Related papers: Solving the Kohn Laplacian on asymptotically flat …

Let $X$ be an abstract not necessarily compact orientable CR manifold of dimension $2n-1$, $n\geqslant2$. Let $\Box^{(q)}_{b}$ be the Gaffney extension of Kohn Laplacian for $(0,q)$-forms. We show that the spectral function of…

Let $X$ be a complex manifold and $M\subset X$ a compact, smooth, pseudoconvex CR manifold of dimension $2n-1$. (Here $n\ge 3$ or, in case $n=2$, it is made the extra assumption that $\dib_b$ has closed range on functions.) Assume that…

Let $(X,T^{1,0}X)$ be a compact strictly pseudoconvex CR manifold which is CR embeddable into the complex Euclidean space. We show that $T^{1,0}X$ can be approximated in $\mathscr{C}^\infty$-topology by a sequence of strictly pseudoconvex…

Let M be a smooth, compact, orientable, weakly pseudoconvex manifold of dimension 3, embedded in C^N (N greater than or equal to 2), of codimension one or more in C^N, and endowed with the induced CR structure. Assuming that the tangential…

We solve $\square_b$ on a class of non-compact 3-dimensional strongly pseudoconvex CR manifolds via a certain conformal equivalence. The idea is to make use of a related $\square_b$ operator on a compact 3-dimensional strongly pseudoconvex…

This note is aimed at simplifying current literature about compactness estimates for the Kohn-Laplacian on CR manifolds. The approach consists in a tangential basic estimate in the formulation given by the first author in \cite{Kh10} which…

In this paper, we consider a non-degenerate CR manifold (M,H(M),J) with a given pseudo-Hermitian 1-form {\theta}, and endow the CR distribution H(M) with any Hermitian metric h instead of the Levi form L_{{\theta}}. This induces a natural…

In this paper, we introduce the notions of $p$-Hermitian-symplectic and $p$-pluriclosed compact complex manifolds as generalisations for an arbitrary positive integer $p$ not exceeding the complex dimension of the manifold of the standard…

Let $M$ be a closed symplectic manifold of dimension $2n$ with non-ellipticity. We can define an almost K\"ahler structure on $M$ by using the given symplectic form. Hence, we have a $\G=\pi_1(M)$-invariant almost K\"ahler structure on the…

Let $(M,\theta)$ be a compact strictly pseudoconvex pseudohermitian manifold which is CR embedded into a complex space. In an earlier paper, Lin and the authors gave several sharp upper bounds for the first positive eigenvalue $\lambda_1$…

Let $X$ be a compact connected CR manifold with a transversal CR $S^1$-action of real dimension $2n-1$, which is only assumed to be weakly pseudoconvex. Let $\Box_b$ be the $\overline{\partial}_b$-Laplacian, with respect to a $T$-rigid…

We develop Berezin-Toeplitz quantization in a non-compact complex geometric setting. Let $(X,\Theta)$ be a Hermitian manifold, $(L,h^L)$ a positive holomorphic line bundle, and $(E,h^E)$ a holomorphic Hermitian vector bundle. Assuming that…

In this paper, we generalize the CR Obata theorem for the Kohn Laplacian to a closed strictly pseudoconvex CR manifold with a weighted volume measure. More precisely, we first derive the weighted CR Reilly's formula associated with the…

We study the eigenvalues of the Kohn Laplacian on a closed embedded strictly pseudoconvex CR manifold as functionals on the set of positive oriented pseudohermitian structures $\mathcal{P}_{+}$. We show that the functionals are continuous…

In this article, we consider a complete, non-compact almost Hermitian manifold whose curvature is asymptotic to that of the complex hyperbolic plane. Under natural geometric conditions, we show that such a manifold arises as the interior of…

Let M^3 be a closed CR 3-manifold. In this paper we derive a Bochner formula for the Kohn Laplacian in which the pseudo-hermitian torsion plays no role. By means of this formula we show that the non-zero eigenvalues of the Kohn Laplacian…

We exhibit examples of compact three-dimensional CR manifolds of positive Webster class, {\em Rossi spheres}, for which the pseudo-hermitian mass as defined in \cite{CMY17} is negative, and for which the infimum of the CR-Sobolev quotient…

Let (M,g,J) be a compact Hermitian manifold with a smooth boundary. Let $\Delta_p$ and $D_p$ be the realizations of the real and complex Laplacians on p forms with either Dirichlet or Neumann boundary conditions. We generalize previous…

We define an ADM-like mass, called p-mass, for an asymptotically flat pseudohermitian manifold. The p-mass for the blow-up of a compact pseudohermitian manifold (with no boundary) is identified with the first nontrivial coefficient in the…

Based on uniform CR Sobolev inequality and Moser iteration, this paper investigates the convergence of closed pseudo-Hermitian manifolds. In terms of the subelliptic inequality, the set of closed normalized pseudo-Einstein manifolds with…