Lower Semicontinuity of the Willmore Functional for Currents

The weak mean curvature is lower semicontinuous under weak convergence of varifolds that is if μk → μ weakly as varifolds then ‖ ~ Hμ ‖Lp(μ)≤ lim infk→∞ ‖ ~ Hμk ‖Lp(μk) . In contrast, if Tk → T weakly as integral currents then μT may not have locally bounded first variation even if ‖ ~ HμTk ‖L∞(μk) is bounded. In 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 Rn+1 using results from [Sch04]. Here we prove it in any dimension and codimension down to the desired exponent p = 2 . For p = n = 2 , this corresponds to the Willmore functional.