A Remark on the Non-Compactness of W2,dW^{2,d} Immersions of dd-Dimensional Hypersurfaces

We consider the continuous $W^{2,d}$ immersions of $d$-dimensional hypersurfaces in $\mathbb{R}^{d+1}$ with second fundamental forms uniformly bounded in $L^d$. Two results are obtained: first, a family of such immersions is constructed, whose limit fails to be an immersion of a manifold. This addresses the endpoint cases in J. Langer and P. Breuning. Second, under the additional assumption that the Gauss map is slowly oscillating, we prove that any family of such immersions subsequentially converges to a set locally parametrised by H\"older functions.Comment: 10p

Optimal regularity for the Pfaff system and isometric immersions in arbitrary dimensions

We prove the existence, uniqueness, and $W^{1,2}$-regularity for the solution to the Pfaff system with antisymmetric $L^2$-coefficient matrix in arbitrary dimensions. Hence, we establish the equivalence between the existence of $W^{2,2}$-isometric immersions and the weak solubility of the Gauss--Codazzi--Ricci equations on simply-connected domains. The regularity assumptions of these results are sharp. As an application, we deduce a weak compactness theorem for $W^{2,2}_{\rm loc}$-immersions.Comment: 10 pages. The proof of Theorem 1.1 has been simplified, as the approximation arguments are unnecessary. Also, Section 6 on the weak rigidity of isometric immersions has been adde