Abstract. We prove stability of rotationally symmetric translating solutions to mean curvature flow. For initial data that converge spatially at infinity to such a soliton, we obtain convergence for large times to that soliton without imposing any decay rates.

Abstract. We study the Ricci flow for initial metrics which are C 0 small perturbations of the Euclidean metric on R n . In the case that this metric is asymptotically Euclidean, we show that a Ricci harmonic map heat flow exists for all times, and converges uniformly to the Euclidean metric as time approaches infinity. In proving this stability result, we introduce a monotone integral quantity which measures the deviation of the evolving metric from the Euclidean metric. We also investigate the convergence of the diffeomorphisms relating Ricci harmonic map heat flow to Ricci flow.

We show that strictly convex surfaces contracting with normal velocity equal to |A| 2 shrink to a point in finite time. After appropriate rescaling, they converge to spheres. We describe our algorithm to find the main test function.

Abstract. We show that strictly convex surfaces expanding by the inverse Gauß curvature flow converge to infinity in finite time. After appropriate rescaling, they converge to spheres. We describe the algorithm to find our main test function.