Pleijel’s theorem for sub-Laplacians on Hn × Rk in the four unresolved cases

Prove that Pleijel’s theorem—i.e., lim sup_{ℓ→∞} v_ℓ(Ω)/ℓ < 1—holds for the Dirichlet realization of the canonical sub-Laplacian on the Heisenberg groups Hn and their products Hn × Rk, for all bounded open sets Ω of finite measure, in the four remaining cases (n,k) ∈ {(1,0), (2,0), (3,0), (1,1)}. Here the sub-Laplacian on Hn × Rk is −Δ_{Hn×Rk} = ∑_{j=1}^n (X_j^2 + Y_j^2) + ∑_{i=1}^k W_i^2 with horizontal vector fields X_j = ∂_{x_j} + 2 y_j ∂_z, Y_j = ∂_{y_j} − 2 x_j ∂_z on Hn and Euclidean derivatives W_i = ∂_{w_i} on Rk, and v_ℓ(Ω) denotes the maximal number of nodal domains of eigenfunctions associated with the ℓ-th Dirichlet eigenvalue.

Background

Pleijel’s theorem provides an asymptotic improvement over Courant’s bound on nodal domains when the dimension is larger than one. The paper develops a sub-Riemannian analogue, reducing the problem to nilpotent groups in Part 1 and establishing many cases for Hn × Rk in Part 2 via Weyl asymptotics and Faber–Krahn bounds.

The authors prove Pleijel’s theorem for all but four pairs (n,k), with additional conditional results assuming Pansu’s conjecture. The remaining unresolved cases involve low-dimensional Heisenberg groups and the product H × R. Establishing the theorem in these cases requires sharper control of the Faber–Krahn (or Sobolev/isoperimetric) constants and/or new spectral or geometric inputs in low homogeneous dimensions.