Published Paper

Inserted: 6 sep 2024
Last Updated: 5 sep 2025

Journal: Proc. Roy. Soc. Edinburgh Sect. A
Year: 2025
Doi: 10.1017/prm.2025.10047
Links: Link arxiv

Abstract:

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.