Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Curvature estimates for minimal hypersurfaces in the Heisenberg group

Gianmarco Giovannardi, Andrea Pinamonti, Simone Verzellesi

TL;DR

This work studies minimal hypersurfaces in sub-Riemannian Heisenberg groups, extending the classical Simons identity and Kato inequalities to the horizontal setting. By deriving a full sub-Riemannian Simons identity and its contracted form, together with improved Kato-type estimates, the authors obtain integral curvature bounds for stable non-characteristic hypersurfaces and formulate a stability framework suited to Carnot groups. Under natural structural assumptions and a volume-growth condition, they prove a Bernstein-type rigidity in ${\mathbb H}^2$, showing that complete, stable non-characteristic hypersurfaces must be vertical hyperplanes, thereby advancing the understanding of sub-Riemannian Bernstein problems. The results provide new analytic tools for curvature control in the Heisenberg sense and offer a pathway toward rigidity phenomena in higher-dimensional sub-Riemannian geometries with potential applications in geometric measure theory on Carnot groups.

Abstract

This paper examines minimal hypersurfaces in sub-Riemannian Heisenberg groups. We extend the celebrated Simons formula and Kato inequality to the sub-Riemannian setting, and we apply them to obtain integral curvature estimates for stable hypersurfaces. These results lead to structural conditions that imply a Bernstein-type rigidity theorem for smooth, non-characteristic hypersurfaces in the second Heisenberg group.

Curvature estimates for minimal hypersurfaces in the Heisenberg group

TL;DR

This work studies minimal hypersurfaces in sub-Riemannian Heisenberg groups, extending the classical Simons identity and Kato inequalities to the horizontal setting. By deriving a full sub-Riemannian Simons identity and its contracted form, together with improved Kato-type estimates, the authors obtain integral curvature bounds for stable non-characteristic hypersurfaces and formulate a stability framework suited to Carnot groups. Under natural structural assumptions and a volume-growth condition, they prove a Bernstein-type rigidity in , showing that complete, stable non-characteristic hypersurfaces must be vertical hyperplanes, thereby advancing the understanding of sub-Riemannian Bernstein problems. The results provide new analytic tools for curvature control in the Heisenberg sense and offer a pathway toward rigidity phenomena in higher-dimensional sub-Riemannian geometries with potential applications in geometric measure theory on Carnot groups.

Abstract

This paper examines minimal hypersurfaces in sub-Riemannian Heisenberg groups. We extend the celebrated Simons formula and Kato inequality to the sub-Riemannian setting, and we apply them to obtain integral curvature estimates for stable hypersurfaces. These results lead to structural conditions that imply a Bernstein-type rigidity theorem for smooth, non-characteristic hypersurfaces in the second Heisenberg group.
Paper Structure (24 sections, 28 theorems, 239 equations)

This paper contains 24 sections, 28 theorems, 239 equations.

Table of Contents

  1. Introduction
  2. An application to the Bernstein problem
  3. Preliminaries
  4. The Heisenberg group
  5. Carnot-Carathéodory structure
  6. Perimeter and perimeter minimizers
  7. Geometric properties of non-characteristic hypersurfaces
  8. Non-characteristic hypersurfaces
  9. Tangent pseudohermitian connection
  10. Second fundamental forms and mean curvature
  11. Eigenvectors
  12. Tangential Laplace-Beltrami operators
  13. The commutation tensor
  14. Gauss-Codazzi equations
  15. Variation formulas
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $S$ be a smooth, immersed, minimal hypersurface in ${\mathbb{H}}^n$. Then on the non characteristic part of $S$, for any $X,Y \in\Gamma({\mathcal{H}} TS)$.

Theorems & Definitions (63)

  • Theorem 1.1: Simons formula
  • Theorem 1.2: Contracted Simons formula
  • Theorem 1.3: Improved Kato inequalities
  • Theorem 1.4: Integral curvature estimates in ${\mathbb{H}}^2$
  • Theorem 1.5
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • ...and 53 more