arXiv research

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

A pseudo-Einstein contact form plays a crucial role in defining some global invariants of closed strictly pseudoconvex CR manifolds. In this paper, we prove that the existence of a pseudo-Einstein contact form is preserved under deformations as a real hypersurface in a fixed complex manifold of complex dimension at lea…

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

The orthogonal decomposition of the Webster curvature provides us a way to characterize some canonical metrics on a pseudo-Hermitian manifold. We derive some subelliptic differential inequalities from the Weitzenböck formulas for the traceless pseudo-Hermitian Ricci tensor and the Chern-Moser tensor of Sasakian manifol…

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…

Normalizes pseudo-Einstein contact forms for easier analysis.

Defines and proves CR invariants on five-manifolds.

The paper calculates variations of Einstein-Hilbert action on CR manifolds.

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 study the problem of prescribing the $\bar Q^{\prime}$-curvature on pseudo-Einstein CR 3-manifolds. In the first stage we study the problem in the compact setting and we show that under natural assumptions, one can prescribe any positive CR pluriharmonic function. In the second stage we study the probl…

Given a three dimensional pseudo-Einstein CR manifold $(M,T^{1,0}M,θ)$, we study the existence of a contact structure conformal to $θ$ for which the logarithmic Hardy-Littlewood-Sobolev (LHLS) inequality holds. Our approach closely follows \cite{Ok1} in the Riemannian setting. For this purpose, we introduce the notion …

We show that any contact form whose Fefferman metric admits a nonzero parallel vector field is pseudo-Einstein of constant pseudohermitian scalar curvature. As an application we compute the curvature groups of the total space of the canonical circle bundle over a CR manifold.

In this paper, we study a general almost Schur Lemma on pseudo-Hermitian (2n+1)-manifolds $(M,J,θ)$ for $n\geq2$. When the equality of almost Schur inequality holds, we derive the contact form $θ$ is pseudo-Einstein and the pseudo-Hermitian scalar curvature is constant.

Let $M$ be a real hypersurface of a complex space form $M^n(c)$, $c\neq0$, $n\geq 3$. We show that the Ricci tensor $S$ of $M$ satisfies $S(X,Y)=ag(X,Y)$ for any vector fields $X$ and $Y$ on the holomorphic distribution, $a$ being a constant, if and only if $M$ is a pseudo-Einstein real hypersurface.

We prove that the total CR $Q$-curvature vanishes for any compact strictly pseudoconvex CR manifold. We also prove the formal self-adjointness of the $P^\prime$-operator and the CR invariance of the total $Q^\prime$-curvature for any pseudo-Einstein manifold without the assumption that it bounds a Stein manifold.

The $Q$-prime curvature is a local invariant of pseudo-Einstein contact forms on integrable strictly pseudoconvex CR manifolds. The transformation law of the $Q$-prime curvature under scaling is given in terms of a differential operator, called the $P$-prime operator, acting on the space of CR pluriharmonic functions. …

The nonnegativity of the CR Paneitz operator plays a crucial role in three-dimensional CR geometry. In this paper, we prove this nonnegativity for embeddable CR manifolds. This result and previous works give an affirmative solution of the CR Yamabe problem for embeddable CR manifolds. We also show the existence of a co…

In this paper, we study contact forms on the three- dimensional Heisenberg manifold with its standard CR structure. We discover that the $Q'$-curvature, introduced by Branson, Fontana and Morpurgo [BFM13] on the CR three-sphere and then generalized to any pseudo-Einstein CR three manifold by Case and Yang [CY95], contr…

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…

Based on uniform CR Sobolev inequality and Moser iteration, this paper investigates the convergence of closed pseudo-Hermitian manifolds. In terms of the subelliptic inequality, the set of closed normalized pseudo-Einstein manifolds with some uniform geometric conditions is compact. Moreover, the set of closed normaliz…

In this note, we first give a criterion of pseudo-Einstein contact forms and then affirm the CR analogue of Frankel conjecture in a closed, spherical, strictly pseudoconvex CR manifold of nonnegative pseudohermitian curvature on the space of smooth representatives of the first Kohn-Rossi cohomology group. Moreover, we …

We introduce curvature-adapted foliations of complex hyperbolic space and study some of their properties. Generalized pseudo-Einstein hypersurfaces of complex hyperbolic space are classified. Analogous results for curvature-adapted hypersurfaces of quaternionic hyperbolic space are also obtained.

We construct contact forms with constant $Q^\prime$-curvature on compact three-dimensional CR manifolds which admit a pseudo-Einstein contact form and satisfy some natural positivity conditions. These contact forms are obtained by minimizing the CR analogue of the $II$-functional from conformal geometry. Two crucial st…

We construct contact forms with constant $Q^\prime$-curvature on compact three-dimensional CR manifolds which admit a pseudo-Einstein contact form and satisfy some natural positivity conditions. These contact forms are obtained by minimizing the CR analogue of the $II$-functional from conformal geometry. Two crucial st…

A closed CR 3-manifold is said to have $C_{0}$-positive pseudohermitian curvature if $(W+C_{0}Tor)(X,X)>0$ for any $0\neq X\in T_{1,0}(M)$. We discover an obstruction for a closed CR 3-manifold to possess $C_{0}$-positive pseudohermitian curvature. We classify closed three-dimensional CR Yamabe solitons according to $C…

Researchers create higher-dimensional $\mathcal{I}^\prime$-curvatures and find counterexamples to the Hirachi conjecture.

Using the methods of moving frames and exterior differential systems, we show that there exist Hopf hypersurfaces in complex hyperbolic space CH^2 with any specified value of the Hopf principal curvature less than or equal to the corresponding value for the horosphere. We give a construction for all such hypersurfaces …

New CR invariant treatment of Rumin complex via differential forms.

The paper generalizes CR invariants using renormalized characteristic forms.

With the help of a generalization of the Fermat principle in general relativity, we show that chains in CR geometry are geodesics of a certain Kropina metric constructed from the CR structure. We study the projective equivalence of Kropina metrics and show that if the kernel distributions of the corresponding 1-forms a…

Study on gradient pseudo-Ricci solitons on real hypersurfaces.

A real hypersurface in the complex quadric $Q^m=SO_{m+2}/SO_mSO_2$ is said to be $\mathfrak A$-principal if its unit normal vector field is singular of type $\mathfrak A$-principal everywhere. In this paper, we show that a $\mathfrak A$-principal Hopf hypersurface in $Q^m$, $m\geq3$ is an open part of a tube around a t…

Extends Tian theorem to Vaisman manifolds for approximations.

In 1972, K. Kenmotsu studied a class of almost contact Riemannian manifolds. Later, such a manifold was called a Kenmotsu manifold. This paper, we studied Kenmotsu manifolds with $(2n+s)$-dimensional $s-$contact metric manifold and this manifold, we have called generalized Kenmotsu manifolds. Necessary and sufficient c…

Study on a new type of manifolds that generalize almost C-manifolds.

Stabilized convex symplectic manifolds are equivalent to flexible Weinstein manifolds.

New class of complex manifolds defined, properties studied.

A selfsimiar manifold is a Riemannian manifold $\left(M,g\right)$ endowed with a homothetic vector field $ξ$. We characterize global selfsimilar manifolds and describe the structure of local selfsimilar manifolds. We prove that any selfsimilar manifold with a potential homothetic vector field is a conical Riemannian ma…

Smoothly embed 3-manifolds in 5-manifolds, simplifying topological to smooth.

The study provides a structure theorem for a new class of noncompact 3-manifolds.

Condition for intersection of real flag manifolds in complex flag manifold.

The paper studies extended quasi-Einstein manifolds with special geometric properties and solitons.

Conditions for flat manifolds as cusp cross-sections in arithmetic hyperbolic manifolds.

Classifies 3-manifolds from simplified (2,0)-trisections of 4-manifolds.

Study on 3D manifolds with specific tensor structures and their properties.

New manifold type PNDP-manifold defined with Einstein warped product structure.

Optimal inequalities for systole, inradius, and volume in hyperbolic 3-manifolds

The paper explores F-manifolds and metrics, constructing canonical structures.