Related papers: Remarks on the higher dimensional Suita conjecture

We study the Bergman metric and introduce the Bergman dual on Cartan-Hartogs (CH) domains. For a bounded domain D in C^n with Bergman kernel K_D, we define the Bergman dual of (D, g_D) as (D*, g_D*), where D* is the maximal domain on which…

In this paper, we investigate the asymptotic behavior of the Bergman kernel at the boundary for some pseudoconvex model domains. This behavior can be described by the geometrical information of the Newton polyhedron of the defining function…

Let $\Omega$ be a bounded pseudoconvex domain in $\mathbb{C}^n$, and let $\phi$ be a strictly plurisubharmonic function on $\Omega$. For each $k\in\mathbb{N}$, we consider determinantal point process $\Lambda_k$ with kernel $K_{k\phi}$,…

A domain in $\C^n$ with Levi-flat boundary near a given point is characterized in terms of the boundary behavior of the Kobayashi or Bergman metrics, or of the Bergman kernel. Some results are given in the case of intermediate values of the…

The Bergman theory of domains $\{ |{z_{1} |^{\gamma}} < |{z_{2}} | < 1 \}$ in $\mathbb{C}^2$ is studied for certain values of $\gamma$, including all positive integers. For such $\gamma$, we obtain a closed form expression for the Bergman…

Let A be the adele ring over a totally real number field F. For cohomological cuspidal automorphic irreducible representations of GSp(4,A) coming from weak endoscopic or Saito-Kurokawa Lifts we determine the local invariant spaces under the…

Comparing Neumann and Dirichlet eigenvalues of the Laplacian on a bounded domain $\Omega\subseteq\Rbb^n$ is a topic that goes back at least to the work of P\'olya \cite{polya}. We study the effect of the isoperimetric ratio of $\Omega$ on…

Hilbert-Kunz multiplicity and F-signature are numerical invariants of commutative rings in positive characteristic that measure severity of singularities: for a regular ring both invariants are equal to one and the converse holds under mild…

In this paper, we study lower bounds for higher eigenvalues of the Dirichlet eigenvalue problem of the Laplacian on a bounded domain $\Omega$ in $\mathbb{R}^n$. It is well known that the $k$-th Dirichlet eigenvalue $\lambda_k$ obeys the…

Denote by $N_{\cal N} (\Omega,\lambda)$ the counting function of the spectrum of the Neumann problem in the domain $\Omega$ on the plane. G. P\'olya conjectured that $N_{\cal N} (\Omega,\lambda) \ge (4\pi)^{-1} |\Omega| \lambda$. We prove…

We prove a conjecture of Lehmann-Tanimoto about the behaviour of the Fujita invariant (or $a$-constant appearing in Manin's conjecture) under pull-back to generically finite covers. As a consequence we obtain results about geometric…

Let M be a closed Riemannian manifold. We consider the inner radius of a nodal domain for a large eigenvalue \lambda. We give upper and lower bounds on the inner radius of the type C/\lambda^k. Our proof is based on a local behavior of…

We associate convex bodies to a wide class of graded G-algebras where G is a connected reductive group. These convex bodies give information about the Hilbert function as well as multiplicities of irreducible representations appearing in…

This paper studies Fefferman's program \cite{F3} of expressing the singularity of the Bergman kernel, for smoothly bounded strictly pseudoconvex domains $\Omega\subset\C^n$, in terms of local biholomorphic invariants of the boundary. By…

We study K_2 of one-dimensional local domains over a field of characteristic 0, introduce a conjecture, and show that this conjecture implies Geller's conjecture. We also show that Berger's conjecture implies Geller's conjecture, and hence…

For a given bounded domain $\Omega\subset {\Bbb R}^n$ with $C^1$-smooth boundary, we prove the P\'olya conjecture for the Neumann eigenvalues. In other words, we prove that \begin{eqnarray*} \mu_{k+1}\le \frac{(2\pi)^2k^{2/n}}{(\omega_n…

We study the detailed properties of Z_2 domain walls in the deconfined high temperature phase of the d=2+1 SU(2) gauge theory. These walls are studied both by computer simulations of the lattice theory and by one-loop perturbative…

We give a parameter version of Graham-Kerzman approximation theorem for bounded holomorphic functions on strictly pseudoconvex domains. As an application, we present some uniform estimates for the boundary behaviour of the Kobayashi and…

For any open hyperbolic Riemann surface $X$, the Bergman kernel $K$, the logarithmic capacity $c_{\beta}$, and the analytic capacity $c_{B}$ satisfy the inequality chain $\pi K \geq c^2_{\beta} \geq c^2_B$; moreover, equality holds at a…

Let $F$ be a real quadratic number field, and let $F_{cyc}$ denote its cyclotomic $\mathbb{Z}_2$-extension. For each integer $n\geq0$, let $F_n$ be the unique intermediate field in $F_{cyc}$ such that $[F_n:F]=2^n$. By studying the $2$-adic…