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Equivalence of sub-Laplacian on Polarized groups

Antoni Kijowski, Sebastiano Nicolussi Golo, Ben Warhurst

TL;DR

This work extends classical Laplacian-symmetry results to sub-Riemannian geometry by showing that if a smooth map between sub-Riemannian Lie groups commutes with sub-Laplacians, then it must be a conformal submersion with a calculable factor. The authors derive an explicit formula for the transformed sub-Laplacian $\Delta_G(u\circ F)$ involving a drift term $b$, and characterize when the commutation reduces to a scaled pullback of $\Delta_H$, tying this to conformal submersion properties. In the Carnot setting, they prove rigidity: the existence of such a map forces a quotient relationship between groups and, in equal dimension, reduces to a dilation-translation-isomorphism, with $C^\infty$ regularity. The paper also develops a symplectic framework to compare Heisenberg-type groups via the symplectic spectrum, providing a complete isometry criterion and coordinate-based illustrations of non-equivalent sub-Laplacians. Overall, the results unify and sharpen the link between sub-Laplacian commutation and the underlying sub-Riemannian structure, with concrete implications for Carnot quotients and Heisenberg group classifications.

Abstract

We characterize smooth maps between sub-Riemannian Lie groups that commute with sub-Laplacians. We show they are sub-Riemannian conformal submersions. Our work clarifies the analysis initiated on Carnot groups in \cite{MR2363343}. In particular, we show that the sub-Laplacian in a Carnot group determines the sub-Riemannian structure.

Equivalence of sub-Laplacian on Polarized groups

TL;DR

This work extends classical Laplacian-symmetry results to sub-Riemannian geometry by showing that if a smooth map between sub-Riemannian Lie groups commutes with sub-Laplacians, then it must be a conformal submersion with a calculable factor. The authors derive an explicit formula for the transformed sub-Laplacian involving a drift term , and characterize when the commutation reduces to a scaled pullback of , tying this to conformal submersion properties. In the Carnot setting, they prove rigidity: the existence of such a map forces a quotient relationship between groups and, in equal dimension, reduces to a dilation-translation-isomorphism, with regularity. The paper also develops a symplectic framework to compare Heisenberg-type groups via the symplectic spectrum, providing a complete isometry criterion and coordinate-based illustrations of non-equivalent sub-Laplacians. Overall, the results unify and sharpen the link between sub-Laplacian commutation and the underlying sub-Riemannian structure, with concrete implications for Carnot quotients and Heisenberg group classifications.

Abstract

We characterize smooth maps between sub-Riemannian Lie groups that commute with sub-Laplacians. We show they are sub-Riemannian conformal submersions. Our work clarifies the analysis initiated on Carnot groups in \cite{MR2363343}. In particular, we show that the sub-Laplacian in a Carnot group determines the sub-Riemannian structure.
Paper Structure (22 sections, 17 theorems, 85 equations)

This paper contains 22 sections, 17 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. General notations
  4. Lie groups
  5. Polarized and Sub-Riemannian Lie groups
  6. Lie differential of order one
  7. Lie differential of order two
  8. Contact maps
  9. Homothetic projections
  10. Conformal submersion of sub-Riemannian Lie groups
  11. Gradient
  12. Haar measure and modular function
  13. Divergence
  14. Sub-Laplacian of a sub-Riemannian Lie group
  15. The computation for conformal submersions
  16. ...and 7 more sections

Key Result

Theorem A

Let $G$ and $H$ be sub-Riemannian Lie groups, with $\Omega_G\subset G$ and $\Omega_H\subset H$ open. Suppose that $F:\Omega_G\to \Omega_H$ is a $C^2$-smooth map, and that $\lambda:\Omega\to(0,+\infty)$, $b:\Omega\to V(H)$, and $c:\Omega\to\mathbb{R}$ are continuous functions. The following statement In the case $\dim(G)=\dim(H)$, then both conditions are equivalent to $F$ being a conformal $C^2$ d

Theorems & Definitions (34)

  • Theorem A
  • proof
  • Theorem B
  • Remark 1.1
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • ...and 24 more