Quantization and reduction studied for CR manifolds with group actions.
problem Quantization and reduction for CR manifolds with group actions.
method Consider a compact torsion free CR manifold X with a G-equivariant rigid CR line bundle L. The high tensor powers of L are studied, and a weighted G-invariant Fourier-Szegő operator projects onto the space of G-invariant CR sections. result Quantization commutes with reduction for sufficiently high tensor powers of the line bundle.
CR invariant differential operators on densities with leading part a power of the sub-Laplacian are derived. One family of such operators is constructed from the ``conformally invariant powers of the Laplacian'' via the Fefferman metric; the powers which arise for these operators are bounded in terms of the dimension. …
Constructs Einstein ACH metrics with CR structures, proving CR GJMS operators exist.
problem Constructing Einstein ACH metrics with specified CR structures.
method Refined Matsumoto's construction, solving Einstein equation to infinite order.
result Self-dual Einstein ACH metrics constructed for CR structures.
The paper derives formulas for CR invariant objects on Sasakian η-Einstein manifolds.
problem Deriving explicit formulas for CR invariant objects on Sasakian η-Einstein manifolds.
method The paper uses the theory of ambient spaces and CR manifolds to derive explicit formulas for CR invariant objects on Sasakian η-Einstein manifolds.
result The paper provides explicit formulas for CR invariant objects on Sasakian η-Einstein manifolds.
Paper connects CR geometry conditions to closed range of ∂ˉ-operator.
problem Establishing closed range of the ∂ˉ-operator on CR manifolds. method Defined third and fourth order CR invariants and used them to show closed range for ∂ˉ-Laplacian. result Third and fourth order CR invariants provide sufficient conditions for closed range of ∂ˉ-operator. We develop the natural tractor calculi associated to conformal and CR structures as a fundamental tool for the study of Fefferman's construction of a canonical conformal class on the total space of a circle bundle over a non--degenerate CR manifold of hypersurface type. In particular we construct and treat the basic ob…
Defines and proves CR invariants on five-manifolds.
problem Defines and studies CR invariants on CR five-manifolds.
method Defines global secondary CR invariants and proves their linear combination.
result Any global secondary CR invariant is a linear combination of total Q′-curvature, total I′-curvature, and a local CR invariant. To any smooth compact manifold M endowed with a contact structure H and partially integrable almost CR structure J, we prove the existence and uniqueness, modulo high-order error terms and diffeomorphism action, of an approximately Einstein ACH (asymptotically complex hyperbolic) metric g on M×(−1,0). W…
The paper constructs global CR invariants from renormalized characteristic forms.
problem Global CR invariants on strictly pseudoconvex domains.
method Renormalized characteristic forms of the Cheng--Yau metric.
result Generalizations of I′-curvature on CR five-manifolds. New invariant for CR maps from spheres discovered.
problem Identifying CR maps from spheres.
method Introducing a CR analogue of the Ahlfors derivative.
result The invariant distinguishes many sphere maps and vanishes for linear embeddings.
Develops a local theory for CR embedded submanifolds, relating their normal bundles and invariants.
problem Understanding the geometry and invariants of CR embedded submanifolds.
method Uses CR tractor calculus and connections to relate submanifold and ambient standard tractor bundles.
result Establishes a CR analogue of the Bonnet theorem for embedded submanifolds.
This paper studies CR geometry of transversal curves in the 3-sphere.
problem Investigating CR geometry of transversal curves in the 3-sphere.
method Using local CR invariants of the 3-sphere, four global invariants are considered: phase anomaly, CR spin, Maslov index, and CR self-linking number.
result Closed critical curves of the simplest CR invariant variational problem for generic transversal curves are studied.
The paper provides formulae for CR invariants in Sasakian η-Einstein manifolds.
problem Calculating CR invariants for a specific class of manifolds.
method Using renormalized characteristic forms, the Burns-Epstein invariant and other CR invariants are derived.
result The derived invariants are algebraically independent.
Innovates contact structure invariant, non-decreasing under operations.
problem Contact structures and CR structures on manifolds.
method Introduced invariant σ_c, proved non-decreasing under handle attaching and connected sum.
result Lower bound on σ_c in a specific case.
Study on CR structures on 3D Lie groups, focusing on equivalence and closed chains.
problem Characterizing CR structures on 3D Lie groups and their chains.
method Analyzing left-invariant CR structures on 3D Lie groups and their equivalence.
result All chains on G are closed if and only if G is CR equivalent to specific spherical CR structures.
Study invariant operators and vanishing theorems in CR geometry.
problem Analyzing invariant operators and vanishing theorems in CR geometry.
method Investigates Kohn-Dirac operators and derives CR invariant twistor operators.
result Proves vanishing theorems for harmonic spinors and Kohn-Rossi groups.
The paper generalizes CR invariants using renormalized characteristic forms.
problem Defining new CR invariants via renormalized characteristic forms.
method Introducing new curvatures for each renormalized characteristic form.
result The new curvatures' integrals match CR invariants constructed by Marugame.
Generalizes Dolbeault cohomology computation to Levi-flat CR structures on compact Lie groups.
problem Computing Dolbeault cohomology for Levi-flat CR structures on compact Lie groups.
method Algebraic classification of left-invariant CR structures combined with Pittie's result on compact Lie groups.
result Generalization of Dolbeault cohomology computation to Levi-flat CR structures.
Study geometric inequalities for CR-submanifolds using curvature invariants.
problem Geometric inequalities for CR-submanifolds in almost Hermitian spaces.
method Comparing mutual curvature invariants with Chen-type invariants and proving geometric inequalities.
result Proved geometric inequalities with intermediate mean curvature squared for CR-submanifolds.
We show that the conformally invariant fractional powers of the sub-Laplacian on the Heisenberg group are given in terms of the scattering operator for an extension problem to the Siegel upper halfspace. Remarkably, this extension problem is different from the one studied, among others, by Caffarelli and Silvestre.
We describe a complete system of invariants for 4-dimensional CR manifolds of CR dimension 1 and codimension 2 with Engel CR distribution by constructing an explicit canonical Cartan connection. We also investigate the relation between the Cartan connection and the normal form of the defining equation of an embedded En…
The study identifies two sources of invariants in 2--nondegenerate CR geometries.
problem Characterizing fundamental invariants of 2--nondegenerate CR geometries.
method Analyzes the harmonic curvature and the difference in complex structures.
result Nontrivial examples of CR geometries can be obtained as deformations of models.
The paper introduces new CR invariants for pseudo-Einstein manifolds.
problem Constructing CR invariants for pseudo-Einstein manifolds.
method Applying Cheeger-Simons differential characters to a modified normal tractor connection.
result Identifies differential characters with renormalized connections and derives formulas for characteristic numbers.
Study on cr-invariant variational problem for Legendrian curves in 3-sphere.
problem Lower-order cr-invariant variational problem for Legendrian curves in 3-sphere.
method Deduced Euler-Lagrange equations, investigated closed critical curves, characterized non-constant cr-curvature curves, proved cr-equivalence classes correspondence to rational points.
result Closed critical curves with non-constant cr-curvature are characterized and their cr-equivalence classes are in one-to-one correspondence with rational points of a connected planar domain.
We relate a recently introduced non-local geometric invariant of compact strictly pseudoconvex Cauchy-Riemann (CR) manifolds of dimension 3 to various eta-invariants in CR geometry: on the one hand a renormalized eta-invariant appearing when considering a sequence of metrics converging to the CR structure by expanding …
The paper introduces a statistical version of contact CR-product for Sasakian statistical manifolds.
problem Characterizing geometric properties of contact CR-submanifolds in Sasakian statistical manifolds.
method Characterization of integrability of invariant and anti-invariant distributions, development of results on specific types of contact CR submanifolds, introduction of statistical contact CR-product.
result Introduction of a statistical version of contact CR-product for Sasakian statistical manifolds.
Equivalence theorem for CR manifolds with circle action.
problem Embedding CR manifolds with circle action.
method Fourier-Szegő kernel analysis and asymptotic expansion.
result Established an equivariant Kodaira embedding theorem.
The paper classifies CR structures on 3D Lie groups, focusing on SL2(R).
problem Classifying CR structures on 3D Lie groups.
method Expository and modern language, using detailed classification results.
result SL2(R) admits two families of left-invariant CR structures: elliptic and hyperbolic.
The CR δ-invariant for CR-submanifolds was introduced in a recent article [B. Y. Chen, An optimal inequality for CR-warped products in complex space forms involving CR δ-invariant, Internat. J. Math. 23} (2012), no. 3, 1250045 (17 pages)]. In this paper, we prove two new optimal inequalities for anti-holomorphic su…
Researchers study surface area functionals in CR manifolds, deducing equations for various cases.
problem Investigating surface area functionals in 3D CR manifolds.
method Deduced Euler-Lagrange equations for energy functionals in various 3D CR manifolds.
result New equations deduced for surface area functionals on disk bundles, Rossi spheres, and 3D tori.
New approach linking CR Yamabe invariant to Sasaki structures.
problem Existence of constant transversal scalar curvature Sasaki structures.
method Drawing on CR Yamabe problem ideas, establishing link between invariant, Sasaki structures, and K-stability.
result CR Yamabe invariant value determines K-semistability of Sasaki manifolds.
Study CR Yamabe constant and CR structures on manifolds.
problem Understanding CR Yamabe constant and its role in CR geometry.
method Developed integral formulae and constructed families of CR structures.
result Found an infinite family of CR structures with varying CR Yamabe constants.
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 …
Models of 2-nondegenerate CR hypersurfaces in C^N are characterized and their defining equations simplified.
problem Characterizing and simplifying the defining equations of 2-nondegenerate CR hypersurfaces.
method Characterization of 2-nondegenerate models, derivation of normal forms, computation of CR invariants, derivation of infinitesimal symmetries.
result The moduli space of 2-nondegenerate CR hypersurfaces in C^N is infinite dimensional for N>3.
CR-harmonic maps defined for pseudoconvex manifolds.
problem Defining CR-harmonic maps in CR geometry.
method Developing renormalized energy and CR covariant subelliptic PDE.
result CR-harmonic maps satisfy a CR covariant subelliptic PDE.
We consider canonical fibrations and algebraic geometric structures on homogeneous CR manifolds, in connection with the notion of CR algebra. We give applications to the classifications of left invariant CR structures on semisimple Lie groups and of CR-symmetric structures on complete flag varieties.
In this paper we give a survey of the constructions in math.DG/0510061 of several new invariants for CR and contact manifolds. The latter extend previous constructions of Hirachi and Boutet de Monvel. In addition, we give simple algebro-geometric arguments proving that Hirachi's invariant vanishes on strictly pseudocon…
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 …
In this paper we provide a characterization of second order fully nonlinear CR invariant equations on the Heisenberg group, which is the analogue in the CR setting of the result proved in the Euclidean setting by A. Li and the first author (2003). We also prove a comparison principle for solutions of second order fully…
Derives a simpler equation for CR-curvature vanishing on certain complex hypersurfaces.
problem Deriving a simpler equation for CR-curvature vanishing on certain complex hypersurfaces.
method Utilizes two invariants discovered by S. Pocchiola to provide an alternative derivation of the CR-curvature vanishing condition.
result Provides an alternative derivation of the CR-curvature vanishing condition equivalent to the Monge equation.
We study a cross-ratio of four generic points of S3 which comes from spherical CR geometry. We construct a homomorphism from a certain group generated by generic configurations of four points in S3 to the pre-Bloch group $\mathcal {P}(\C)$. If M is a 3-dimensional spherical CR manifold with a CR triangulation…
This paper consists of two parts. In the first part we show that in odd dimension, as well as in even dimension below the critical weight (i.e. half the dimension), the logarithmic singularities of Schwartz kernels and Green kernels of conformal invariant pseudodifferential operators are linear combinations of Weyl con…
The paper computes CR GJMS operators and CR tractor calculus for pseudo-Einstein forms.
problem Computing CR GJMS operators and CR tractor calculus for pseudo-Einstein forms.
method Algorithm using CR tractors to compute CR GJMS operators, P' operator, and Q' curvature.
result Explicit factorization of CR GJMS operators and P' operator, and constant Q' curvature.
It was shown by Samelson and Wang that each compact Lie group K of even dimension admits left-invariant complex structures. When K has odd dimension it admits a left-invariant CR-structure of maximal dimension. This has been proved recently by Charbonnel and Khalgui who have also given a complete algebraic description …
Study shows CR Q-curvature vanishes and CR P' operator is formally self-adjoint.
problem CR Q-curvature and P' operator properties on compact manifolds.
method Analytical proofs for CR Q-curvature and P' operator properties.
result CR Q-curvature vanishes for compact strictly pseudoconvex CR manifolds and P' operator is formally self-adjoint.
Study minimizes CR surfaces in Heisenberg group with rotational symmetry.
problem Minimizing CR surfaces with vanishing CR invariant energy E1 in Heisenberg group. method Proved local uniqueness, classified global surfaces with rotational symmetry, computed second variation.
result Clifford torus is not a local minimizer of E1. Study CR geometry surface area elements and singular Yamabe problem solutions.
problem Solving the singular CR Yamabe problem in 3D CR geometry.
method Expressed CR invariant surface area elements, deduced Euler-Lagrange equations, provided solutions.
result One energy functional coefficient is shown to be proportional to the log term in volume renormalization.
The Q-curvature has been playing a central role in conformal geometry since its discovery by T. Branson. It has natural analogy in CR geometry, however, the CR Q-curvature vanishes on the boundary of a strictly pseudoconvex domain in C^{n+1} with a natural choice of contact form. This fact enables us to define a "secon…