Abstract. We study the phenomenon of Type-II curvature blow-up in mean curvature flows of rotationally symmetric noncompact embedded hypersurfaces. Using analytic techniques based on formal matched asymptotics and the construction of upper and lower barrier solutions enveloping formal solutions with prescribed behavior, we show that for each initial hypersurface considered, a mean curvature flow solution exhibits the following behavior near the "vanishing" time T : (1) The highest curvature concentrates at the tip of the hypersurface (an umbilic point), and for each choice of the parameter γ > 1/2, there is a solution with the highest curvature blowing up at the rate (T − t) −(γ+1/2) . (2) In a neighborhood of the tip, the solution converges to a translating soliton which is a higher-dimensional analogue of the "Grim Reaper" solution for the curve-shortening flow. (3) Away from the tip, the flow surface approaches a collapsing cylinder at a characteristic rate dependent on the parameter γ.

We discuss the center of mass of asymptotically flat manifolds. Our main result is that for a class of metrics that includes those which near infinity are conformally flat with vanishing scalar curvature and positive mass, the Huisken–Yau geometric center of mass agrees with the center of mass defined by the ADM formulation of the initial-value problem for Einstein's equation.

We present a numerical study of the local stability of mean curvature flow (MCF) of rotationally symmetric, complete noncompact hypersurfaces with type-II curvature blowup. Our numerical analysis employs a novel overlap method that constructs ‘numerically global’ (i.e., with spatial domain arbitrarily large but finite) flow solutions with initial data covering analytically distinct regions. Our numerical results show that for certain prescribed families of perturbations, there are two classes of initial data that lead to distinct behaviours under MCF. Firstly, there is a ‘near’ class of initial data which lead to the same singular behaviour as an unperturbed solution; in particular, the curvature at the tip of the hypersurface blows up at a type-II rate no slower than (T − t)−1. Secondly, there is a ‘far’ class of initial data which lead to solutions developing a local type-I nondegenerate neckpinch under MCF. These numerical findings further suggest the existence of a ‘critical’ class of initial data which conjecturally lead to MCF of noncompact hypersurfaces forming local type-II degenerate neckpinches with the highest curvature blowup rate strictly slower than (T − t)−1.

We prove that the space of smooth Riemannian metrics on the three-ball with non-negative Ricci curvature and strictly convex boundary is path-connected; and, moreover, that the associated moduli space (i.e., modulo orientation-preserving diffeomorphisms of the threeball) is contractible. As an application, using results of Maximo, Nunes, and Smith [MNS], we show the existence of properly embedded free boundary minimal annulus on any three-ball with non-negative Ricci curvature and strictly convex boundary.Let Ric(g s ) denote the Ricci curvature tensor of the metric g s . Then, as in [MNS, Section 6.5] and by Lemma 3.1, we haveMoreover, again by Lemma 3.1, we have ∂ s | s=0 Ric(g s ) = Hess g f + ∆ g f g > 0 on (−ε, 0] × D.So for s 1 sufficiently small, g s will have positive Ricci curvature on (−ε, 0]×D for all s ∈ [0, s 1 ].4 Since given any real number A, there exists a = a(A) > 0 such that for x ∈ (0, a], the function p(x) = exp −x −2 satisfies p ′′ (x) − Ap ′ (x) > 0. David J. Wraith, On the moduli space of positive Ricci curvature metrics on homotopy spheres, Geom. Topol. 15

In each dimension N ≥ 3 and for each real number λ ≥ 1, we construct a family of complete rotationally symmetric solutions to Ricci flow on R N which encounter a global singularity at a finite time T . The singularity forms arbitrarily slowly with the curvature blowing up arbitrarily fast at the rate (T − t) −(λ+1) . Near the origin, blow-ups of such a solution converge uniformly to the Bryant soliton. Near spatial infinity, blow-ups of such a solution converge uniformly to the shrinking cylinder soliton. As an application of this result, we prove that there exist standard solutions of Ricci flow on R N whose blow-ups near the origin converge uniformly to the Bryant soliton.