The weak mean curvature is lower semicontinuous under weak convergence of varifolds, that is, if ¹k ! ¹ weakly as varifolds then k ~H¹ kLp(¹)· lim infk!1 k ~H¹k kLp(¹k). In contrast, if Tk ! T weakly as integral currents, then ¹T may not have a locally bounded ¯rst variation even if k ~H¹Tk kL1(¹k) is bounded. In 1999, Luigi Ambrosio asked the question whether lower semi- continuity of the weak mean curvature is true when T is assumed to be smooth. This was proved in [AmMa03] for p > n = dim T in Rn+1 using results from [Sch04]. Here we prove this in any dimension and codimension down to the desired exponent p = 2. For p = n = 2, this corresponds to the Willmore functional. In a forthcoming joint work [RoSch06], main steps of the pre- sent article are used to prove a modi¯ed conjecture of De Giorgi that the sum of the area and the Willmore functional is the ¡-limit of a di®use Landau-Ginzburg approximation.

We consider an embedded convex compact ancient solution t to the curve shortening flow in R2. We prove that t is either a family of contracting circles, which is a type I ancient solution, or a family of evolving Angenent ovals, which is of type II.

No abstract uploaded!

Let ϕ be an analytic function defined on the unit disk$D$, with ϕ($D$)⊂$D$, ϕ(0)=0, and ϕ′(0)=λ≠0. Then by a classical result of G. Kœnigs, the sequence of normalized iterates Φ_{$n$}/λ^{$n$}converges uniformly on compact subsets of$D$to a function σ analytic in$D$which satisfies$σ$°φ=λ$σ$. It is of interest in the study of composition operators to know if, whenever σ belongs to a Hardy space$H$_{$p$}, the sequence Φ_{$n$}/λ^{$n$}converges to σ in the norm of$H$_{$p$}. We show that this is indeed the case, generalizing a result of P. Bourdon obtained under the assumption that ϕ is univalent.

Cwikel's bound is extended to an operator-valued setting. One application of this result is a semi-classical bound for the number of negative bound states for Schrödinger operators with operator-valued potentials. We recover Cwikel's bound for the Lieb-Thirring constant$L$_{0,3}which is far worse than the best available by Lieb (for scalar potentials). However, it leads to a uniform bound (in the dimension$d$≥3) for the quotient$L$_{0,d}/L^{cl}_{0,d}is the so-called classical constant. This gives some improvement in large dimensions.