A Remark on the Non-Compactness of $W^{2,d}$ Immersions of $d$-Dimensional Hypersurfaces

TL;DR

This paper investigates the limits of $W^{2,d}$ immersions of hypersurfaces, showing that limits can fail to be immersions, but under certain conditions, convergence to H"older parametrized sets is guaranteed.

Contribution

It constructs examples where limits of $W^{2,d}$ immersions are not immersions and proves convergence to H"older parametrized sets under a slow oscillation condition on the Gauss map.

Findings

Constructed a family of $W^{2,d}$ immersions with non-immersive limits.

Proved convergence to H"older parametrized sets under slow oscillation of the Gauss map.

Addressed endpoint cases in the theory of $W^{2,d}$ immersions.

Abstract

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.