Study CR Yamabe constant, flow, and soliton on CR manifolds.
problem Analyzing CR Yamabe constant, flow, and soliton on CR manifolds.
method Using maximum principles, proving uniqueness for CR Yamabe flow, and studying properties of CR Yamabe soliton.
result Uniqueness theorem for CR Yamabe flow and properties of CR Yamabe soliton.
The study classifies CR solitons based on C0-positivity and negativity.
problem Classifying CR solitons based on curvature positivity.
method Analyzing C0-positive pseudohermitian curvature and CR torsion flow. result Closed three-dimensional CR torsion solitons are the standard Sasakian space form.
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 …
In this paper, we show that the Webster scalar curvature of any compact CR Yamabe soliton must be constant.
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 study provides estimates for curvature of gradient Ricci solitons.
problem Estimating curvature of gradient Ricci solitons.
method Hamilton-Ivey estimates applied to steady and expanding gradient solitons.
result Quantitative lower bounds on curvature for specific solitons.
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.
Paper proves nonnegativity of CR Paneitz operator for embeddable CR manifolds.
problem Nonnegativity of CR Paneitz operator for embeddable CR manifolds.
method Analytical proof and geometric analysis.
result Affirmative solution to CR Yamabe problem for embeddable CR manifolds.
Paper shows CR Q-curvature orthogonal to CR pluriharmonic functions.
problem Understanding CR Q-curvature and pluriharmonic functions on CR manifolds. method Obtained a cohomological expression for the integral of CR Q-curvature and pluriharmonic functions. result CR Q-curvature is orthogonal to CR pluriharmonic functions. Connected sum of CR manifolds with positive CR Yamabe constant is possible.
problem Establishing the existence of a CR structure with positive CR Yamabe constant for connected sums of CR manifolds.
method Analyzing the properties of connected sums of CR manifolds with positive CR Yamabe constant.
result The connected sum of M1 and M2 admits a CR structure with positive CR Yamabe constant. CR structure on S³ with non-compact solutions to CR Yamabe problem.
problem Existence of non-compact solutions to CR Yamabe problem.
method Deforming standard CR structure of S³, using Lyapunov-Schmidt method.
result Existence of a blowing-up sequence of solutions.
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.
Study on CR Paneitz operator on non-embeddable CR manifolds.
problem Understanding the CR Paneitz operator on non-embeddable CR manifolds.
method Analysis of the CR Paneitz operator's properties on non-embeddable CR manifolds.
result CR Paneitz operator on the Rossi sphere has infinitely many negative eigenvalues.
CR Q-curvature flow solves CR manifold curvature conjecture.
problem Proving existence and convergence of CR Q-curvature flow in CR 3-manifolds.
method Deforming a contact form according to CR Q-curvature flow.
result Existence and smooth asymptotic convergence of CR Q-curvature flow.
The paper improves embedding results for CR submanifolds, showing that compact deformations stay embeddable.
problem Embedding compactly supported deformations of CR submanifolds in Cn+d. method Proving new embedding results for compactly supported CR deformations of 2-pseudoconcave CR submanifolds. result Compact deformations of 2-pseudoconcave CR submanifolds stay embeddable in Cn+d. Survey on CR Paneitz operator and manifold embeddability.
problem Global embeddability of CR manifolds.
method Survey of recent developments.
result Relations between CR Paneitz operator and embeddability.
In this note, we mainly focus on the existence of pseudo-Einstein contact forms, an upper bound eigenvalue estimate for the CR Paneitz operator and its applications to the uniformization theorem for Sasakian space form in an embeddable closed strictly pseudoconvex CR 3-manifold. Firstly, the existence of pseudo-Einstei…
We construct examples of nondegenerate CR manifolds with Levi form of signature (p,q), 2≤p≤q, which are compact, not locally CR flat, and admit essential CR vector fields. We also construct an example of a noncompact nondegenerate CR manifold with signature (1,n−1) which is not locally CR flat and admits …
New CR-structures lemma simplifies CR-manifold deformation proof.
problem Deformation unobstructedness of CR-manifolds.
method New Tian-Todorov lemma applied to CR-manifolds.
result Reproved deformation unobstructedness of CR-manifolds.
Study of CR-submanifolds in various Lorentzian manifolds.
problem Exploring CR-submanifolds in different Lorentzian structures.
method Analyzing properties and results of CR-submanifolds in LCS, LP-cosymplectic, S, and GKM manifolds.
result Obtained results on totally umbilical and geodesic CR-submanifolds.
Suppose M1 and M2 are two closed (compact with no boundary) spherical CR manifolds with positive CR Yamabe constant. In this note, we show that the connected sum of M1 and M2 also admits a spherical CR structure with positive CR Yamabe constant.
Improved CR Sobolev inequalities on CR sphere established.
problem Establishing CR Sobolev inequalities on CR sphere.
method Nice commutator identities involving CR intertwining operators.
result Simpler proof of existence and classification of minimizers.
Defines Lewy curves in para-CR geometry and characterizes their path geometries.
problem Characterizing path geometries defined by para-CR Lewy curves.
method Definition and characterization of para-CR Lewy curves in various dimensions.
result Lewy curves determine the para-CR structure up to sign in flat cases.
Study Szegő kernel on non-compact CR manifolds with specific conditions.
problem Analyzing Szegő kernel on non-compact CR manifolds.
method Establish Szegő kernel asymptotic expansions on non-compact strictly pseudoconvex CR manifolds with transversal CR R-action under natural geometric conditions. result Szegő kernel asymptotic expansions established on non-compact CR manifolds.
CR Killing operator derived from tractor calculus for CR structures.
problem Analyzing CR structures and their deformations.
method Tractor calculus and BGG operators applied to compatible almost CR structures.
result CR Killing operator is a first BGG operator for the modified adjoint tractor connection.
We study the minimality of an isometric immersion of a Riemannian manifold into a strictly pseudoconvex CR manifold endowed with the Webster metric hence formulate a version of the CR Yamabe problem for CR manifolds-with-boundary. This is shown to be a nonlinear subelliptic problem of variational origin.
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.
Study infinitesimal CR symmetries of accidental CR structures.
problem Accidental CR structures and their infinitesimal CR automorphisms.
method Complete explicit lists of infinitesimal CR automorphisms for specific CR models.
result Explicit generators of infinitesimal CR symmetries in extrinsic holomorphic coordinates.
We extend the notions of CR GJMS operators and Q-curvature to the case of partially integrable CR structures. The total integral of the CR Q-curvature turns out to be a global invariant of compact nondegenerate partially integrable CR manifolds equipped with an orientation of the bundle of contact forms, which is nontr…
The CR Frankel conjecture is proven for spherical CR manifolds.
problem Proving the CR Frankel conjecture in spherical CR manifolds.
method Criterion of pseudo-Einstein contact forms and analysis of first Kohn-Rossi cohomology group.
result CR Frankel conjecture affirmed for spherical CR manifolds.
Study CR manifolds focusing on Levi and contact-nondegeneracy.
problem Investigate CR algebras and their properties.
method Locally homogeneous case analysis, cross-marked painted root diagrams.
result Establish correspondences with CR-algebra properties and combinatorics.
CR Yamabe flow fails to converge on small deformations of the standard CR three-sphere.
problem CR Yamabe flow convergence
method Constructing a contact form with negative pseudohermitian mass
result CR Yamabe flow fails to converge on small deformations of the standard CR three-sphere
Study proves curvature flow existence on CR manifolds.
problem Existence of curvature flow solutions on CR manifolds.
method Prescribed Webster scalar curvature flow approach.
result Proves existence of curvature flow solutions on 3D CR manifolds.
Analyzes the Levi form on CR manifolds of any dimension.
problem Understanding the Levi form on CR manifolds of varying dimensions and codimensions.
method Analytical and geometrical study of the Levi form.
result Comprehensive insights into the Levi form on CR manifolds.
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.
We prove several theorems concerning the connection between the local CR embeddability of 3-dimensional CR manifolds, and the existence of algebraically special Maxwell and gravitational fields. We reduce the Einstein equations for spacetimes associated with such fields to a system of CR invariant equations on a 3-dime…
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.
We show that various notions of local homogeneity for CR-manifolds are equivalent. In particular, if germs at any two points of a CR-manifold are CR-equivalent, there exists a transitive local Lie group action by CR-automorphisms near every point.
The paper constructs CR manifolds with arbitrary Levi nondegeneracy.
problem Creating CR manifolds with specific Levi nondegeneracy properties.
method Using CR algebras from su(2) representations, studying iterated Levi forms, and local model equations. result Explicit construction and analysis of homogeneous CR manifolds with arbitrary Levi nondegeneracy.
Characterizes chains in 3D CR and para-CR structures.
problem Determining when a 3D path geometry comes from CR or para-CR chains.
method Provides necessary and sufficient conditions for a 3D path geometry to arise from chains of CR or para-CR 3-manifolds, and verifies computationally.
result Characterization of chains in 3D CR and para-CR structures.
We develop a complete local theory for CR embedded submanifolds of CR manifolds in a way which parallels the Ricci calculus for Riemannian submanifold theory. In particular, we establish the subtle relationship between the submanifold and ambient standard tractor bundles, allowing us to relate the respective normal Car…
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. 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.
We give a condition which ensures that the Paneitz operator of an embedded three-dimensional CR manifold is nonnegative and has kernel consisting only of the CR pluriharmonic functions. Our condition requires uniform positivity of the Webster scalar curvature and the stability of the CR pluriharmonic functions for a re…
We consider a class of stratified groups with a CR structure and a compatible control distance. For these Lie groups we show that the space of conformal maps coincide with the space of CR and anti-CR diffeomorphisms. Furthermore, we prove that on products of such groups, all CR and anti-CR maps are product maps, up to …
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.
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 describe a general procedure for constructing new Sasaki metrics of constant scalar curvature from old ones. Explicitly, we begin with a regular Sasaki metric of constant scalar curvature on a 2n+1-dimensional compact manifold M and construct a sequence, depending on four integer parameters, of rays of constant scal…