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A characterization of horizontally totally geodesic hypersurfaces in Heisenberg groups

Andrea Pinamonti, Simone Verzellesi

Abstract

In this paper we achieve a first concrete step towards a better understanding of the so-called Bernstein problem in higher dimensional Heisenberg groups. Indeed, in the sub-Riemannian Heisenberg group $\mathbb{H}^n$, with $n\geq 2$, we show that the only entire hypersurfaces with vanishing horizontal symmetric second fundamental form are hyperplanes. This result relies on a sub-Riemannian characterization of a higher dimensional ruling property, as well as on the study of sub-Riemannian geodesics on Heisenberg hypersurfaces.

A characterization of horizontally totally geodesic hypersurfaces in Heisenberg groups

Abstract

In this paper we achieve a first concrete step towards a better understanding of the so-called Bernstein problem in higher dimensional Heisenberg groups. Indeed, in the sub-Riemannian Heisenberg group , with , we show that the only entire hypersurfaces with vanishing horizontal symmetric second fundamental form are hyperplanes. This result relies on a sub-Riemannian characterization of a higher dimensional ruling property, as well as on the study of sub-Riemannian geodesics on Heisenberg hypersurfaces.
Paper Structure (12 sections, 23 theorems, 197 equations)

This paper contains 12 sections, 23 theorems, 197 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Heisenberg group $\mathbb{H}^n$
  4. Sub-Riemannian structure on $\mathbb{H}^n$
  5. Carnot-Carathéodory structure on $\mathbb{H}^n$
  6. Horizontal perimeter and horizontal gradient
  7. Hypersurfaces in $\mathbb H^n$
  8. Intrinsic graphs
  9. Higher dimensional ruled hypersurfaces
  10. Ruled hypersurfaces with countable characteristic set
  11. Horizontal second fundamental forms on ${\mathbb{H}}^n$
  12. Local existence of geodesics on hypersurfaces

Key Result

Theorem 1.1

Let $S\subseteq{\mathbb{H}}^n$ be a hypersurface without boundary of class $C^2$. The following are equivalent: If in addition $n\geqslant 2$ and $S$ is (topologically) closed, then $(i)$ and $(ii)$ hold if and only if $S$ is a hyperplane.

Theorems & Definitions (52)

  • Theorem 1.1: Main Theorem
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 3.1: Local ruling property
  • Definition 3.1: Global ruling property
  • Proposition 3.2
  • proof
  • Proposition 3.2
  • ...and 42 more