For two-dimensional, immersed closed surfaces $f:\Sigma \to {ℝ}^n$, we study the curvature functionals ${ℰ}^p\left(f\right)$ and ${𝒲}^p\left(f\right)$ 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 ${𝒲}^p$-bounded sequences. In the case of ${ℰ}^p$ this is just Langer’s theorem [16], while for ${𝒲}^p$ we have to impose a bound for the Willmore energy strictly below $8\pi$ as an additional condition....

We study vanishing theorems for Killing vector fields on complete stable hypersurfaces in a hyperbolic space ${ℍ}^{n+1}(-1)$. We derive vanishing theorems for Killing vector fields with bounded L²-norm in terms of the bottom of the spectrum of the Laplace operator.

In this paper, for complete Riemannian manifolds with radial Ricci or sectional curvature bounded from below or above, respectively, with respect to some point, we prove several volume comparison theorems, which can be seen as extensions of already existing results. In fact, under this radial curvature assumption, the model space is the spherically symmetric manifold, which is also called the generalized space form, determined by the bound of the radial curvature, and moreover, volume comparisons...

Nous étudions des analogues en dimension supérieure de l’inégalité de Burago $A\left(S\right)\le R^{2}T\left(S\right)$, avec $S$ une surface fermée de classe $C^2$ immergée dans ${ℝ}^3$, $A\left(S\right)$ son aire et $T\left(S\right)$ sa courbure totale. Nous donnons un exemple explicite qui prouve qu’une inégalité analogue de la forme $\mathrm{vol}\left(M\right)\le C_n{R}^{n}T\left(M\right)$, avec $C_n\>0$ une constante, ne peut être vraie pour une hypersurface fermée $M$ de classe $C^2$ dans ${ℝ}^{n+1}$, $n\ge 3$. Nous mettons toutefois en évidence une condition suffisante sur la courbure de Ricci sous laquelle l’inégalité est vérifiée en dimension $n=3$. En dimension...

Let $𝒴$ be a smooth oriented Riemannian manifold which is compact, connected, without boundary and with second homology group without torsion. In this paper we characterize the sequential weak closure of smooth graphs in $B^n\times 𝒴$ with equibounded Dirichlet energies, $B^n$ being the unit ball in ${ℝ}^n$. More precisely, weak limits of graphs of smooth maps $u_k:B^n\to 𝒴$ with equibounded Dirichlet integral give rise to elements of the space ${cart}^{2,1}(B^n\times 𝒴)$ (cf. [4], [5], [6]). In this paper we prove that every element $T$ in ${cart}^{2,1}(B^n\times 𝒴)$ is the weak limit...