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.

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We give a complete diffeomorphism classification of 1-connected closed manifolds M with integral homology H¤(M) ∼= Z ⊕ Z ⊕ Z, provided that dim(M) 6= 4.

We further develop the asymptotic analytic approach to the study of scattering diagrams. We do so by analyzing the asymptotic behavior of Maurer--Cartan elements of a differential graded Lie algebra constructed from a (not-necessarily tropical) monoid-graded Lie algebra. In this framework, we give alternative differential geometric proofs of the consistent completion of scattering diagrams, originally proved by Kontsevich--Soibelman, Gross--Siebert and Bridgeland. We also give a geometric interpretation of theta functions and their wall-crossing. In the tropical setting, we interpret Maurer--Cartan elements, and therefore consistent scattering diagrams, in terms of the refined counting of tropical disks. We also describe theta functions, in both their tropical and Hall algebraic settings, in terms of distinguished flat sections of the Maurer--Cartan-deformed differential. In particular, this allows us to give a combinatorial description of Hall algebra theta functions for acyclic quivers with non-degenerate skew-symmetrized Euler forms.