Ph.D. Thesis
Inserted: 10 apr 2025
Last Updated: 24 apr 2025
Year: 2025
Abstract:
In this thesis, we prove that the proximal unit normal bundle of the graph of a $ W^{2,n} $-function in $n$-variables carries a natural structure of Legendrian cycle. We then generalize Alexandrov's sphere theorems for higher-order mean curvature functions to hypersurfaces in $ \mathbb{R}^{n+1} $ which are locally graphs of arbitrary $ W^{2,n} $-functions, under a general degenerate ellipticity condition. The proof relies on extending the Montiel-Ros argument to this class of hypersurfaces and on the existence of the aforementioned Legendrian cycles. We also prove the existence of $n$-dimensional Legendrian cycles with $2n$-dimensional support, thus answering a question posed by Rataj and Zähle. Furthermore, we extend some of these results to Sobolev-type manifolds, representable as finite unions of $W^{2,n}$-regular graphs, and generalize Reilly's variational formulas in this context. Finally, we provide a very general version of the umbilicality theorem for Sobolev-type hypersurfaces.
Keywords: Legendrian cycles, $W^{2,n}$-functions, Lusin $(N)$-property, sphere theorem.
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