Lower semicontinuity of the willmore functional for currents

Reiner SchÄatzle Mathematisches Institut der Eberhard-Karls-UniversitÄat TÄubingen

Differential Geometry mathscidoc:1609.10104

Journal of Differential Geometry, 81, (2), 437-456, 2009
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.
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@inproceedings{reiner2009lower,
  title={LOWER SEMICONTINUITY OF THE WILLMORE FUNCTIONAL FOR CURRENTS},
  author={Reiner SchÄatzle},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160911180032634892761},
  booktitle={Journal of Differential Geometry},
  volume={81},
  number={2},
  pages={437-456},
  year={2009},
}
Reiner SchÄatzle. LOWER SEMICONTINUITY OF THE WILLMORE FUNCTIONAL FOR CURRENTS. 2009. Vol. 81. In Journal of Differential Geometry. pp.437-456. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160911180032634892761.
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