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Alexandrov sphere theorems for $ W^{2,n} $-hypersurfaces

Mario Santilli, Paolo Valentini

TL;DR

The paper generalizes Alexandrov's sphere theorem to bounded $W^{2,n}$-domains under a degenerate ellipticity condition for higher-order mean curvatures, replacing the classical smooth machinery with a Sobolev-analytic, Legendrian-currents framework. It develops the proximal unit normal bundle as a Legendrian carrier, extends Reilly-type variational formulae and Minkowski/Heintze–Karcher inequalities to $W^{2,n}$-domains, and proves an Alexandrov-type sphere rigidity result in this non-smooth setting. A key outcome is the construction of Legendrian cycles whose supports may have dimension $2n$, answering questions posed by Rataj–Zaehle, and a broad Nabelpunktsatz for Sobolev graphs, linking almost-umbilic graphs to planar or spherical geometry. Collectively, these results extend classical curvature rigidity to Sobolev regularity, providing new geometric-measure-theoretic tools for analyzing higher-order curvatures in non-smooth hypersurfaces with potential applications in geometric analysis and PDEs.

Abstract

In this paper we extend Alexandrov's sphere theorems for higher-order mean curvature functions to $ W^{2,n} $-regular hypersurfaces under a general degenerate elliptic condition. The proof is based on the extension of the Montiel-Ros argument to the aforementioned class of hypersurfaces and on the existence of suitable Legendrian cycles over them. Using the latter we can also prove that there are $ n $-dimensional Legendrian cycles with $ 2n $-dimensional support, hence answering a question by Rataj and Zaehle. Finally we provide a very general version of the umbilicality theorem for Sobolev-type hypersurfaces.

Alexandrov sphere theorems for $ W^{2,n} $-hypersurfaces

TL;DR

The paper generalizes Alexandrov's sphere theorem to bounded -domains under a degenerate ellipticity condition for higher-order mean curvatures, replacing the classical smooth machinery with a Sobolev-analytic, Legendrian-currents framework. It develops the proximal unit normal bundle as a Legendrian carrier, extends Reilly-type variational formulae and Minkowski/Heintze–Karcher inequalities to -domains, and proves an Alexandrov-type sphere rigidity result in this non-smooth setting. A key outcome is the construction of Legendrian cycles whose supports may have dimension , answering questions posed by Rataj–Zaehle, and a broad Nabelpunktsatz for Sobolev graphs, linking almost-umbilic graphs to planar or spherical geometry. Collectively, these results extend classical curvature rigidity to Sobolev regularity, providing new geometric-measure-theoretic tools for analyzing higher-order curvatures in non-smooth hypersurfaces with potential applications in geometric analysis and PDEs.

Abstract

In this paper we extend Alexandrov's sphere theorems for higher-order mean curvature functions to -regular hypersurfaces under a general degenerate elliptic condition. The proof is based on the extension of the Montiel-Ros argument to the aforementioned class of hypersurfaces and on the existence of suitable Legendrian cycles over them. Using the latter we can also prove that there are -dimensional Legendrian cycles with -dimensional support, hence answering a question by Rataj and Zaehle. Finally we provide a very general version of the umbilicality theorem for Sobolev-type hypersurfaces.
Paper Structure (18 sections, 35 theorems, 289 equations)

This paper contains 18 sections, 35 theorems, 289 equations.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Legendrian cycles and sphere theorem
  4. The support of Legendrian cycles
  5. The Nabelpunktsatz
  6. Acknowledgments
  7. Notation and background
  8. Basic notions from geometric measure theory
  9. Differential forms and currents
  10. Legendrian currents
  11. The proximal unit normal bundle
  12. $W^{2,n}$-functions
  13. Legendrian cycles over $W^{2,n}$-graphs
  14. The support of Legendrian cycles
  15. Reilly-type variational formulae for $W^{2,n}$-domains
  16. ...and 3 more sections

Key Result

Theorem 1

A bounded and connected $C^2$-domain $\Omega \subseteq \mathbf{R}^{n+1}$ must be a round ball, provided there exist a $C^1$ function $\varphi : \mathbf{R}^n \rightarrow \mathbf{R}$ and $\lambda \in \mathbf{R}$ such that and for every $p \in \partial \Omega$. Here ${\raisebox{\depth}{$\chi$}}_{\Omega,1} \leq \ldots \leq {\raisebox{\depth}{$\chi$}}_{\Omega,n}$ are the principal curvatures of $\par

Theorems & Definitions (98)

  • Theorem : Alexandrov
  • Theorem A: cf. Theorem \ref{['W2n functions normal cycles']} and Theorem \ref{['W2n domains']}
  • Theorem B
  • Theorem C: cf. Theorem \ref{['Alexandrov']} and Remark \ref{['Alexandrov rmk']}
  • Theorem D
  • Theorem E: cf. Theorem \ref{['Nabelpunktsatz']}
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • ...and 88 more