A remark on the non-compactness of 𝑊^{2,𝑑}-immersions of 𝑑-dimensional hypersurfaces

Li, Siran

doi:10.1090/proc/14710

A remark on the non-compactness of $W^{2,d}$-immersions of $d$-dimensional hypersurfaces
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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, we construct a family of such immersions whose limit fails to be an immersion of a manifold. This addresses the endpoint cases in J. Langer [Math. Ann. 270 (1985), pp. 223–234], and P. Breuning [J. Geom. Anal. 25 (2015), pp. 1344–1386]. 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 Hölder functions.

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