzebruch surfaces assuming that the initial metric is invariant under a maximal compact subgroup of the automorphism group. I will describe how, in the sense of Gromov-Hausdorff, the flow either shrinks to a point, collapses to P^1 or contracts an exceptional divisor. This confirms a conjecture of Feldman-Ilmanen-Knopf. This is a joint work with Jian Song.

We will discuss the existence and uniqueness of the foliations by stable spheres with constant mean curvature for asymptotically flat manifolds satisfying the parity condition at infinity. The concept of center of mass in general relativity will also be discussed. This work generalizes the earlier results of Huisken-Yau, R. Ye, and Metzger.

We discuss 3-Sasakian 7-manifolds carrying metrics of positive curvature and an example of this kind on a new diffeomorphism type.

Singularities occur in Ricci flow because of curvature blowup. For dimensional reasons, when approaching a singularity, one expects the curvature to blow up like the inverse of the time to the singularity. If this does not happen, the singularity is said to be type II. The first example of a type II singularity, studied by Daskalopoulos-Del Pino-Hamilton-Sesum, occurs on a noncompact surface which is the result of capping off a hyperbolic cusp. The analysis in the surface case uses isothermal coordinates. It is not immediately clear whether it extends to higher dimensions. We look at the Ricci flow on finite-volume metrics that live on the complement of a divisor in a compact Khler manifold. We compute the blowup time in terms of cohomological data and give sufficient conditions for a type II singularity to emerge. This is joint work with Zhou Zhang.