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, D-79104, FREIBURG, 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