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.