Related papers: The Szeg\"o kernel for certain non-pseudoconvex do…

We study the Kohn-Laplacian and its fundamental solution on some model domains in $\mathbb C^{n+1}$, and further discuss the explicit kernel of the Cauchy-Szeg\"o projections on these model domains using the real analysis method. We further…

We investigate problems related with the existence of square integrable holomorphic functions on (unbounded) balanced domains. In particular, we solve the problem of Wiegerinck for balanced domains in dimension two. We also give a…

We prove that there exists a bounded convex domain $\Omega \subset \mathbf{R}^3$ of fixed volume that minimizes the first positive curl eigenvalue among all other bounded convex domains of the same volume. We show that this optimal domain…

We introduce the notion of extremal basis of tangent vector fields at a boundary point of finite type of a pseudo-convex domain in $\mathbb{C}^n$. Then we define the class of geometrically separated domains at a boundary point, and give a…

A wealth of information on multiloop string amplitudes is encoded in fermionic two-point functions known as Szeg\"o kernels. In this paper we show that cyclic products of any number of Szeg\"o kernels on a Riemann surface of arbitrary genus…

In analogy to complex function theory we introduce a Szeg\"o metric in the context of hypercomplex function theory dealing with functions that take values in a Clifford algebra. In particular, we are dealing with Clifford algebra valued…

For $g\geq 2$, let $\Gamma\subset\mathrm{Sp}(2g,\mathbb{R})$ be a discrete subgroup, which is either a cocompact subgroup or an arithmetic subgroup without torsion elements, and let $\mathbb{H}_{g}$ denote the Siegel upper half space of…

In this paper we study the microlocal properties of the Szeg\H{o} kernel of a given compact connected orientable CR orbifold whose Kohn Laplacian has closed range. This last assumption is satisfied if certain geometric conditions hold true,…

Let $\mu_2(\Omega)$ be the first positive eigenvalue of the Neumann Laplacian in a bounded domain $\Omega\subset\mathbb{R}^N$. It was proved by Szeg\H{o} for $N=2$ and by Weinberger for $N \geq 2$ that among all equimeasurable domains…

We give a precise estimate of the Bergman kernel for the model domain defined by $\Omega_F=\{(z,w)\in \mathbb{C}^{n+1}:{\rm Im}w-|F(z)|^2>0\},$ where $F=(f_1,...,f_m)$ is a holomorphic map from $\mathbb{C}^n$ to $\mathbb{C}^m$, in terms of…

Following recent work by Ghomi, Solomon and Tabachnikov, we study geometry and topology of skew branes. A skew brane is a codimension 2 submanifold in affine space such that the tangent spaces at any pair of distinct points are not…

Let $X$ be a compact connected strongly pseudoconvex CR manifold of dimension $2n+1, n \ge 1$ with a transversal CR $S^1$ action on $X$. We establish an asymptotic expansion for the $m$-th Fourier component of the Szeg\H{o} kernel function…

Non-Hermitian random matrices with symplectic symmetry provide examples for Pfaffian point processes in the complex plane. These point processes are characterised by a matrix valued kernel of skew-orthogonal polynomials. We develop their…

In this paper we study the Bergman kernel and projection on the unbounded worm domain $$ \mathcal{W}_\infty = \big\{(z_1,z_2)\in\mathbb{C}^2 : \big|z_1-e^{i\log|z_2|^2}\big|^2<1, z_2\neq0\big\}. $$ We first show that the Bergman space of…

A property of smooth convex domains $\Omega \subset \mathbb{R}^n$ is that if two points on the boundary $x, y \in \partial \Omega$ are close to each other, then their normal vectors $n(x), n(y)$ point roughly in the same direction and this…

We consider a compact connected CR manifold with a transversal CR locally free $\mathbb R$-action endowed with a rigid positive CR line bundle. We prove that a certain weighted Fourier-Szeg\H{o} kernel of the CR sections in the high tensor…

We consider the cubic Szeg\"o equation i u_t=Pi(|u|^2u) in the Hardy space on the upper half-plane, where Pi is the Szeg\"o projector on positive frequencies. It is a model for totally non-dispersive evolution equations and is completely…

We prove nontangential asymptotic limits of the Bergman kernel on the diagonal, and the Bergman metric and its holomorphic sectional curvature at exponentially flat infinite type boundary points of smooth bounded pseudoconvex domains in…

Estimates of the Bergman kernel and the Bergman and Kobayashi metrics on pseudoconvex domains near boundaries with constant Levi ranks are given.

We obtain new explicit formulas for the Bergman kernel function on two families of Hartogs domains. To do so, we first compute the Bergman kernels on the slices of these Hartogs domains with some coordinates fixed, evaluate these kernel…