Related papers: The Bergman kernel in constant curvature

An effective formula for the Bergman kernel on $\mathbb{H}_{\gamma} = \{|z_1|^\gamma < |z_2| < 1 \}$ is obtained for rational $\gamma = \frac{m}{n} >1$. The formula depends on arithmetic properties of $\gamma$, which uncovers new symmetries…

Let $M$ be an $n$-dimensional Lagrangian submanifold of a complex space form. We prove a pointwise inequality $$\delta(n_1,\ldots,n_k) \leq a(n,k,n_1,\ldots,n_k) \|H\|^2 + b(n,k,n_1,\ldots,n_k)c,$$ with on the left hand side any…

Let $\{M_j\}$ be a sequence of complete Riemannian surfaces which converges in the sense of Cheeger-Gromov to a complete Riemannian surface $M$. We study the convergence of the Bergman kernel $K_{M_j}$ of $M_j$ by using isoperimetric…

Kernel-based methods in Numerical Analysis have the advantage of yielding optimal recovery processes in the "native" Hilbert space $\calh$ in which they are reproducing. Continuous kernels on compact domains have an expansion into…

Let L be a holomorphic line bundle with a positively curved singular Hermitian metric over a complex manifold X. One can define naturally the sequence of Fubini-Study currents associated to the space of square integrable holomorphic…

Devoted to multi-task learning and structured output learning, operator-valued kernels provide a flexible tool to build vector-valued functions in the context of Reproducing Kernel Hilbert Spaces. To scale up these methods, we extend the…

We consider linear approximation based on function evaluations in reproducing kernel Hilbert spaces of certain analytic weighted power series kernels and stationary kernels on the interval $[-1,1]$. Both classes contain the popular Gaussian…

We give a mathematical exposition of the Page metric, and introduce an efficient coordinate system for it. We carefully examine the submanifolds of the underlying smooth manifold, and show that the Page metric does not have positive…

We study bounded domains $\Omega\subset\mathbb{C}^n$ whose Bergman metric is locally symmetric, i.e. its Riemannian curvature tensor is parallel with respect to the Levi-Civita connection. Following the strategy developed in…

We prove a conjecture of Bernstein that the superconvexity of the heat kernel on hyperbolic space holds in all dimensions and, hence, there is an analog of Huisken's monotonicity formula for mean curvature flow in hyperbolic space of all…

In this article, we consider Bergman kernels related to modules at boundary points for singular hermitian metrics on holomorphic vector bundles, and obtain a log-subharmonicity property of the Bergman kernels. As applications, we obtain a…

We analyze the Nystr\"om approximation of a positive definite kernel associated with a probability measure. We first prove an improved error bound for the conventional Nystr\"om approximation with i.i.d. sampling and singular-value…

We compute the first four coefficients of the asymptotic off-diagonal expansion of the Bergman kernel for the N-th power of a positive line bundle on a compact Kaehler manifold, and we show that the coefficient b_1 of the N^{-1/2} term…

The aim of the present paper is three folds. Firstly, we complete the study of the weighted hyperholomorphic Bergman space of the second kind on the ball of radius $R$ centred at the origin. The explicit expression of its Bergman kernel is…

We show the asymptotics of the Bergman kernel function near the smooth divisor at infinity of the Cheng-Yau metric on quasi-projective manifolds. In particular, we show that there is a quantum phenomenon for the points very close to the…

We show that the Bergman kernel of a finite-volume quotient of a Hermitian manifold $\widetilde{X}$ with bounded geometry by a discrete group $\Gamma$ of its isometries is the same as the averaging over $\Gamma$ of the Bergman kernel on…

In this article our main result is a more complete version of the statements obtained in {\rm [6]}. One of the important technical point of our proof is an $\displaystyle L^{2\over m}$ extension theorem of Ohsawa-Takegoshi type, which is…

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…

Using the techniques developed in \cite{SunSun}, we give estimations of the Bergman kernel of the punctured disk with the standard complete Poincar\'{e} metric. As an application, we improve the result of \cite{AMM} on the Bergman kernels…

We study the asymptotics of Ohsawa-Takegoshi extension operator and orthogonal Bergman projector associated with high tensor powers of a positive line bundle. More precisely, for a fixed complex submanifold in a complex manifold, we…