# 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 $f:\Sigma \to \mathbb R^n$, 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.

## 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