Two-dimensional curvature functionals with superquadratic growth

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.