Concentration-Compactness Principle for Trudinger–Moser’s Inequalities on Riemannian Manifolds and Heisenberg Groups: A Completely Symmetrization-Free Argument

The concentration-compactness principle for the Trudinger–Moser-type inequality in the Euclidean space was established crucially relying on the Pólya–Szegő inequality which allows to adapt the symmetrization argument. As far as we know, the first concentration-compactness principle of Trudinger–Moser type in non-Euclidean settings, such as the Heisenberg (and more general stratified) groups where the Pólya–Szegő inequality fails, was found in [J. Li, G. Lu and M. Zhu, Concentration-compactness principle for Trudinger–Moser inequalities on Heisenberg groups and existence of ground state solutions, Calc. Var. Partial Differential Equations 57 2018, 3, Paper No. 84] by developing a nonsmooth truncation argument. In this paper, we establish the concentration-compactness principle of Trudinger–Moser type on any compact Riemannian manifolds as well as on the entire complete noncompact Riemannian manifolds with Ricci curvature lower bound. Our method is a symmetrization-free argument on Riemannian manifolds where the Pólya–Szegő inequality fails. This method also allows us to give a completely symmetrization-free argument on the entire Heisenberg (or stratified) groups which refines and improves a proof in the paper of Li, Lu and Zhu. Our results also show that the bounds for the suprema in the concentration-compactness principle on compact manifolds are continuous and monotone increasing with respect to the volume of the manifold.

Critical nonlocal Schrödinger-Poisson system on the Heisenberg group

In this paper, we are concerned with the following a new critical nonlocal Schrödinger-Poisson system on the Heisenberg group: − a − b ∫ Ω | ∇ H u | 2 d ξ Δ H u + μ ϕ u = λ | u | q − 2 u + | u | 2 u , in Ω , − Δ H ϕ = u 2 , in Ω , u = ϕ = 0 , on ∂ Ω , $$\begin{equation*}\begin{cases} -\left(a-b\int_{\Omega}|\nabla_{H}u|^{2}d\xi\right)\Delta_{H}u+\mu\phi u=\lambda|u|^{q-2}u+|u|^{2}u,\quad &\mbox{in} \, \Omega,\\ -\Delta_{H}\phi=u^2,\quad &\mbox{in}\, \Omega,\\ u=\phi=0,\quad &\mbox{on}\, \partial\Omega, \end{cases} \end{equation*}$$ where Δ H is the Kohn-Laplacian on the first Heisenberg group H 1 $ \mathbb{H}^1 $ , and Ω ⊂ H 1 $ \Omega\subset \mathbb{H}^1 $ is a smooth bounded domain, a, b > 0, 1 < q < 2 or 2 < q < 4, λ > 0 and μ ∈ R $ \mu\in \mathbb{R} $ are some real parameters. Existence and multiplicity of solutions are obtained by an application of the mountain pass theorem, the Ekeland variational principle, the Krasnoselskii genus theorem and the Clark critical point theorem, respectively. However, there are several difficulties arising in the framework of Heisenberg groups, also due to the presence of the non-local coefficient (a − b∫Ω∣∇ H u∣2 dx) as well as critical nonlinearities. Moreover, our results are new even on the Euclidean case.

Complexity and randomness in the Heisenberg groups (and beyond)

By studying the commuting graphs of conjugacy classes of the sequence of Heisenberg groups $H_{2n+1}(p)$ and their limit $H_\infty(p)$ we find pseudo-random behavior (and the random graph in the limiting case). This makes a nice case study for transfer of information between finite and infinite objects. Some of this behavior transfers to the problem of understanding what makes understanding the character theory of the uni-upper-triangular group (mod p) “wild.” Our investigations in this paper may be seen as a meditation on the question: is randomness simple or is it complicated?

Geometry of left-invariant Randers metric on the Heisenberg groups

PurposeIn this paper, we consider the Heisenberg groups which play a crucial role in both geometry and theoretical physics.Design/methodology/approachIn the first part, we retrieve the geometry of left-invariant Randers metrics on the Heisenberg group H2n+1, of dimension 2n + 1. Considering a left-invariant Randers metric, we give the Levi-Civita connection, curvature tensor, Ricci tensor and scalar curvature and show the Heisenberg groups H2n+1 have constant negative scalar curvature.FindingsIn the second part, we present our main results. We show that the Heisenberg group H2n+1 cannot admit Randers metric of Berwald and Ricci-quadratic Douglas types. Finally, the flag curvature of Z-Randers metrics in some special directions is obtained which shows that there exist flags of strictly negative and strictly positive curvatures.Originality/valueIn this work, we present complete Reimannian geometry of left invarint-metrics on Heisenberg groups. Also, some geometric properties of left-invarainat Randers metrics will be studied.

HW2,2 loc-regularity for p-harmonic functions in Heisenberg groups

Let 1 < p ≤ 4 {1<p\leq 4} when n = 1 {n=1} , and 1 < p < 3 + 1 n - 1 {1<p<3+\frac{1}{n-1}} when n ≥ 2 {n\geq 2} . We obtain the second-order horizontal Sobolev HW loc 2 , 2 {\operatorname{HW}^{2,2}_{\mathrm{loc}}} -regularity of p-harmonic functions in the Heisenberg group ℍ n {{\mathbb{H}}^{n}} . This improves the known range of p obtained by Domokos and Manfredi in 2005.