arXiv research

Formal Normal Form created for special CR singularities.

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

We describe the automorphisms of a singular multicontact structure, that is a generalisation of the Martinet distribution. Such a structure is interpreted as a para-CR structure on a hypersurface M of a direct product space R^2 x R^2. We introduce the notion of a finite type singularity analogous to CR geometry and, al…

This is an extensive (published) survey on CR geometry, whose major themes are: formal analytic reflection principle; generic properties of Systems of (CR) vector fields; pairs of foliations and conjugate reflection identities; Sussmann's orbit theorem; local and global aspects of holomorphic extension of CR functions;…

This paper continues the previous studies in two papers of Huang-Yin [HY3-4] on the flattening problem of a CR singular point of real codimension two sitting in a submanifold in ${\mathbb C}^{n+1}$ with $n+1\ge 3$, whose CR points are non-minimal. Partially based on the geometric approach initiated in [HY3] and a forma…

It is constructed a normal form for a class of real-smooth surfaces M\subset\mathbb{C}^{2} defined near a degenerate CR singularity.

Study on Hausdorff dimension of singular CR Yamabe problem.

Proves a theorem similar to Moser's using a normalization method.

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 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…

We express two CR invariant surface area elements in terms of quantities in pseudohermitian geometry. We deduce the Euler-Lagrange equations of the associated energy functionals. Many solutions are given and discussed. In relation to the singular CR Yamabe problem, we show that one of the energy functionals appears as …

We construct a formal normal form for a real 2-codimensional submanifold $M\subset\mathbb{C}^{N+1}$ near a CR singularity approximating the sphere. This result gives a higher dimensional extension of Huang-Yin's normal form in $\mathbb{C}^{2}$.

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.

We give a geometric derivation of Branson's Q-curvature in terms of the ambient metric associated with conformal structures; it naturally follows from the ambient metric construction of conformally invariant operators and can be applied to a large class of invariant operators. This procedure can be also applied to CR g…

The purpose of this paper is to give a counterexample of Theorem 10.4 in [Ann. of Math. 102 (1975), 223-290]. In the Harvey-Lawson paper, a global result is claimed, but only a local result is proven. This theorem has had a big impact on CR geometry for almost a quarter of a century because one can use the theory of is…

In this paper we prove geometric residue theorems for bundle maps over a compact manifold. The theory developed associates residues to the singularity submanifolds of the map for any invariant polynomial. The theory is then applied to a variety of settings: smooth maps between equidimensional manifolds, CR-singularitie…

There are proven few analogues of the Theorem of Moser using The Approximation Theorem of Artin.

A local uniqueness property of holomorphic functions on real-analytic nowhere minimal CR submanifolds of higher codimension is investigated. A sufficient condition called almost minimality is given and studied. A weaker necessary condition, being contained a possibly singular real-analytic Levi-flat hypersurface is stu…

CR 3-sphere rigidity proven through curvature invariant.

Let $(X, T^{1,0}X)$ be a compact connected orientable CR manifold of dimension $2n+1$ with non-degenerate Levi curvature. Assume that $X$ admits a connected compact Lie group action $G$. Under certain natural assumptions about the group action $G$, we show that the $G$-invariant Szegö kernel for $(0,q)$ forms is a comp…

We construct a family of analytic discs attached to a real submanifold M \subset $\mathbb{C}^{N+1}$ of codimension $2$ defined near a CR singularity.

We consider the sphere $\Sph^{2n+1}$ equipped with its standard CR structure. In this paper we construct explicit contact forms on $\Sph^{2n+1}\setminus \Sph^{2k+1}$, which are conformal to the standard one and whose related Webster metrics have constant Webster curvature; in particular the curvature is positive if $2k…

We address the problem of existence and uniqueness of a Levi-flat hypersurface $M$ in $C^n$ with prescribed compact boundary $S$ for $n\ge3$. The situation for $n\ge3$ differs sharply from the well studied case $n=2$. We first establish necessary conditions on $S$ at both complex and CR points, needed for the existence…

We consider local CR-immersions of a strictly pseudoconvex real hypersurface $M\subset\bC^{n+1}$, near a point $p\in M$, into the unit sphere $\mathbb S\subset\bC^{n+d+1}$ with $d>0$. Our main result is that if there is such an immersion $f\colon (M,p)\to \mathbb S$ and $d < n/2$, then $f$ is {\em rigid} in the sense t…

Study CR Yamabe constant and CR structures on manifolds.

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

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

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

Study on CR Paneitz operator on non-embeddable CR manifolds.

Survey on CR Paneitz operator and manifold 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 study here limit spaces $(M_α,g_α,p_α)\stackrel{GH}{\rightarrow} (Y,d_Y,p)$, where the $M_α$ have a lower Ricci curvature bound and are volume noncollapsed. Such limits $Y$ may be quite singular, however it is known that there is a subset of full measure $\cR(Y)\subseteq Y$, called {\it regular} points, along with c…

We construct examples of nondegenerate CR manifolds with Levi form of signature $(p,q)$, $2\leq p\leq q$, 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 …

Study of CR-submanifolds in various Lorentzian manifolds.

Suppose $M_{1}$ and $M_{2}$ are $3$-dimensional closed (compact without boundary) CR manifolds with positive CR Yamabe constant. In this note, we show that the connected sum of $M_{1}$ and $M_{2}$ also admits a CR structure with positive CR Yamabe constant.

In this paper we prove new embedding results for compactly supported deformations of $CR$ submanifolds of $\mathbb{C}^{n+d}$: We show that if $M$ is a $2$-pseudoconcave $CR$ submanifold of type $(n,d)$ in $\mathbb{C}^{n+d}$, then any compactly supported $CR$ deformation stays in the space of globally $CR$ embeddable in…

Suppose $M_{1}$ and $M_{2}$ 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 $M_{1}$ and $M_{2}$ also admits a spherical CR structure with positive CR Yamabe constant.

Improved CR Sobolev inequalities on CR sphere established.

Let $X$ be a compact connected strongly pseudoconvex $CR$ manifold of real dimension $2n-1$ in $\mathbb{C}^{N}$. For $n\ge 3$, Yau solved the complex Plateau problem of hypersurface type by checking a bunch of Kohn-Rossi cohomology groups in 1981. In this paper, we generalize Yau's conjecture on some numerical invarian…

We develop the notion of renormalized energy in CR geometry, for maps from a strictly pseudoconvex pseudohermitian manifold to a Riemannian manifold. This energy is a CR invariant functional, whose critical points, which we call CR-harmonic maps, satisfy a CR covariant subelliptic partial differential equation. The cor…

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

Study Szegő kernel on non-compact CR manifolds with specific conditions.

CR Killing operator derived from tractor calculus for CR structures.

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.

Study infinitesimal CR symmetries of accidental CR structures.

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…

We establish cross-ratio invariants for surfaces in 4-space in an analogous way to Uribe-Vargas's work for surfaces in 3-space. We study the geometric locii of local and multi-local singularities of ortogonal projections of the surface. The cross-ratio invariants at $P_3(c)$-points are used to recover two moduli in the…