Szegö kernel, regular quantizations and spherical CR-structures

We compute the Szegö kernel of the unit circle bundle of a negative line bundle dual to a regular quantum line bundle over a compact Kähler manifold. As a corollary we provide an infinite family of smoothly bounded strictly pseudoconvex domains on complex manifolds (disk bundles over homogeneous Hodge manifolds) for which the log-terms in the Fefferman expansion of the Szegö kernel vanish and which are not locally CR-equivalent to the sphere. We also give a proof of the fact that, for homogeneous Hodge manifolds, the existence of a locally spherical CR-structure on the unit circle bundle alone implies that the manifold is biholomorphic to a projective space. Our results generalize those obtained by Engli? (Math Z 264(4):901–912, 2010) for Hermitian symmetric spaces of compact type.     

On balanced Hermitian structures on Lie groups

We present several methods for the construction of balanced Hermitian structures on Lie groups. In our methods a partial differential equation is involved so that the resulting structures are in general non homogeneous. In particular, we prove that for 3-step nilpotent Lie groups G of dimension 6, any left-invariant complex structure on G admits a balanced Hermitian metric. Starting from normal almost contact structures, we construct balanced metrics on 6-dimensional manifolds, generalizing warped products. Finally, explicit balanced Hermitian structures are also given on solvable Lie groups defined as semidirect products ${\mathbb{R}^k \ltimes \mathbb{R}^{2n-k}}$ .     

Dirichlet problems for plurisubharmonic functions on compact sets

In this paper we solve the Dirichlet problems for different classes of plurisubharmonic functions on compact sets in ${\mathbb C^n}$ including continuous, pluriharmonic and maximal functions.     

The selfnegadual properties of generalised quadratic Boolean functions

We define and characterise selfnegadual generalised quadratic Boolean functions by establishing a link, both to the multiplicative order of symmetric binary matrices, and also to the Hermitian self-dual ${\mathbb{F}_4}$ -linear codes. This facilitates a novel way to classify Hermitian self-dual ${\mathbb{F}_4}$ -linear codes.     

Koppelman formulas on flag manifolds and harmonic forms

We construct Koppelman formulas on manifolds of flags in ${\mathbb{C}^N}$ for forms with values in any holomorphic line bundle as well as in the tautological vector bundles and their duals. As an application we obtain new explicit proofs of some vanishing theorems of the Bott–Borel–Weil type by solving the corresponding ${\bar{\partial}}$ -equation. We also construct reproducing kernels for harmonic (p, q)-forms in the case of Grassmannians.     

Compact pseudo-Riemannian manifolds with parallel Weyl tensor

It is shown that in every dimension n = 3j + 2, j = 1, 2, 3, . . ., there exist compact pseudo-Riemannian manifolds with parallel Weyl tensor, which are Ricci-recurrent, but neither conformally flat nor locally symmetric, and represent all indefinite metric signatures. The manifolds in question are diffeomorphic to nontrivial torus bundles over the circle. They all arise from a construction that a priori yields bundles over the circle, having as the fibre either a torus, or a 2-step nilmanifold with a complete flat torsionfree connection; our argument only realizes the torus case.     

Kernel based quadrature on spheres and other homogeneous spaces

Quadrature formulas for spheres, the rotation group, and other compact, homogeneous manifolds are important in a number of applications and have been the subject of recent research. The main purpose of this paper is to study coordinate independent quadrature (or cubature) formulas associated with certain classes of positive definite and conditionally positive definite kernels that are invariant under the group action of the homogeneous manifold. In particular, we show that these formulas are accurate—optimally so in many cases—and stable under an increasing number of nodes and in the presence of noise, provided the set $X$ of quadrature nodes is quasi-uniform. The stability results are new in all cases. In addition, we may use these quadrature formulas to obtain similar formulas for manifolds diffeomorphic to $\mathbb S ^n$ , oblate spheroids for instance. The weights are obtained by solving a single linear system. For $\mathbb S ^2$ , and the restricted thin plate spline kernel $r^2\log r$ , these weights can be computed for two-thirds of a million nodes, using a preconditioned iterative technique introduced by us.     

Geodesics on the Moduli Space of Oriented Circles in \mathbb{S}^{3}

Let ${\mathbb{Q}^3}$ be the moduli space of oriented circles in the three dimensional unit sphere ${\mathbb{S}^3}$ . Given a natural complex structure such space becomes a three dimensional complex manifold, with a M?bius invariant Hermitian metric h of type (2, 1). Up to M?bius transformations, all geodesics with respect to the Lorentz metric g = Re(h) on ${\mathbb{Q}^3}$ are determined to form a one-parameter family of circles on a helicoid in a space form ${\mathbb{R}^3, \mathbb{H}^3}$ or ${\mathbb{S}^{3}}$ , resp. We show also that any two oriented circles in ${\mathbb{S}^3}$ are connected by countably infinitely many geodesics in ${\mathbb{Q}^3}$ .     

Bi-Lipschitz extension from boundaries of certain hyperbolic spaces

Tukia and Väisälä showed that every quasi-conformal map of ${\mathbb{R}^n}$ extends to a quasi-conformal self-map of ${\mathbb{R}^{n+1}}$ . The restriction of the extended map to the upper half-space ${\mathbb{R}^n \times \mathbb{R}_+}$ is, in fact, bi-Lipschitz with respect to the hyperbolic metric. More generally, every simply connected homogeneous negatively curved manifold decomposes as ${M = N \rtimes \mathbb{R}_+}$ where N is a nilpotent group with a metric on which ${\mathbb{R}_+}$ acts by dilations. We show that under some assumptions on N, every quasi-symmetry of N extends to a bi-Lipschitz map of M. The result applies to a wide class of manifolds M including non-compact rank one symmetric spaces and certain manifolds that do not admit co-compact group actions. Although M must be Gromov hyperbolic, its curvature need not be strictly negative.     

The spectrum of the twisted Dirac operator on K?hler submanifolds of the complex projective space

We establish an upper estimate for the small eigenvalues of the twisted Dirac operator on K?hler submanifolds in K?hler manifolds carrying K?hlerian Killing spinors. We then compute the spectrum of the twisted Dirac operator of the canonical embedding ${{\mathbb C}P^d\rightarrow {\mathbb C}P^n}$ in order to test the sharpness of the upper bounds.     

Hankel Operators on Holomorphic Hardy–Orlicz Spaces

We characterize the symbols of Hankel operators that extend into bounded operators from the Hardy–Orlicz ${\mathcal H^{\Phi_1}(\mathbb B^n)}$ into ${\mathcal H^{\Phi_2}(\mathbb B^n)}$ in the unit ball of ${\mathbb C^n}$ , in the case where the growth functions ${\Phi_1}$ and ${\Phi_2}$ are either concave or convex. The case where the growth functions are both concave has been studied by Bonami and Sehba. We also obtain several weak factorization theorems for functions in ${\mathcal H^{\Phi}(\mathbb B^n)}$ , with concave growth function, in terms of products of Hardy–Orlicz functions with convex growth functions.     

Reproducing Kernel Hilbert Spaces Supporting Nontrivial Hermitian Weighted Composition Operators

We characterize those generating functions ${k(z) = \sum_{j=0}^\infty z^j/\beta(j)^2}$ that produce weighted Hardy spaces H ²(β) of the unit disk ${\mathbb D}$ supporting nontrivial Hermitian weighted composition operators. Our characterization shows that the spaces associated with the “classical reproducing kernels” ${z \mapsto (1 - \bar{w}z)^{-\eta}}$ , where ${w \in \mathbb D}$ and η > 0, as well as certain natural extensions of these spaces, are precisely those that are hospitable to Hermitian weighted composition operators. It also leads to a refinement of a necessary condition for a weighted composition to be Hermitian, obtained recently by Cowen, Gunatillake, and Ko, into one that is both necessary and sufficient.     

Extrinsic radius pinching in space forms of nonnegative sectional curvature

We first give new estimates for the extrinsic radius of compact hypersurfaces of the Euclidean space and the open hemisphere in terms of high order mean curvatures. Then we prove pinching results corresponding to these estimates. We show that under a suitable pinching condition, M is diffeomorphic and almost isometric to an n-dimensional sphere.      

Codes defined by forms of degree 2 on non-degenerate Hermitian varieties in $${\mathbb{P}^{4}(\mathbb{F}_q)}$$

We study the functional codes of second order on a non-degenerate Hermitian variety as defined by G. Lachaud. We provide the best possible bounds for the number of points of quadratic sections of . We list the first five weights, describe the corresponding codewords and compute their number. The paper ends with two conjectures. The first is about minimum distance of the functional codes of order h on a non-singular Hermitian variety . The second is about distribution of the codewords of first five weights of the functional codes of second order on a non-singular Hermitian variety .      

Exponents of polar codes using algebraic geometric code kernels

Reed–Solomon and BCH codes were considered as kernels of polar codes by Mori and Tanaka (IEEE Information Theory Workshop, 2010, pp 1–5) and Korada et al. (IEEE Trans Inform Theory 56(12):6253–6264, 2010) to create polar codes with large exponents. Mori and Tanaka showed that Reed–Solomon codes over the finite field \(\mathbb {F}_q\) with \(q\) elements give the best possible exponent among all codes of length \(l \le q\) . They also stated that a Hermitian code over \(\mathbb {F}_{2^r}\) with \(r \ge 4\) , a simple algebraic geometric code, gives a larger exponent than the Reed–Solomon matrix over the same field. In this paper, we expand on these ideas by employing more general algebraic geometric (AG) codes to produce kernels of polar codes. Lower bounds on the exponents are given for kernels from general AG codes, Hermitian codes, and Suzuki codes. We demonstrate that both Hermitian and Suzuki kernels have larger exponents than Reed–Solomon codes over the same field, for \(q \ge 3\) ; however, the larger exponents are at the expense of larger kernel matrices. Comparing kernels of the same size, though over different fields, we see that Reed–Solomon kernels have larger exponents than both Hermitian and Suzuki kernels. These results indicate a tradeoff between the exponent, kernel matrix size, and field size.     

New extremal domains for the first eigenvalue of the Laplacian in flat tori

We prove the existence of nontrivial compact extremal domains for the first eigenvalue of the Laplacian in manifolds ${\mathbb{R}^{n}\times \mathbb{R}{/}T\, \mathbb{Z}}$ with flat metric, for some T > 0. These domains are close to the cylinder-type domain ${B_1 \times \mathbb{R}{/}T\, \mathbb{Z}}$ , where B ₁ is the unit ball in ${\mathbb{R}^{n}}$ , they are invariant by rotation with respect to the vertical axe, and are not invariant by vertical translations. Such domains can be extended by periodicity to nontrivial and noncompact domains in Euclidean spaces whose first eigenfunction of the Laplacian with 0 Dirichlet boundary condition has also constant Neumann data at the boundary.     

On Some Holomorphic Cohomological Equations

Let Σ be a non compact Riemann surface and ${\gamma :\Sigma \longrightarrow \Sigma}$ an automorphism acting freely and properly such that the quotient M = Σ/γ is a non compact Riemann surface. Using the fact that Σ and M are Stein manifolds, we prove that, for any holomorphic function ${g : \Sigma \longrightarrow {\mathbb C}}$ and any ${\lambda \in {\mathbb C}}$ , there exists a holomorphic function ${f:\Sigma \longrightarrow {\mathbb C}}$ which is a solution of the holomorphic cohomological equation ${f \circ \gamma - \lambda f = g}$ .     

An Area Preserving Projection from the Regular Octahedron to the Sphere

In this paper, we propose an area preserving bijective map from the regular octahedron to the unit sphere ${\mathbb{S}^2}$ , both centered at the origin. The construction scheme consists of two steps. First, each face F _i of the octahedron is mapped to a curved planar triangle ${\mathcal{T}_i}$ of the same area. Afterwards, each ${\mathcal{T}_i}$ is mapped onto the sphere using the inverse Lambert azimuthal equal area projection with respect to a certain point of ${\mathbb{S}^2}$ . The proposed map is then used to construct uniform and refinable grids on a sphere, starting from any triangular uniform and refinable grid on the triangular faces of the octahedron.     

Scattering of wave maps from {{\mathbb {R}^{2+1}}} to general targets

We show that smooth, radially symmetric wave maps U from ${\mathbb {R}^{2+1}}$ to a compact target manifold (N, where ? _r U and ? _t U have compact support for any fixed time, scatter. The result will follow from the work of Christodoulou and Tahvildar-Zadeh, and Struwe, upon proving that for ${(\lambda^{\prime} \in (0,1),}$ energy does not concentrate in the set $$K_{\frac{5}{8}T,\frac{7}{8}T}^{\lambda^{\prime}} = \{(x,t) \in \mathbb{R}^{2+1} \vert \quad|x| \leq \lambda^{\prime} t, t \in [(5/8)T,(7/8)T] \}.$$