Weak immersions with second fundamental form in a critical Sobolev space

Abstract

We develop the analysis of Lipschitz immersions of $n$-dimensional manifolds into $\mathbb{R}^d$ having their second fundamental forms bounded in the critical Sobolev space $W^{\frac{n}{2}-1,2}$ in dimension $n\geq 4$ even and any codimension. We prove that, while such a weak immersion is not necessary $C^1$, it generates a $C^1$ differential structure on the domain. More precisely, for any such an immersion, there is an atlas in which the first fundamental form is continuous and the transition maps are $C^1$. We prove that this $C^1$ structure is diffeomorphic to the original one. This result is the starting point of the analysis of the behavior of sequences of weak immersions with second fundamental forms uniformly bounded in the critical Sobolev space $W^{\frac{n}{2}-1,2}$. In the second part of the paper we establish a weakly sequential closure theorem for such sequences. This analysis is motivated by the study of conformally invariant Lagrangian of immersions in dimension larger than two such as generalized Willmore energies, for instance the Graham--Reichert functional obtained in the computation of renormalized volumes of five-dimensional minimal submanifolds of the hyperbolic space $\mathbb{H}^{d+1}$.