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A second-order operator for horizontal quasiconvexity in the Heisenberg group and application to convexity preserving for horizontal curvature flow

Antoni Kijowski, Qing Liu, Ye Zhang, Xiaodan Zhou

TL;DR

The paper develops a nonlinear second-order operator $L$ for horizontal quasiconvexity in the Heisenberg group ${\mathbb H}$ and shows that $L[f]\ge 0$ in the viscosity sense is necessary for h-quasiconvexity, while $L[f]>0$ is sufficient. It extends the Euclidean Barron–Goebel–Jensen framework to the sub-Riemannian setting, carefully addressing singularities at characteristic points via envelopes $L^*$ and $\overline{L}$. A uniform notion of $h$-quasiconvexity is introduced and linked to preservation under horizontal curvature flow, with a game-theoretic approximation used to construct and control solutions. The paper also provides practical methods to generate admissible initial data (e.g., star-shaped sets via Minkowski-type functionals) and verifies h-quasiconvexity preservation in star-shaped and rotationally symmetric scenarios, advancing PDE-based convexity theory in sub-Riemannian geometry. The results have potential implications for geometric flows and convexity hull constructions in the Heisenberg group, contributing a PDE-based toolkit for sub-Riemannian convexity analysis.

Abstract

This paper is concerned with a PDE approach to horizontally quasiconvex (h-quasiconvex) functions in the Heisenberg group based on a nonlinear second order elliptic operator. We discuss sufficient conditions and necessary conditions for upper semicontinuous, h-quasiconvex functions in terms of the viscosity subsolution to the associated elliptic equation. Since the notion of h-quasiconvexity is equivalent to the horizontal convexity (h-convexity) of the function's sublevel sets, we further adopt these conditions to study the h-convexity preserving property for horizontal curvature flow in the Heisenberg group. Under the comparison principle, we show that the curvature flow starting from a star-shaped h-convex set preserves the h-convexity during the evolution.

A second-order operator for horizontal quasiconvexity in the Heisenberg group and application to convexity preserving for horizontal curvature flow

TL;DR

The paper develops a nonlinear second-order operator for horizontal quasiconvexity in the Heisenberg group and shows that in the viscosity sense is necessary for h-quasiconvexity, while is sufficient. It extends the Euclidean Barron–Goebel–Jensen framework to the sub-Riemannian setting, carefully addressing singularities at characteristic points via envelopes and . A uniform notion of -quasiconvexity is introduced and linked to preservation under horizontal curvature flow, with a game-theoretic approximation used to construct and control solutions. The paper also provides practical methods to generate admissible initial data (e.g., star-shaped sets via Minkowski-type functionals) and verifies h-quasiconvexity preservation in star-shaped and rotationally symmetric scenarios, advancing PDE-based convexity theory in sub-Riemannian geometry. The results have potential implications for geometric flows and convexity hull constructions in the Heisenberg group, contributing a PDE-based toolkit for sub-Riemannian convexity analysis.

Abstract

This paper is concerned with a PDE approach to horizontally quasiconvex (h-quasiconvex) functions in the Heisenberg group based on a nonlinear second order elliptic operator. We discuss sufficient conditions and necessary conditions for upper semicontinuous, h-quasiconvex functions in terms of the viscosity subsolution to the associated elliptic equation. Since the notion of h-quasiconvexity is equivalent to the horizontal convexity (h-convexity) of the function's sublevel sets, we further adopt these conditions to study the h-convexity preserving property for horizontal curvature flow in the Heisenberg group. Under the comparison principle, we show that the curvature flow starting from a star-shaped h-convex set preserves the h-convexity during the evolution.
Paper Structure (15 sections, 18 theorems, 142 equations)

This paper contains 15 sections, 18 theorems, 142 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Necessary and sufficient conditions for horizontal quasiconvexity
  4. Application to horizontal curvature flow
  5. Notations
  6. Second order quasiconvexity operator in the Euclidean space
  7. Horizontal quasiconvexity in the Heisenberg group
  8. Second order h-quasiconvexity operator
  9. Necessary condition for h-quasiconvexity
  10. Sufficient condition for h-quasiconvexity
  11. Uniformly h-quasiconvex functions
  12. Application to convexity preserving for horizontal curvature flow
  13. The game-theoretic approximation
  14. H-quasiconvexity preserving property
  15. Construction of initial functions

Key Result

Theorem 1.1

Let $\Omega$ be an h-convex open set in ${\mathbb H}$. Let $f\in USC(\Omega)$. Then $L[f]\geq 0$ in $\Omega$ holds in the viscosity sense if $f$ is h-quasiconvex in $\Omega$. Moreover, $f$ is h-quasiconvex in $\Omega$ if $L[f]>0$ in $\Omega$ holds in the viscosity sense.

Theorems & Definitions (47)

  • Theorem 1.1: Characterization of H-quasiconvex functions
  • Theorem 1.2: H-quasiconvexity preserving property
  • Definition 2.1: Subsolutions associated to quasiconvexity operator
  • Theorem 2.2: BGJ13
  • Theorem 2.3: BGJ13
  • Theorem 2.4: BGJ13
  • Remark 2.5: Characterization with upper semicontinuous envelop
  • Theorem 2.6: An improved sufficient condition for quasiconvexity
  • proof
  • Definition 3.1: H-convex sets
  • ...and 37 more