arXiv research

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.

In this paper, we study the torsion flow which is served as the CR analogue of the Ricci flow in a closed pseudohermitian manifold. We show that there exists a unique smooth solution to the CR torsion flow in a small time interval with the CR pluriharmonic function as an initial data. In spirit, it is the CR analogue o…

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

We study the fillability (or embeddability) of $CR$ structures under the gauge-fixed Cartan flow. We prove that if the initial $CR$ structure is fillable with nowhere vanishing Tanaka-Webster curvature and free torsion, then it keeps having the same property after a short time. In the Appendix, we show the uniqueness o…

We propose a definition for analytic torsion of the contact complex on contact manifolds. We show it coincides with Ray-Singer torsion on any 3-dimensional CR Seifert manifold equipped with a unitary representation. In this particular case we compute it and relate it to dynamical properties of the Reeb flow. In fact th…

CR Q-curvature flow solves CR manifold curvature conjecture.

We study the fillability (or embeddability) of 3-dimensional $CR$ structures under the geometric flows. Suppose we can solve a certain second order equation for the geometric quantity associated to the flow. Then we prove that if the initial $CR$ structure is fillable, then it keeps having the same property as long as …

Study of spectral invariants on CR contact manifolds with circle action.

We deform the contact form by the amount of the Tanaka-Webster curvature on a closed spherical $CR$ three-manifold. We show that if a contact form evolves with free torsion and positive Tanaka-Webster curvature as initial data, then a certain Harnack inequality for the Tanaka-Webster curvature holds.

Let $X$ be a compact connected strongly pseudoconvex CR manifold of dimension $2n+1, n \ge 1$ with a transversal CR $S^1$-action on $X$. We introduce the Fourier components of the Ray-Singer analytic torsion on $X$ with respect to the $S^1$-action. We establish an asymptotic formula for the Fourier components of the an…

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

Quantization and reduction studied for CR manifolds with group actions.

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

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

The paper solves heat kernel asymptotics on non-degenerate CR manifolds.

CR Yamabe flow fails to converge on small deformations of the standard CR three-sphere.

Study proves curvature flow existence on CR manifolds.

We establish Bochner-type formulas for operators related to $CR$ automorphisms and spherical $CR$ structures. From such formulas, we draw conclusions about rigidity by making assumptions on the Tanaka-Webster curvature and torsion.

This paper mainly focuses on the CR analogue of the three-circle theorem in a complete noncompact pseudohermitian manifold of vanishing torsion being odd dimensional counterpart of Kähler geometry. In this paper, we show that the CR three-circle theorem holds if its pseudohermitian sectional curvature is nonnegative. A…

New CR almost Schur Lemma estimates curvature on compact manifolds.

This is the very first paper to focus on the CR analogue of Yau's uniformization conjecture in a complete noncompact pseudohermitian $(2n+1)$-manifold of vanishing torsion (i.e. Sasakian manifold) which is an odd dimensional counterpart of Kähler geometry. In this paper, we mainly deal with the problem of the sharp dim…

Characterizes CR manifolds as critical points of an energy functional.

We consider the CR Yamabe flow on a compact strictly pseudoconvex CR manifold $M$ of real dimension $2n+1$. We prove convergence of the CR Yamabe flow when $n=1$ or $M$ is spherical.

In this paper, we solve the so-called CR Poincaré-Lelong equation by solving the CR Poisson equation on a complete noncompact CR $(2n+1)$-manifold with nonegative pseudohermitian bisectional curvature tensors and vanishing torsion which is an odd dimensional counterpart of Kähler geometry. With applications of this sol…

The paper studies a flow to prescribe curvature on CR manifolds.

In this paper, we study the CR Yamabe flow with zero CR Yamabe invariant. We use the CR Poincaré inequality and a Gagliardo-Nirenberg type interpolation inequality to show that this flow has long time solution and the solution converges to a contact form with flat pseudo-Hermitian scalar curvature exponentially.

In this paper, we investigate the geometry and classification of three-dimensional CR Yamabe solitons. In the compact case, we show that any 3-dimensional CR Yamabe soliton must have constant Tanaka-Webster scalar curvature; we also obtain a classification under the assumption that their potential functions are in the …

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…

This is the first of two papers, in which we prove some properties of the Webster scalar curvature flow. More precisely, we establish the long-time existence, L^p convergence and the blow-up analysis for the solution of the flow. As a by-product, we prove the convergence of the CR Yamabe flow on the CR sphere. The resu…

We introduce a global Cauchy-Riemann($CR$)-invariant and discuss its behavior on the moduli space of $CR$-structures. We argue that this study is related to the Smale conjecture in 3-topology and the problem of counting complex structures. Furthermore, we propose a contact-analogue of Ray-Singer's analytic torsion. Thi…

Researchers study surface area functionals in CR manifolds, deducing equations for various cases.

We characterise the integrability of any co-CR quaternionic structure in terms of the curvature and a generalized torsion of the connection. Also, we apply this result to obtain, for example, the following. (1) New co-CR quaternionic structures built on vector bundles over a quaternionic manifold M, whose twistor space…

In this paper, we first obtain the sub-Laplacian comparison theorem in a complete noncompact pseudohermitian manifold of vanishing torsion (i.e. Sasakian manifold). Secondly, we derive the sub-gradient estimate for positive pseudoharmonic functions in a complete noncompact pseudohermitian manifold which satisfies the C…

We prove a CR Obata type result that if the first positive eigenvalue of the sub-Laplacian on a compact strictly pseudoconvex pseudohermitian manifold with a divergence free pseudohermitian torsion takes the smallest possible value then, up to a homothety of the pseudohermitian structure, the manifold is the standart S…

In this paper, we derive the CR Reilly's formula and its applications to studying of the first eigenvalue estimate for CR Dirichlet eigenvalue problem and embedded p-minimal hypersurfaces. In particular, we obtain the first Dirichlet eigenvalue estimate in a compact pseudohermitian (2n+1)-manifold with boundary and the…

Let $Ω$ be a bounded strictly pseudoconvex domain in $C^2$ with a smooth, connected and compact boundary M and having a CR structure $J_0$ induced from $C^2$. Assume this CR structure has zero Webster torsion. Then if we deform the CR structure through real-analytic dependence on the deformation parameter and such that…

In this paper, by applying a linear trace Li-Yau-Hamilton inequality for a positive (1,1)-form solution of the CR Hodge-Laplace heat equation and monotonicity of the heat equation deformation, we obtain an optimal gap theorem for a complete strictly pseudocovex CR manifold with nonnegative pseudohermitian bisectional c…

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…

We study a cross-ratio of four generic points of $S^3$ which comes from spherical CR geometry. We construct a homomorphism from a certain group generated by generic configurations of four points in $S^3$ to the pre-Bloch group $\mathcal {P}(\C)$. If $M$ is a $3$-dimensional spherical CR manifold with a CR triangulation…

In this paper we study the topology of pseudo convex CR manifolds whose Reeb flow preserves the Levi metric.

There are only some exceptional CR dimensions and codimensions such that the geometries enjoy a discrete classification of the pointwise types of the homogeneous models. The cases of CR dimensions $n$ and codimensions $n^2$ are among the very few possibilities of the so called parabolic geometries. Indeed, the homogene…

This is the second of two papers, in which we study the problem of prescribing Webster scalar curvature on the CR sphere as a given function f. Using the Webster scalar curvature flow, we prove an existence result under suitable assumptions on the Morse indices of f.

Let $X$ be a compact connected strongly pseudoconvex CR manifold of dimension $2n+1, n\ge 1$ with a transversal CR $S^1$-action on $X$. In this paper we introduce the Quillen metric on the determinant line of the Fourier components of the Kohn-Rossi cohomology on $X$ with respect to the $S^1$-action. We study the behav…

The general theory of parabolic geometries is applied to the study of the normal Cartan connections for all hyperbolic and elliptic 6-dimensional CR-manifolds of codimension two. The geometric meaning of the individual components of the torsion is explained and the chains of dimensions one and two are discussed. This s…

New dynamical torsion for contact Anosov flows connects to Reidemeister torsion.

The paper examines torsional rigidity bounds under geometric flows.

We establish an algorithm which computes formulae for the CR GJMS operators, the $P^\prime$-operator, and the $Q^\prime$-curvature in terms of CR tractors. When applied to torsion-free pseudo-Einstein contact forms, this algorithm both gives an explicit factorisation of the CR GJMS operators and the $P^\prime$-operator…

In this paper, we generalize the Cao-Yau's gradient estimate for the sum of squares of vector fields up to higher step under assumption of the generalized curvature-dimension inequality. With its applications, by deriving a curvature-dimension inequality, we are able to obtain the Li-Yau gradient estimate for the CR he…