CR Functions Research Articles

On CR maps between hyperquadrics and Winkelmann hypersurfaces

In this paper, we study CR maps between hyperquadrics and Winkelmann hypersurfaces. Based on a previous study on the CR Ahlfors derivative of Lamel–Son and a recent result of Huang–Lu–Tang–Xiao on CR maps between hyperquadrics, we prove that a transversal CR map from a hyperquadric into a hyperquadric or a Winkelmann hypersurface extends to a local holomorphic isometric embedding with respect to certain Kähler metrics if and only if the Hermitian part of its CR Ahlfors derivative vanishes on an open set of the source. Our proof is based on relating the geometric rank of a CR map into a hyperquadric and its CR Ahlfors derivative.

Holomorphic support functions for uniformly pseudoconvex hypersurfaces, with an application to CR maps

We construct holomorphic support functions for smooth weakly pseudoconvex hypersurfaces with Levi form of constant rank. These are then applied to show that formal holomorphic curves which are tangential to infinite order to such a hypersurface must be formally contained in its Levi foliation. As a consequence, we obtain a holomorphic deformation theorem for nowhere smooth CR maps into smooth pseudoconvex hypersurfaces with one-dimensional Levi foliation, strengthening a very general result of Lamel and Mir about formal deformations in this particular case.

Regularity of CR maps into uniformly pseudoconvex hypersurfaces and applications to proper holomorphic maps

We study regularity properties of CR maps in positive codimension valued in pseudoconvex manifolds which carry a nontrivial Levi foliation. We introduce an invariant which can be used to deduce that any sufficiently regular CR map from a minimal manifold into such a foliated target is either generically smooth or geometrically highly constrained, and to show generic smoothness of sufficiently regular CR transversal CR maps between pseudoconvex hypersurfaces. As an application, we discuss boundary regularity of proper holomorphic maps into bounded symmetric domains.

Rational CR maps

Given a (vector-valued) polynomial numerator, we determine explicit formulas for the denominator of a rational sphere map when the denominator is of degree at most [Formula: see text] and indicate how to obtain them in general.

Creatine chemical exchange saturation transfer (CEST) CMR imaging reveals myocardial early involvement in idiopathic inflammatory myopathy at 3T: feasibility and initial experience.

To measure creatine distribution in idiopathic inflammatory myopathy (IIM) patients' myocardial segments and investigate whether cardiovascular magnetic resonance (CMR) chemical exchange saturation transfer (CEST) creatine mapping can detect subclinical myocardial changes, CEST's ability was further compared with other conventional CMR mapping sequences. Forty IIM patients (53.5 ± 10.5 years, 26 males) and eight healthy controls (35.4 ± 6 years, 5 males) underwent CMR scans on a 3.0-T MR scanner. Patients with IIM were further classified into two subgroups according to cardiac troponin T (cTn-T) values: the elevated cTn-T subgroup (n = 14) and the normal cTn-T subgroup (n = 26). Cine imaging, T2 SPAIR, LGE imaging, T1 mapping, T2 mapping, and Cr (creatine) CEST were performed. Cr mapping showed significantly reduced creatine in IIM patients among global myocardium (IIM: 0.109 ± 0.063, controls: 0.121 ± 0.021, p < 0.05), and decreased creatine signals were detected in all 16 cardiac segments (p < 0.05). Patients also had significantly prolonged native T1 and decreased enhanced T1 values in each cardiac segment (p < 0.05). There was no significant difference of LVEF and T2 values between IIM patients and controls. Between the two subgroups, elevated cTn-T was linked with creatine and extracellular volume fraction (ECV) values, providing a global average creatine signal of 0.107 vs 0.112 (p < 0.05) and 24.7 vs 32.4 (p < 0.05). Creatine CEST mapping can detect early-stage heart involvement with negative LGE findings in IIM. Compared with T1 mapping, CEST provides increased sensitivity to ECV measurement, making it significantly better than T1, and a promising CMR sequence for screening subclinical myocardial damage. • IIM patients with potential or ongoing heart involvement, elevated ECV, and reduced Cr CEST values could provide valuable information. • ECV and Cr CEST values were closely related to elevated cTn-T.

Constructing group-invariant CR mappings

We construct CR mappings between spheres that are invariant under actions of finite unitary groups. In particular, we combine a tensoring procedure with D’Angelo’s construction of a canonical group-invariant CR mapping to obtain new invariant mappings. We also explore possible gap phenomena in this setting.

Two decades of finite jet determination of CR mappings

This is a survey paper discussing the developments around the so-called finite jet determination problem for CR maps over the past twenty years.

Cartan uniqueness theorem on nonopen sets

Cartan’s uniqueness theorem does not hold in general for CR mappings, but it does hold under certain conditions guaranteeing extendibility of CR functions to a fixed neighborhood. These conditions can be defined naturally for a wide class of sets such as local real-analytic subvarieties or subanalytic sets, not just submanifolds. Suppose that V is a locally connected and locally closed subset of $${\mathbb {C}}^n$$ such that the hull constructed by contracting analytic discs close to arbitrarily small neighborhoods of a point always contains the point in the interior. Then, restrictions of holomorphic functions uniquely extend to a fixed neighborhood of the point. Using this extension, we obtain a version of Cartan’s uniqueness theorem for such sets. When V is a real-analytic subvariety, we can generalize the concept of infinitesimal CR automorphism and also prove an analogue of the theorem. As an application of these two results we show that, for circular subvarieties, the only automorphisms, CR or infinitesimal, are linear.

The CR Ahlfors derivative and a new invariant for spherically equivalent CR maps

We study a CR analogue of the Ahlfors derivative for conformal immersions of Stowe [23] that generalizes the CR Schwarzian derivative studied earlier by the second-named author [21]. This notion possesses several important properties similar to those of the conformal counterpart and provides a new invariant for spherically equivalent CR maps from strictly pseudoconvex CR manifolds into a sphere. The invariant is computable and distinguishes many well-known sphere maps. In particular, it vanishes precisely when the map is spherically equivalent to the linear embedding of spheres.

The finite jet determination problem for CR maps of positive codimension into Nash manifolds

We prove the first general finite jet determination result in positive codimension for CR maps from real-analytic minimal submanifolds M ⊂ C N $M\subset \mathbb {C}^N$ into Nash (real) submanifolds M ′ ⊂ C N ′ $M^{\prime }\subset \mathbb {C}^{N^{\prime }}$ . For a sheaf S $\mathcal {S}$ of C ∞ $\mathcal {C}^\infty$ -smooth CR maps from M $M$ into M ′ $M^{\prime }$ , we show that the non-existence of so-called 2-approximate CR S $\mathcal {S}$ -deformations from M $M$ into M ′ $M^{\prime }$ implies the following strong finite jet determination property: There exists a map ℓ : M → Z + $\ell \colon M\rightarrow {\mathbb {Z}}_+$ , bounded on compact subsets of M $M$ , such that for every point p ∈ M $p\in M$ , whenever f , g $f,g$ are two elements of S p $\mathcal {S}_p$ with j p ℓ ( p ) f = j p ℓ ( p ) g $j^{\ell (p)}_pf=j^{\ell (p)}_pg$ , then f = g $f=g$ . Applying the deformation point of view allows a unified treatment of a number of classes of target manifolds, which includes, among others, strictly pseudoconvex, Levi–non-degenerate, but also some particularly important Levi-degenerate targets, such as boundaries of classical domains.

On the relative Kuo condition and the second relative Kuo condition

The Kuo condition and the second Kuo condition are known as criteria for an r-jet to be V-sufficient in Cr mappings and Cr+1 mappings, respectively. In [2] we considered the notions of V-sufficiency of jets and these Kuo conditions in the relative case to a given closed set Σ, and showed that the relative Kuo condition is a criterion for a relative r-jet to be Σ-V-sufficient in Cr mappings and the second relative Kuo condition is a sufficient condition for a relative r-jet to be Σ-V-sufficient in Cr+1 mappings. In this paper we discuss several conditions equivalent to the relative Kuo condition or the second relative Kuo condition.

Equivariant embeddings of strongly pseudoconvex Cauchy–Riemann manifolds

Let X be a CR manifold with transversal, proper CR action of a Lie group G. We show that the quotient X/G is a complex space such that the quotient map is a CR map. Moreover the quotient is universal, i.e. every invariant CR map into a complex manifold factorizes uniquely over a holomorphic map on X/G. We then use this result and complex geometry to prove an embedding theorem for (non-compact) strongly pseudoconvex CR manifolds with transversal G rtimes S^1-action. The methods of the proof are applied to obtain a projective embedding theorem for compact CR manifolds.

G-invariant Szegő kernel asymptotics and CR reduction

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 G action. Under certain natural assumptions about the group G action, we show that the G-invariant Szegő kernel for (0, q) forms is a complex Fourier integral operator, smoothing away \(\mu ^{-1}(0)\) and there is a precise description of the singularity near \(\mu ^{-1}(0)\), where \(\mu \) denotes the CR moment map. We apply our result to the case when X admits a transversal CR \(S^1\) action and deduce an asymptotic expansion for the mth Fourier component of the G-invariant Szegő kernel for (0, q) forms as \(m\rightarrow +\infty \) and when \(q=0\), we recover Xiaonan Ma and Weiping Zhang’s result about the existence of the G-invariant Bergman kernel for ample line bundles. As an application, we show that if m large enough, quantization commutes with reduction.

Unique jet determination and extension of germs of CR maps into spheres

We provide a new way of simultaneously parametrizing arbitrary local CR maps from real-analytic generic manifolds M ⊂ C N M\subset \mathbb {C}^N into spheres S 2 N ′ − 1 ⊂ C N ′ \mathbb {S}^{2N’-1}\subset \mathbb {C}^{N’} of any dimension. The parametrization is obtained as a composition of universal rational maps with a holomorphic map depending only on M M . As applications, we obtain rigidity results of different flavours such as unique jet determination and global extension of local CR maps.

Deformations of CR maps and applications

We study the deformation theory of CR maps in the positive codimensional case. In particular, we study structural properties of the {\em mapping locus} $E$ of (germs of nondegenerate) holomorphic maps $H \colon (M,p) \to M'$ between generic real submanifolds $M \subset \mathbb C^N$ and $M' \subset \mathbb C^{N'}$, defined to be the set of points $p' \in M'$ which admit such a map with $H(p) = p'$. We show that this set $E$ is semi-analytic and provide examples for which $E$ posseses (prescribed) singularities.

High-sensitivity CEST mapping using a spatiotemporal correlation-enhanced method.

To obtain high-sensitivity CEST maps by exploiting the spatiotemporal correlation between CEST images. A postprocessing method accomplished by multilinear singular value decomposition (MLSVD) was used to enhance the CEST SNR by exploiting the correlation between the Z-spectrum for each voxel and the low-rank property of the overall CEST data. The performance of this method was evaluated using CrCEST in ischemic mouse brain at 11.7 tesla. Then, MLSVD CEST was applied to obtain Cr, amide, and amine CEST maps of the ischemic mouse brain to demonstrate its general applications. Complex-valued Gaussian noise was added to CEST k-space data to mimic a low SNR situation. MLSVD CEST analysis was able to suppress the noise, recover the degraded CEST peak, and provide better CrCEST quality compared to the smoothing and singular value decomposition (SVD)-based denoising methods. High-resolution Cr, amide, and amine CEST maps of an ischemic stroke using MLSVD CEST suggest that CrCEST is also a sensitive pH mapping method, and a wide range of pH changes can be detected by combing CrCEST with amine CEST at high magnetic fields. MLSVD CEST provides a simple and efficient way to improve the SNR of CEST images.

Conformal and CR mappings on Carnot groups

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 a permutation isomorphism, and affine in each component. As examples, we consider free groups on two generators, and show that these admit very simple polynomial embeddings in C N \mathbb {C}^N that induce their CR structure.

Stationary discs and finite jet determination for CR mappings in higher codimension

We discuss stationary discs for generic CR manifolds and apply them to the problem of finite jet determination for CR mappings. We prove that a C2-smooth CR diffeomorphism of two C4-smooth strictly pseudoconvex Levi generating CR manifolds is uniquely determined by its 2-jet at a given point. A new key element of the proof is the existence of non-defective stationary discs.

On the Levi-flat Plateau problem

We solve the Levi-flat Plateau problem in the following case. Let $M \subset {\mathbb C}^{n+1}$, $n \geq 2$, be a connected compact real-analytic codimension-two submanifold with only nondegenerate CR singularities. Suppose $M$ is a diffeomorphic image via a real-analytic CR map of a real-analytic hypersurface in ${\mathbb C}^n \times {\mathbb R}$ with only nondegenerate CR singularities. Then there exists a unique compact real-analytic Levi-flat hypersurface, nonsingular except possibly for self-intersections, with boundary $M$. We also study boundary regularity of CR automorphisms of domains in ${\mathbb C}^n \times {\mathbb R}$.

Sufficient and necessary conditions for local rigidity of CR mappings and higher order infinitesimal deformations

In this paper we continue our study of local rigidity for maps of CR submanifolds of the complex space. We provide a linear sufficient condition for local rigidity of finitely nondegenerate maps between minimal CR manifolds. Furthermore, we show higher order infinitesimal conditions can be used to give a characterization of local rigidity.