Related papers: Remarks on the higher dimensional Suita conjecture

We study the boundary behaviour of the Fefferman--Szeg\"o metric and several associated invariants in a $C^\infty$-smoothly bounded strictly pseudoconvex domain.

In convex geometry, the Blaschke surface area measure on the boundary of a convex domain can be interpreted in terms of the complexity of approximating polyhedra. In response to a question raised by D. Barrett, this approach is formulated…

In this paper we study optimization problems for Neumann eigenvalues $\mu_k$ among convex domains with a constraint on the diameter or the perimeter. We work mainly in the plane, though some results are stated in higher dimension. We study…

Under a potential-theoretical hypothesis named $f$-Property with $f$ satisfying $\displaystyle\int_t^\infty \dfrac{da}{a f(a)}<\infty$, we show that the Kobayashi metric $K(z,X)$ on a weakly pseudoconvex domain $\Om$, satisfies the estimate…

In the spirit of Kobayashi's applications of methods of invariant metrics to questions of projective geometry, we introduce a projective analogue of the complex squeezing function. Using Frankel's work, we prove that for convex domains it…

We consider the subharmonicity property of the logarithm of Azukawa pseudometrics of pseudoconvex domains under pseudoconvex variations. We prove that such a property holds for the variation of balanced domains. We also give a non-balanced…

We prove the nontangential asymptotic limits of the Bergman canonical invariant, Ricci and Scalar curvatures of the Bergman metric, as well as the Kobayashi--Fuks metric, at exponentially flat infinite type boundary points of smooth bounded…

Given a bounded strongly pseudoconvex domain $D$ in $\mathbb{C}^n$ with smooth boundary, we give a characterization through products of functions in weighted Bergman spaces of $(\lambda,\gamma)$-skew Carleson measures on $D$, with…

The Fantappi\`e and Laplace transforms realize isomorphisms between analytic functionals supported on a convex compact set $K\subset{\mathbb C}^n$ and certain spaces of holomorphic functions associated with $K$. Viewing the Bergman space of…

In this paper, we first establish the localization of the Bergman kernels for unbounded pseudoconvex domains near a D'Angelo finite type boundary point. This result was proved by Engli\v{s} more than twenty years ago for bounded…

Our main result introduces a new way to characterize two-dimensional finite ball quotients by algebraicity of their Bergman kernels. This characterization is particular to dimension two and fails in higher dimensions, as is illustrated by a…

In the analysis on self-similar fractal sets, the Kusuoka measure plays an important role (cf. \cite{kusuoka2}, \cite{kajino}, \cite{str3}). Here we investigate the Kusuoka measure from an ergodic theoretic viewpoint, seen as an invariant…

We study the motion of a random string in a convex domain $O$ in $\R^d$, namely the solution of a vector-valued stochastic heat equation, confined in the closure of $O$ and reflected at the boundary of $O$. We study the structure of the…

We use the Suita conjecture (now a theorem) to prove that for any domain $\Omega \subset \mathbb{C}$ its Bergman kernel $K(\cdot, \cdot)$ satisfies $K(z_0, z_0) = \hbox{Volume}(\Omega)^{-1}$ for some $z_0 \in \Omega$ if and only if $\Omega$…

We establish for $2 \le k \le n-1$ the strict concavity of the function $f_k(\lambda)=\log(\sigma_k(\lambda))$ on a subset of the positive cone $\Gamma_n=\{\lambda=(\lambda_{1}, \lambda_{2}, \cdots,\lambda_{n})\in \mathbb{R}^n;…

The random beta-transformation K is isomorphic to a full shift. This relation gives an invariant measure for K that yields the Bernoulli convolution by projection. We study the local dimension of the invariant measure for K for special…

We express the Kodaira-Iitaka dimension and the multiplicity of graded linear series in terms of the intersection theory of the plurisubharmonic envelope associated with the linear series, and obtain two refined versions of these formulas…

Let $\Omega \subset \mathbb{R}^d$ be a quasiconvex Lipschitz domain and $A(x)$ be a $d \times d$ uniformly elliptic, symmetric matrix with Lipschitz coefficients. Assume a nontrivial $u$ solves $-\nabla \cdot (A(x) \nabla u) = 0$ in…

This paper deals with the existence and limiting behavior of invariant measures of the stochastic Landau-Lifshitz-Bloch equation driven by linear multiplicative noise and additive noise defined in the entire space $\mathbb{R}^d$ for…

In [7], Dong and I proved that the domains $D \subset \mathbb{C}$ of finite volume whose on-diagonal Bergman kernels $K(\cdot, \cdot)$ satisfy $K(z_0, z_0) = Volume(D)^{-1}$ are disks minus closed polar sets. We utilized the solution of the…