For an ancient solution of the mean curvature flow, we show that each time slice M t is contained in an affine subspace with dimension bounded in terms of the density and the dimension of the evolving submanifold. Recall that an ancient solution is a family M t that evolves under mean curvature flow for all negative time t. IntroductionThis paper deals with ancient solutions of mean curvature flow. An ancient solution is a family (M t ) of n-dimensional submanifold of R n+k that moves by mean curvature flow for all negative time t (or in general, for all times t < T for some fixed T ). We prove that each M t is contained in an affine subspace of bounded dimension. The bound on the dimension depends only on a bound on the density and on the dimension of the evolving manifold.A family (M t ) t∈(a,b) of n-dimensional submanifolds of R n+k moves by mean curvature if there exist immersions x t = x(·, t) : M n −→ R n+k of an n-dimensional manifold M n with images M t = x t (M n ) satisfying the evolution equationHere H(p, t) denotes the mean curvature vector of M t at x(p, t) for (p, t) ∈ M n × (a, b). The space-time track of the family (M t ) is the setIn particular, a minimal n-dimensional submanifold M of R n+k is a stationary solution of the evolution equation (1), because in M we have H = 0.Mean curvature flow can be defined not only for smooth manifolds, but also for more general objects. In particular, in [B] Brakke defines mean curvature flow for integral varifolds. A varifold is a measure-theoretic generalization of a manifold that can have singularities. Often a smooth solution of (1) develops singularities in finite time, and after that it becomes a varifold solution (also called a weak solution). Most of what we state in this paper for smooth solutions of mean curvature flow is also true for weak solutions.

Abstract. In this paper, we generalize a well-known estimate of ColdingMinicozzi for the extinction time of convex hypersurfaces in Euclidean space evolving by their mean curvature to a much broader class of parabolic curvature flows.It is well known that a convex closed hypersurface in R n+1 evolving by its mean curvature remains convex and vanishes in finite time. In [1], B. Andrews showed that the same holds for a much broader class of evolutions. T. H. Colding and W. P. Minicozzi in [7] give a bound on extinction time for mean curvature flow in terms of an invariant of the initial hypersurface that they call the width. In this paper, we generalize this estimate to the class evolutions considered by Andrews: All functions F in this class are assumed to be either concave or convex. In the concave case, we require an a priori pinching condition on the initial hypersurface M 0 . In the case of convexity a pinching condition is not needed, and we can take C 0 = 1 above. This class is of interest since it contains many classical flows as particular examples, such as the n-th root of the Gauss curvature, mean curvature, and hyperbolic mean curvature flows. The key to arriving at our estimate on extinction time is a uniform estimate on the rate of change of the width, for which preservation of convexity of the evolving hypersurface is fundamental.We also study flows which are similar to those considered by Andrews in [1], except that we allow for higher degrees of homogeneity. The motivation for this consideration is that the degree 1 homogeneity condition on the flows given in Andrews' paper excludes some naturally arising evolutions, such as powers of mean curvature, for which preservation convexity and extinction time estimates are known

We show that there exists a metric with positive scalar curvature on S 2 × S 1 and a sequence of embedded minimal cylinders that converges to a minimal lamination that, in a neighborhood of a strictly stable 2-sphere, is smooth except at two helicoid-like singularities on the 2-sphere. The construction is inspired by a recent example by D. Hoffman and B. White.

We consider a two-dimensional model of a porous medium where circular grains are uniformly distributed in a squared container. We assume that such medium is partially filled with water and that the stationary interface separating the water phase from the air phase is described by the balance of capillarity and gravity. Taking the unity as the average distance between grains, we identify four asymptotic regimes that depend on the Bond number and the size of the container. We analyse, in probabilistic terms, the possible global interfaces that can form in each of these regimes. In summary, we show that in the regimes where gravity dominates the probability of configurations of grains allowing solutions close to the horizontal solution is close to one. Moreover, in such regimes where the size of the container is sufficiently large we can describe deviations from the horizontal in probabilistic terms. On the other hand, when capillarity dominates while the size of the container is sufficiently large, we find that the probability of finding interfaces close to the graph of a given smooth curve without self-intersections is close to one.