We compute the leading coefficient in the asymptotic expansion of the eigenvalue counting function for the Kohn Laplacian on the spheres. We express the coefficient as an infinite sum and as an integral.

In this paper, we first derive the CR volume doubling property, CR Sobolev inequality, and the mean value inequality. We then apply them to prove the CR analogue of Yau’s conjecture on the space consisting of all pseudoharmonic functions of polynomial growth of degree at most $d$ in a complete noncompact pseudohermitian $(2n+1)$-manifold. As a by-product, we obtain the CR analogue of the volume growth estimate and the Gromov precompactness theorem.

In this paper, we give some rigidity results for both harmonic and pseudoharmonic maps from pseudo-Hermitian manifolds into Riemannian manifolds or Kähler manifolds. Some foliated results, pluriharmonicity and Siu–Sampson type results are established for both harmonic maps and pseudoharmonic maps.

We consider the spectral behavior and noncommutative geometry of commutators $[P,f]$, where $P$ is an operator of order 0 with geometric origin and $f$ a multiplication operator by a function. When $f$ is Hölder continuous, the spectral asymptotics is governed by singularities. We study precise spectral asymptotics through the computation of Dixmier traces; such computations have only been considered in less singular settings. Even though a Weyl law fails for these operators, and no pseudodifferential calculus is available, variations of Connes’ residue trace theorem and related integral formulas continue to hold. On the circle, a large class of nonmeasurable Hankel operators is obtained from Hölder continuous functions $f$, displaying a wide range of nonclassical spectral asymptotics beyond the Weyl law. The results extend from Riemannian manifolds to contact manifolds and noncommutative tori.

We study the behavior of the eigenvalues of a sublaplacian ${{\Delta }_{b}}$ on a compact strictly pseudoconvex $\text{CR}$ manifold $M$, as functions on the set ${{\mathcal{P}}_{+}}$ of positively oriented contact forms on $M$ by endowing ${{\mathcal{P}}_{+}}$ with a natural metric topology.

We prove the CR version of the Obata's result for the first eigenvalue of the sub-Laplacian in the setting of a compact strictly pseudoconvex pseudohermitian three-dimensional manifold with non-negative CR-Paneitz operator which satisfies a Lichnerowicz-type condition. We show that if the first positive eigenvalue of the sub-Laplacian takes the smallest possible value, then, up to a homothety of the pseudohermitian structure, the manifold is the standard Sasakian three-dimensional unit sphere.

In this paper we construct analytic jet parametrizations for the germs of real analytic CR automorphisms of some essentially finite CR manifolds on their finite jet at a point. As an application we show that the stability groups of such CR manifolds have Lie group structure under composition with the topology induced by uniform convergence on compacta.