CR embeddability of quotients of the Rossi sphere via spectral theory

We look at the action of finite subgroups of SU(2) on [Formula: see text], viewed as a CR manifold, both with the standard CR structure as the unit sphere in [Formula: see text] and with a perturbed CR structure known as the Rossi sphere. We show that quotient manifolds from these actions are indeed CR manifolds, and relate the order of the subgroup of SU(2) to the asymptotic distribution of the Kohn Laplacian’s eigenvalues on the quotient. We show that the order of the subgroup determines whether the quotient of the Rossi sphere by the action of that subgroup is CR embeddable. Finally, in the unperturbed case, we prove that we can determine the size of the subgroup by using the point spectrum.

Equivariant embeddings of strongly pseudoconvex Cauchy–Riemann manifolds

Let X be a CR manifold with transversal, proper CR action of a Lie group G. We show that the quotient X/G is a complex space such that the quotient map is a CR map. Moreover the quotient is universal, i.e. every invariant CR map into a complex manifold factorizes uniquely over a holomorphic map on X/G. We then use this result and complex geometry to prove an embedding theorem for (non-compact) strongly pseudoconvex CR manifolds with transversal $$G \rtimes S^1$$ G ⋊ S 1 -action. The methods of the proof are applied to obtain a projective embedding theorem for compact CR manifolds.

The complex green operator with Sobolev estimates up to a finite order

The purpose of this paper is to explore the geometry of a smooth CR manifold of hypersurface type and its relationship to the higher regularity properties of the complex Green operator on [Formula: see text]-forms in the [Formula: see text]-Sobolev space [Formula: see text] for a fixed [Formula: see text] and [Formula: see text].

Second eigenvalue of the CR Yamabe operator

Let [Formula: see text] be a compact, connected, strictly pseudo-convex CR manifold. In this paper, we give some properties of the CR Yamabe Operator [Formula: see text]. We present an upper bound for the Second CR Yamabe Invariant, when the First CR Yamabe Invariant is negative, and the existence of a minimizer for the Second CR Yamabe Invariant, under some conditions.

Regularity of CR mappings of abstract CR structures

We study the [Formula: see text] regularity problem for CR maps from an abstract CR manifold [Formula: see text] into some complex Euclidean space [Formula: see text]. We show that if [Formula: see text] satisfies a certain condition called the microlocal extension property, then any [Formula: see text]-smooth CR map [Formula: see text], for some integer [Formula: see text], which is nowhere [Formula: see text]-smooth on some open subset [Formula: see text] of [Formula: see text], has the following property: for a generic point [Formula: see text] of [Formula: see text], there must exist a formal complex subvariety through [Formula: see text], tangent to [Formula: see text] to infinite order, and depending in a [Formula: see text] and CR manner on [Formula: see text]. As a consequence, we obtain several [Formula: see text] regularity results generalizing earlier ones by Berhanu–Xiao and the authors (in the embedded case).

The asymptotics of the analytic torsion on CR manifolds with S1 action

Let [Formula: see text] be a compact connected strongly pseudoconvex CR manifold of dimension [Formula: see text], [Formula: see text] with a transversal CR [Formula: see text]-action on [Formula: see text]. We introduce the Fourier components of the Ray–Singer analytic torsion on [Formula: see text] with respect to the [Formula: see text]-action. We establish an asymptotic formula for the Fourier components of the analytic torsion with respect to the [Formula: see text]-action. This generalizes the asymptotic formula of Bismut and Vasserot on the holomorphic Ray–Singer torsion associated with high powers of a positive line bundle to strongly pseudoconvex CR manifolds with a transversal CR [Formula: see text]-action.

On the Sobolev–Poincaré Inequality of CR-manifolds

The purpose is to study the CR-manifold with a contact structure conformal to the Heisenberg group. In our previous work [22], we have proved that if the $Q^{\prime }$-curvature is nonnegative and the integral of $Q^{\prime }$-curvature is below the dimensional bound $c_1^{\prime }$, then we have the isoperimetric inequality. In this paper, we manage to deal with general contact structure conformal to the Heisenberg group, removing the condition that $Q^{\prime }$-curvature is nonnegative. We prove that the volume form $e^{4u}$ is a strong $A_{\infty }$ weight. As a corollary, we prove the Sobolev–Poincaré inequality on a class of CR-manifolds with integrable $Q^{\prime }$-curvature.

Szegö kernel asymptotic expansion on strongly pseudoconvex CR manifolds with S1 action

Let [Formula: see text] be a compact connected strongly pseudoconvex Cauchy–Riemann (CR) manifold of real dimension [Formula: see text] with a transversal CR [Formula: see text] action on [Formula: see text]. We establish an asymptotic expansion for the [Formula: see text]th Fourier component of the Szegö kernel function as [Formula: see text], where the expansion involves a contribution in terms of a distance function from lower dimensional strata of the [Formula: see text] action. We also obtain explicit formulas for the first three coefficients of the expansion.

Skew semi-invariant submanifolds of generalized quasi-Sasakian manifolds

In the present paper, we study a new class of submanifolds of a generalized Quasi-Sasakian manifold, called skew semi-invariant submanifold. We obtain integrability conditions of the distributions on a skew semi-invariant submanifold and also find the condition for a skew semi-invariant submanifold of a generalized Quasi-Sasakian manifold to be mixed totally geodesic. Also it is shown that a skew semi-invariant submanifold of a generalized Quasi-Sasakian manifold will be anti-invariant if and only if $A_{\xi}=0$; and the submanifold will be skew semi-invariant submanifold if $\nabla w=0$. The equivalence relations for the skew semi-invariant submanifold of a generalized Quasi-Sasakian manifold are given. Furthermore, we have proved that a skew semi-invariant $\xi^\perp$-submanifold of a normal almost contact metric manifold and a generalized Quasi-Sasakian manifold with non-trivial invariant distribution is $CR$-manifold. An example of dimension 5 is given to show that a skew semi-invariant $\xi^\perp$ submanifold is a $CR$-structure on the manifold.