Scalar curvature

About: Scalar curvature is a research topic. Over the lifetime, 12701 publications have been published within this topic receiving 296040 citations. The topic is also known as: Ricci scalar.

The nature of singularities in mean curvature flow of mean-convex sets

Abstract: Let K be a compact subset of R, or, more generally, of an (n+1)-dimensional riemannian manifold. We suppose that K is mean-convex. If the boundary of K is smooth and connected, this means that the mean curvature of ∂K is everywhere nonnegative (with respect to the inward unit normal) and is not identically 0. More generally, it means that Ft(K) is contained in the interior of K for t > 0, where Ft(K) is the set obtained by letting K evolve for time t under the level set mean curvature flow. As K evolves, it traces out a closed set K of spacetime: K = {(x, t) ∈ R ×R : x ∈ Ft(K)}. Also, there is associated to K a Brakke flow M : t 7→Mt of rectifiable varifolds. We call the pair (M,K) a mean-convex flow. Let X = (x, t) be a point in spacetime with t > 0. Suppose (xi, ti) is a sequence of points converging to X and λi is a sequence of positive numbers tending to infinity. Translate the pair M and K in spacetime by (y, τ) 7→ (y − xi, τ − ti) and then dilate parabolically by (y, τ) 7→ (λiy, λi τ) to get new flows Mi and Ki. The sequence (Mi,Ki) is called a blow-up sequence at X . General compactness theorems guarantee that this sequence will have subsequential limits. A subsequential limit (M′,K′) is called a limit flow. Here M′ : t ∈ (−∞,∞) 7→M ′ t