The mean curvature at the first singular time of the mean curvature flow

Consider a family of smooth immersions $F(\cdot,t): M^n\to \mathbb{R}^{n+1}$ of closed hypersurfaces in $\mathbb{R}^{n+1}$ moving by the mean curvature flow $\frac{\partial F(p,t)}{\partial t} = -H(p,t)\cdot \nu(p,t)$, for $t\in [0,T)$. We prove that the mean curvature blows up at the first singular time $T$ if all singularities are of type I. In the case $n = 2$, regardless of the type of a possibly forming singularity, we show that at the first singular time the mean curvature necessarily blows up provided that either the Multiplicity One Conjecture holds or the Gaussian density is less than two. We also establish and give several applications of a local regularity theorem which is a parabolic analogue of Choi-Schoen estimate for minimal submanifolds

Parabolic methods for the construction of spacelike slices of prescribed mean curvature in cosmological spacetimes

Spacelike hypersurfaces of prescribed mean curvature in cosmological space times are constructed as asymptotic limits of a geometric evolution equation. In particular, an alternative, constructive proof is given for the existence of maximal and constant mean curvature slices

Convex ancient solutions of the mean curvature flow

We study solutions of the mean curvature flow which are defined for all negative curvature times, usually called ancient solutions. We give various conditions ensuring that a closed convex ancient solution is a shrinking sphere. Examples of such conditions are: a uniform pinching condition on the curvatures, a suitable growth bound on the diameter or a reverse isoperimetric inequality. We also study the behaviour of uniformly k-convex solutions, and consider generalizations to ancient solutions immersed in a sphere

On the classical geometry of embedded manifolds in terms of Nambu brackets

We prove that many aspects of the differential geometry of embedded Riemannian manifolds can be formulated in terms of a multi-linear algebraic structure on the space of smooth functions. In particular, we find algebraic expressions for Weingarten's formula, the Ricci curvature and the Codazzi-Mainardi equations.Comment: 14 page