arXiv research

The paper proves existence and eigenvalue bounds for CR structures, leading to uniformization theorems.

CR Q-curvature flow solves CR manifold curvature conjecture.

CR singularities in 3-manifolds can be cancelled by an isotopy supported in an arbitrarily small neighborhood of a Seifert surface.

We classify the normal CR structures on $S^3$ and their automorphism groups. Together with [3], this closes the classification of normal CR structures on contact 3-manifolds. We give a criterion to compare 2 normal CR structures, and we show that the underlying contact structure is, up to homotopy, unique.

3-manifolds are CR uniformized on spheres, proving a conjecture.

We compute a recently introduced geometric invariant of stricly pseudoconvex CR 3-manifolds for certain circle invariant spherical CR structures on Seifert manifolds. We give applications to the problem of filling the CR manifold by a complex hyperbolic manifold, and more generally by a Kaehler-Einstein or an Einstein …

Uniformizes CR structure on a specific 3-manifold.

The study classifies CR solitons based on $C_0$-positivity and negativity.

Defines Lewy curves in para-CR geometry and characterizes their path geometries.

The paper solves CR curvature prescription on pseudo-Einstein 3-manifolds.

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 are bounded below by a positive constant provided the CR Paneitz operator is non-n…

Characterizes chains in 3D CR and para-CR structures.

Introduces a new CR invariant for co-oriented contact structures on closed 3-manifolds

We consider the discrete representations of 3-manifold groups into $PU(2,1)$ that appear in the Falbel-Koseleff-Rouillier census, such that the peripheral subgroups have cyclic unipotent holonomy. We show that two of these representations have conjugate images, even though they represent different 3-manifold groups. Th…

In this paper we define the torsion flow, a CR analogue of the Ricci flow. For homogeneous CR manifolds we give explicit solutions to the torsion flow illustrating various kinds of behavior. We also derive monotonicity formulas for CR entropy functionals. As an application, we classify torsion breathers.

We study normal CR compact manifolds in dimension 3. For a choice of a CR Reeb vector field, we associate a Sasakian metric on them, and we classify those metrics. As a consequence, the underlying manifolds are topologically finite quotiens of the 3-sphere or of a circle bundle over a Riemann surface of positive genus.…

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 expansion of the Green function for the CR Laplacian. We deduce an integral formu…

In this paper we develop a method to compute the Burns-Epstein invariant of a spherical CR homology sphere, up to an integer, from its holonomy representation. As application, we give a formula for the Burns-Epstein invariant, modulo an integer, of a spherical CR structure on a Seifert fibered homology sphere in terms …

Let $M^{2n-1}$ be the smooth boundary of a bounded strongly pseudo-convex domain $Ω$ in a complete Stein manifold $V^{2n}$. Then (1) For $n \ge 3$, $M^{2n-1}$ admits a pseudo-Eistein metric; (2) For $n \ge 2$, $M^{2n-1}$ admits a Fefferman metric of zero CR Q-curvature; and (3) for a compact strictly pseudoconvex CR em…

We prove that any real analytic strictly pseudoconvex CR 3-manifold is the boundary (at infinity) of a unique selfdual Einstein metric defined in a neighborhood. The proof uses a new construction of twistor space based on singular rational curves.

The paper embeds CR manifolds into twistor spaces and constructs neutral hyperkähler metrics.

The Fefferman metric connects CR manifolds to conformal geodesics in 3D.

Complex hyperbolic triangle groups yield specific 3-manifolds at infinity.

New invariant distinguishes tight contact structures on 3-tori.

The metrics of S. Y. Cheng and S.-T. Yau are considered on a strictly pseudoconvex domains in a complex manifold. Such a manifold carries a complete Kähler-Einstein metric if and only if its canonical bundle is positive. We consider the restricted case in which the CR structure on $\partial M$ is normal. In this case M…

The paper classifies CR structures on 3D Lie groups, focusing on SL2(R).

New CR representations are found and shown to be redundant.

Let $(\hat{X}, T^{1,0} \hat{X})$ be a compact orientable CR embeddable three dimensional strongly pseudoconvex CR manifold, where $T^{1,0} \hat{X}$ is a CR structure on $\hat{X}$. Fix a point $p \in \hat{X}$ and take a global contact form $\hatθ$ so that $\hatθ$ is asymptotically flat near $p$. Then $(\hat{X}, T^{1,0} …

Unified CR-twistor spaces for $G_2$ and $Spin(7)$ structures.

Defines flag structures on real 3-manifolds and proves null curvature models.

This paper studies CR manifolds and embeddability in complex spaces.

Study reveals hidden accidental parabolics in complex hyperbolic geometry.

Let $(\mathbf{M}^{3},J,θ_{0})$ be a closed pseudohermitian 3-manifold. Suppose the associated torsion vanishes and the associated $Q$-curvature has no kernel part with respect to the associated Paneitz operator. On such a background pseudohermitian 3-manifold, we study the change of the contact form according to a cert…

Engel structures on bundles over 3-manifolds in complex 3-space.

Slim curves on 3-sphere help spherical CR uniformizations.

We prove that the classical set of moves for standard spines of 3-manifolds (i.e. the MP-move and the V-move) does not suffice to relate to each other any two standard skeleta of a 3-manifold with marked boundary. We also describe a condition on the 3-manifold with marked boundary that tells whether the generalised set…

Study on pseudo-Einstein 3-manifolds, calculating determinant changes under conformal transformations.

Study on pseudo-Einstein 3-manifolds for a specific inequality, introducing Robin mass.

Study CR Yamabe constant and CR structures on manifolds.

Paper proves nonnegativity of CR Paneitz operator for embeddable CR manifolds.

Paper shows CR $Q$-curvature orthogonal to CR pluriharmonic functions.

Study CR Yamabe constant, flow, and soliton on CR manifolds.

Connected sum of CR manifolds with positive CR Yamabe constant is possible.

Connected sum of CR manifolds preserves positive CR Yamabe constant.

Solves CR Poincaré-Lelong equation on CR manifolds, revealing structures and solitons.

CR Yamabe flow converges on specific CR manifolds.

Counterexamples found for CR manifolds with mixed signature.

CR structure on S³ with non-compact solutions to CR Yamabe problem.