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


For two-dimensional, immersed closed surfaces f:ΣRnf:\Sigma \to \mathbb R^n, we study the curvature functionals Ep(f)\mathcal{E}^p(f) and Wp(f)\mathcal{W}^p(f) with integrands (1+A2)p/2(1+|A|^2)^{p/2} and (1+H2)p/2(1+|H|^2)^{p/2}, respectively. Here AA is the second fundamental form, HH is the mean curvature and we assume p>2p > 2. Our main result asserts that W2,pW^{2,p} critical points are smooth in both cases. We also prove a compactness theorem for Wp\mathcal{W}^p-bounded sequences. In the case of Ep\mathcal{E}^p this is just Langer's theorem [16], while for Wp\mathcal{W}^p we have to impose a bound for the Willmore energy strictly below 8π8\pi 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