We will present joint work with Gerhard Huisken on a fully nonlinear flow for hypersurfaces in Riemannian manifolds. Unlike mean curvature flow, this flow preserves two-convexity in a general ambient manifold. For this fully nonlinear flow, we establish a convexity estimate, a cylindrical estimate, and a pointwise curvature derivative estimate. These estimates allow us to extend the flow beyond singularities by a surgery procedure, similar to the ones developed by Hamilton and Perelman for the Ricci flow and by Huisken and Sinestrari for mean curvature flow

Let $(M,g_0)$ be a compact n-dimensional Riemannian manifold with a finite number of singular points, where at each singular point the metric is asymptotic to a cone over a compact (n-1)-dimensional manifold with curvature operator greater or equal to one. We show that there exists a smooth Ricci flow starting from such a metric with curvature decaying like C/t. The initial metric is attained in Gromov-Hausdorff distance and smoothly away from the singular points. To construct this solution, we desingularize the initial metric by glueing in expanding solitons with positive curvature operator, each asymptotic to the cone at the singular point, at a small scale s. Localizing a recent stability result of Deruelle-Lamm for such expanding solutions, we show that there exists a solution from the desingularized initial metric for a uniform time T>0, independent of the glueing scale s. The solution is then obtained by letting s->0. We also show that the so obtained limiting solution has the corresponding expanding soliton as a forward tangent flow at each initial singular point. This is joint work with P. Gianniotis

We show that n-manifolds with a lower volume bound v and upper diameter D bound whose curvature operator is bounded below by −ε(n,v,D) also admit metrics with nonnegative curvature operator. The proof relies on heat kernel estimates for the Ricci flow and shows that various smoothing properties of the Ricci flow remain valid if an upper curvature bound is replaced by a lower volume bound

We study the limiting behavior of the Ricci flow on the 2-sphere with marked points. We show that the normalized Ricci flow will always converge to a unique constant curvature metric or a shrinking gradient soliton metric. In the semi-stable and unstable cases of the 2-sphere with more than two marked points, the limiting metric space carries a different conical and the complex structure from the initial structure. We also study the blow-up behavior of the flow in the semi-stable and unstable cases. This is a joint work with Phong, Sturm and Wang

The Teichmueller harmonic map flow is a gradient flow of the classical harmonic map energy, in which both a map from a surface and the metric on that surface are allowed to evolve. In principle, the flow wants to find minimal immersions. However, in general, the domain metric might degenerate in finite time. In this talk we show how to flow beyond finite time singularities, and this allows us to decompose a general map into a collection of minimal immersions.

This is forthcoming work joint with Melanie Rupflin

We prove a gap theorem for homogeneous spaces : If the norm of the Riemannian curvature is one, then the norm of the Ricci curvature is larger than $\ep(n)$, where $\ep(n)$ is a positive constant depending only on the dimension $n$ of the homogeneous space. This is used to show that solutions with finite extinction time are Type I, immortal solutions are Type III and ancient solutions are Type I, where all the constants involved depend  only on the dimension $n$.

This is joint work with Christoph Böhm (University of Münster), Ramiro Lafuente (University of Münster)

Let X be a compact Kaehler manifold. I will show that the Kaehler-Ricci flow can be run from a degenerate initial data, (more precisely, from an arbitrary positive closed current) and that it is immediately smooth in a Zariski open subset of X. Moreover, if the initial data has positive Lelong number we indeed have propagation of singularities for short time. Finally, I will prove a uniqueness result in the case of zero Lelong numbers.

(This is a joint work with Chinh Lu)