The mean curvature flow by parallel hypersurfaces

Constant mean curvature hypersurfaces foliated by spheres ∗

We ask when a constant mean curvature n-submanifold foliated by spheres in one of the Euclidean, hyperbolic and Lorentz–Minkowski spaces (En+1, Hn+1 or Ln+1), is a hypersurface of revolution. In En+1 and Ln+1 we will assume that the spheres lie in parallel hyperplanes and in the case of hyperbolic space Hn+1, the spheres will be contained in parallel horospheres. Finally, Riemann examples in L3...

Hypersurfaces with Constant Mean Curvature in a Lorentzian Space Form

Partial Regularity of Mean-Convex Hypersurfaces Flowing by Mean Curvature

In this paper we announce various new results about singularities in the mean curvature flow. Some results apply to any weak solution (i.e., any Brakke flow of integral varifolds.) Our strongest results, however, are for initially regular mean-convex hypersurfaces. (We say a hypersurface is mean-convex if it bounds a region such that the mean curvature with respect to the inward unit normal is ...

Hyperbolic flow by mean curvature

A hyperbolic flow by mean curvature equation, l t #cv"i, for the evolution of interfaces is studied. Here v, i and l t are the normal velocity, curvature and normal acceleration of the interface. A crystalline algorithm is developed for the motion of closed convex polygonal curves; such curves may exhibit damped oscillations and their shape appears to rotate during the evolutionary process. The...

Mean Curvature Blowup in Mean Curvature Flow

In this note we establish that finite-time singularities of the mean curvature flow of compact Riemannian submanifolds M t →֒ (N, h) are characterised by the blow up of the mean curvature.