Two-dimensional curvature functionals with superquadratic growth

  • Ernst Kuwert

    Universität Freiburg, Germany
  • Tobias Lamm

    Karlsruhe Institute of Technology (KIT), Germany
  • Yuxiang Li

    Tsinghua University, Beijing, China

Abstract

For two-dimensional, immersed closed surfaces , we study the curvature functionals and with integrands and , respectively. Here is the second fundamental form, is the mean curvature and we assume . Our main result asserts that critical points are smooth in both cases. We also prove a compactness theorem for -bounded sequences. In the case of this is just Langer's theorem [16], while for we have to impose a bound for the Willmore energy strictly below as an additional condition. Finally, we establish versions of the Palais–Smale condition for both functionals.

Cite this article

Ernst Kuwert, Tobias Lamm, Yuxiang Li, Two-dimensional curvature functionals with superquadratic growth. J. Eur. Math. Soc. 17 (2015), no. 12, pp. 3081–3111

DOI 10.4171/JEMS/580