Lower semicontinuity of the Willmore functional for currents
Reiner Michael Schätzle · Journal of Differential Geometry · 2009
The weak mean curvature is lower semicontinuous under weak convergence of varifolds, that is, if µ k → µ weakly as varifolds thenIn contrast, if T k → T weakly as integral currents, then µ T may not have a locally bounded first variation even ifIn 1999, Luigi Ambrosio asked the question whether lower semicontinuity 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 R n+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 present article are used to prove a modified conjecture of De Giorgi that the sum of the area and the Willmore functional is the Γ-limit of a diffuse Landau-Ginzburg approximation.