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

Author: Siran Li
Journal: Proc. Amer. Math. Soc. 148 (2020), 2245-2255
MSC (2010): Primary 58D10
Published electronically: January 28, 2020
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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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Siran Li
Affiliation: Department of Mathematics, Rice University, MS 136 P.O. Box 1892, Houston, Texas 77251-1892; and Department of Mathematics, McGill University, Burnside Hall, 805 Sherbrooke Street West, Montreal, Quebec, H3A 0B9, Canada

Keywords: Immersions, hypersurface, chord-arc surface, second fundamental form, Gauss map, compactness, bounded mean oscillations (BMO), finiteness theorems, Riemannian geometry
Received by editor(s): July 1, 2018
Received by editor(s) in revised form: March 17, 2019
Published electronically: January 28, 2020
Communicated by: Deane Yang
Article copyright: © Copyright 2020 American Mathematical Society