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A note on the Pleijel theorem for $H$-type groups

Yaozhong W. Qiu

TL;DR

This work extends Pleijel-type nodal-domain bounds to subriemannian Laplacians on $H$-type groups, where the homogeneous dimension is $Q(n,m)=2n+2m$. The authors combine the sharp $L^2$-Sobolev constant from Yang (2024) with Weyl asymptotics, using a fiberwise diagonalisation after a partial Fourier transform to obtain an explicit bound $\widetilde{\gamma}_{n,m}$ governing $\limsup_{\ell\to\infty} \frac{\nu_\ell(\Omega)}{\ell}$. They show $\widetilde{\gamma}_{n,m}<1$ for all $(n,m)$ except the four pairs $(1,1),(2,1),(3,1),(2,2)$, with the remaining case $(2,3)$ treated separately, hence Pleijel's theorem holds for these groups. This generalises prior results for $\mathbb{H}_n\times\mathbb{R}^k$ and points toward extensions to Métivier groups and connections with isoperimetric questions in the subriemannian setting.

Abstract

We continue the program initiated by [J. Éc. Polytech., Math. 12, 1083-1160 (2025)] and show that the Pleijel theorem holds unconditionally on all but four $H$-type groups.

A note on the Pleijel theorem for $H$-type groups

TL;DR

This work extends Pleijel-type nodal-domain bounds to subriemannian Laplacians on -type groups, where the homogeneous dimension is . The authors combine the sharp -Sobolev constant from Yang (2024) with Weyl asymptotics, using a fiberwise diagonalisation after a partial Fourier transform to obtain an explicit bound governing . They show for all except the four pairs , with the remaining case treated separately, hence Pleijel's theorem holds for these groups. This generalises prior results for and points toward extensions to Métivier groups and connections with isoperimetric questions in the subriemannian setting.

Abstract

We continue the program initiated by [J. Éc. Polytech., Math. 12, 1083-1160 (2025)] and show that the Pleijel theorem holds unconditionally on all but four -type groups.
Paper Structure (2 sections, 2 theorems, 31 equations, 2 figures)

This paper contains 2 sections, 2 theorems, 31 equations, 2 figures.

Table of Contents

  1. Introduction and main result
  2. Proof of main result

Key Result

Theorem 1

Let $\mathbb{G} \cong \mathbb{R}^{2n}_x \times \mathbb{R}^m_t$ be a $H$-type group. Then Pleijel's theorem $\gamma(\mathbb{G}) < 1$ holds for all but $(n, m) \in \{(1, 1), (2, 1), (3, 1), (2, 2)\}$.

Figures (2)

  • Figure 1: Values of $\widetilde{\gamma}_{n, m}$ for $1 \leq n, m \leq 10$.
  • Figure 2: Values of $\overline{\gamma}_{n, m}$ for $1 \leq n, m \leq 10$.

Theorems & Definitions (4)

  • Theorem 1
  • Remark 1
  • Proposition 1
  • proof