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Alexandrov sphere theorems for
$ W^{2,n} $-hypersurfaces
- Part of
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics , First View
- Published online by Cambridge University Press:
- 26 August 2025, pp. 1-49
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- Article
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We prove that the proximal unit normal bundle of the subgraph of a
$W^{2,n} $-function carries a natural structure of Legendrian cycle. This result is used to obtain an Alexandrov-type sphere theorem for hypersurfaces in 
$ \mathbf{R}^{n+1} $, which are locally graphs of arbitrary 
$W^{2,n} $-functions. We also extend the classical umbilicality theorem to 
$ W^{2,1} $-graphs, under the Lusin (N) condition for the graph map.
