Journal of the European Mathematical Society

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Volume 17, Issue 12, 2015, pp. 3081–3111
DOI: 10.4171/JEMS/580

Two-dimensional curvature functionals with superquadratic growth

Ernst Kuwert[1], Tobias Lamm[2] and Yuxiang Li[3]

(1) Mathematisches Institut, Universität Freiburg, Eckerstraße 1, 79104, Freiburg i. Br., Germany
(2) Institut für Analysis, Karlsruhe Institute of Technology (KIT), Englerstrasse 2, 76131, Karlsruhe, Germany
(3) Department of Mathematical Sciences, Tsinghua University, 100084, Beijing, China

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

Keywords: Curvature functionals, Palais–Smale condition

Kuwert Ernst, Lamm Tobias, Li Yuxiang: Two-dimensional curvature functionals with superquadratic growth. J. Eur. Math. Soc. 17 (2015), 3081-3111. doi: 10.4171/JEMS/580